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GROWTH  AND  FORM 


Ah  instantdnedus  photograph  of  a  'splash'  of  milk.  From  Harold  E. 
'   '"*'       KdgertbTiTirlassachusetts  Institute  of  Technology 


See  p.  390 


ON  94 

GROWTH  AND  FORM  ^^ 


BY 


D'ARCY  WENTWORTH  THOMPSON 


A  new  edition 

MARINE 

BIOLOGICAL 

LABORATORY 

LIBRARY 

WOODS  HOLE,  MASS. 
W.  H.  0.  1. 

CAMBRIDGE:  AT  THE  UNIVERSIT 
NEW  YORK:  THE  MACMILLAN  CC 

Y  PRESS 
)MPANY 

1945 

"The  reasonings  about  the  wonderful  and  intricate  operations 
of  Nature  are  so  full  of  uncertainty,  that,  as  the  Wise-man  truly 
observes,  hardly  do  we  guess  aright  at  the  things  jhat  are  upon 
earth,  and  with  labour  do  we  find  the  things  that  are  before  us.'' 
Stephen  Hales,  Vegetable  Staticks  (1727),  p.  318,  1738. 

"Ever  since  I  have  been  enquiring  into  the  works  of  Nature 
I  have  always  loved  and  admired  the  Simplicity  of  her  Ways." 
Dr  George  Martine  (a  pupil  of  Boorhaave's),  in  Medical  Essays  and 
Observations,  Edinburgh,  1747. 


PREFATORY  NOTE 

THIS  book  of  mine  has  little  need  of  preface,  for  indeed  it  is 
"all  preface"  from  beginning  to  end.  I  have  written  it  as 
an  easy  introduction  to  the  study  of  organic  Form,  by  methods 
which  are  the  common-places  of  physical  science,  which  are  by 
no  means  novel  in  their  application  to  natural  history,  but  which 
nevertheless  naturalists  are  little  accustomed  to  employ. 

It  is  not  the  biologist  with  an  inkling  of  mathematics,  but 
the  skilled  and  learned  mathematician  who  must  ultimately  deal 
with  such  problems  as  are  sketched  and  adumbrated  here.  I  pretend 
to  no  mathematical  skill,  but  I  have  made  what  use  I  could  of 
what  tools  I  had;  I  have  dealt  with  simple  cases,  and  the  mathe- 
matical methods  which  I  have  introduced  are  of  the  easiest  and 
simplest  kind.  Elementary  as  they  are,  my  book  has  not  been 
written  without  the  help — the  indispensable  help^ — of  many  friends. 
Like  Mr  Pope  translating  Homer,  when  I  felt  myself  deficient  I 
sought  assistance!  And  the  experience  which  Johnson  attributed 
to  Pope  has  been  mine  also,  that  men  of  learning  did  not  refuse 
to  help  me. 

I  wrote  this  book  in  wartime,  and  its  revision  has  employed 
me  during  another  war.  It  gave  me  solace  and  occupation,  when 
service  was  debarred  me  by  my  years. 

Few  are  left  of  the  friends  who  helped  me  write  it,  but  I  do  not 
forget  the  debt  I  owe  them  all.  Let  me  add  another  to  these 
kindly  names,  that  of  Dr  G.  T.  Bennett,  of  Emmanuel  College, 
Cambridge;  he  has  never  wearied  of  collaboration  with  me,  and 
his  criticisms  have  been  an  education  to  receive. 

D.  W.  T. 
1916-1941. 


American   edition   published August,  1942 

Reprinted    January,  1943 

Reprinted    May,  1944 

Reprinted    May,  1945 


PRINTED    IN    THE    UNITED    STATES    OF    AMERICA 


CONTENTS 

CHAP.  PAGE 

I.  Introductory 1 

II.  On  Magnitude 22 

III.  The  Rate  of  Growth 78 

IV.  On  the  Internal  Form  and  Structure  of  the  Cell    .         .  286 

V.  The  Forms  of  Cells 346 

VI.  A  Note  on  Adsorption 444 

VII.  The  Forms  of  Tissues,  or  Cell-aggregates          .        .        .  465 

VIII.  The  same  (continued) .         .  566 

IX.  On  Concretions,  Spicules,  and  Spicular  Skeletons   .        .  645 

X.  A  Parenthetic  Note  on  Geodetics 741 

XI.  The  Equiangular  Spiral 748 

XII.      The  Spiral  Shells  of  the  Foraminifera      ....  850 

XIII.  The  Shapes  of  Horns,   and   of  Teeth   or  Tusks:  with 

a  Note  on  Torsion 874 

XIV.  On  Leaf-arrangement,  or  Phyllotaxis       ....  912 

XV.  On  the  Shapes  of  Eggs,  and  of  certain  other  Hollow 

Structures 934, 

XVI.  On  Form  and  Mechanical  Efficiency           ....  958 

XVII.  On  the  Theory  of  Transformations,  or  the  Comparison 

OF  Related  Forms                    1026 

Epilogue 1093 

Index 1095 

Plates 

A  Splash  of  Milk Frontispiece 

The  Latter  Phase  of  a  Splash         .         .  "      .         .         .         .    facing  page  390 


"The  mathematicians  are  well  acquainted  with  the  difference 
between  pure  science,  which  has  to  do  only  with  ideas,  and  the 
application  of  its  laws  to  the  use  of  life,  in  which  they  are  con- 
strained to  submit  to  the  imperfections  of  matter  and  the  influence 
of  accident."  Dr  Johnson,  in  the  fourteenth  Rambler,  May  5,  1750. 

"Natural  History. .  .is  either  the  beginning  or  the  end  of  physical 
science."  Sir  John  Herschel,  in  The  Study  of  Natvml  Philosophy, 
p.  221,  1831. 

"I  believe  the  day  must  come  when  the  biologist  will — without 
being  a  mathematician — not  hesitate  to  use  mathematical  analysis 
when  he  requires  it."     Karl  Pearsonf  in  Nature,  January  17,  1901. 


CHAPTER  I 

INTRODUCTORY 

Of  the  chemistry  of  his  day  and  generation,  Kant  declared  that  it 
was  a  science,  but  not  Science — eine  Wissenschaft,  aber  nicht  Wissen- 
schaft — for  that  the  criterion  of  true  science  lay  in  its  relation  to 
mathematics*.  This  was  an  old  story :  for  Roger  Bacon  had  called 
mathematics  porta  et  clavis  scientiarum,  and  Leonardo  da  Vinci  had 
said  much  the  samef.  Once  again,  a  hundred  years  after  Kant, 
Du  Bois  Reymond,  profound  student  of  the  many  sciences  on  which 
physiology  is  based,  recalled  the  old  saying,  and  declared  that 
chemistry  would  only  reach  the  rank  of  science,  in  the  high  and 
strict  sense,  when  it  should  be  found  possible  to  explain  che^dcal 
reactions  in  the  light  of  their  causal  relations  to  the  velocities, 
tensions  and  conditions  of  equihbrium  of  the  constituent  molecules ; 
that,  in  short,  the  chemistry  of  the  future  must  deal  with  molecular 
mechanics  by  the  methods  and  in  the  strict  language  of  mathematics, 
as  the  astronomy  of  Newtoii  and  Laplace  dealt  with  the  stars  in 
their  courses.  We  know  how  great  a  step  was  made  towards  this 
distant  goal  as  Kant  defined  it,  when  van't  Hoif  laid  the  firm 
foundations  of  a  mathematical  chemistry,  and  earned  his  proud 
epitaph — Physicam  chemiae  adiunxitX- 

We  need  not  wait  for  the  full  reahsation  of  Kant's  desire,  to  apply 
to  the  natural  sciences  the  principle  which  he  laid  down.  Though 
chemistry  fall  short  of  its  ultimate  goal  in  mathematical  mechanics  §, 
nevertheless  physiology  is   vastly  strengthened   and   enlarged   by 

*  "  Ich  behaupte  nur  dass  in  jeder  besonderen  Naturlehre  nur  so  viel  eigentliche 
Wissenschaft  angetroffen  konne  als  darin  Mathematik  anzutreffen  ist" :  Gesammelte 
Schriften,  iv,  p.  470. 

t  "Nessuna  humana  investigazione  si  puo  dimandare  vera  scienzia  s'essa  non 
passa  per  le  matematiche  dimostrazione." 

X  Cf.  also  Crum  Brown,  On  an  application  of  Mathematics  to  Chemistry,  Trans. 
R.S.E.  XXIV,  pp.  691-700,  1867. 

§  Ultimate,  for,  as  Francis  Bacon  tells  us:  Mathesis  philosophiam  naturalem 
terminare  debet,  non  generare  aut  procreare. 


2  INTRODUCTORY  <^i,J^^H 

making  use  of  the  chemistry,  and  of  the  physics,  of  the  age.  Littl 
by  httle  it  draws  nearer  to  our  conception  of  a  true  science  with 
each  branch  of  physical  science  which  it  brings  into  relation  with 
itself:  with  every  physical  law  and  mathematical  theorem  which  it 
learns  to  take  into  its  employ*.  Between  the  physiology  of  Haller, 
fine  as  it  was,  and  that  of  Liebig,  Helmholtz,  Ludwig,  Claude 
Bernard,  there  was  all  the  difference  in  the  worldf. 

As  soon  as  we  adventure  on  the  paths  of  the  physicist,  we  learn 
to  weigh  and  to  measure,  to  deal  with  time  and  space  and  mass  and 
their  related  concepts,  and  to  find  more  and  more  our  knowledge 
expressed  and  our  needs  satisfied  through  the  concept  of  number, 
as  in  the  dreams  and  visions  of  Plato  and  Pythagoras ;  for  modern 
chemistry  would  have  gladdened  the  hearts  of  those  great  philo- 
sophic dreamers.  Dreams  apart,  numerical  precision  is  the  very 
soul  of  science,  and  its  attainment  affords  the  best,  perhaps,  the 
only  criterion  of  the  truth  of  theories  and  the  correctness  of  experi- 
mentsj:.  So  said  Sir  John  Herschel,  a  hundred  years  ago;  and 
Kant  had  said  that  it  was  Nature  herself,  and  not  the  mathematician, 
who  brings  mathematics  into  natural  philosophy. 

But  the  zoologist  or  morphologist  has  been  slow,  where  the 
physiologist  has  long  been  eager,  to  invoke  the  aid  of  the  physical 
or  mathematical  sciences;  and  the  reasons  for  this  difference  lie 
deep,  and  are  partly  rooted  in  old  tradition  and  partly  in  the 
diverse  minds  and  temperaments  of  men.  To  treat  the  living  body 
as  a  mechanism  was  repugnant,  and  seemed  even  ludicrous,  to 
Pascal  §;  and  Goethe,  lover  of  nature  as  he  was,  ruled  mathematics 
out  of  place  in  natural  history.  Even  now  the  zoologist  has  scarce 
begun  to  dream  of  defining  in  mathematical  language  even  the 
simplest  organic  forms.     When  he  meets  with  a  simple  geometrical 

*  "Sine  profunda  Mechanices  Scientia  nil  veri  vos  intellecturos,  nil  boni  pro- 
laturos  aliis":   Boerhaave,  De  usu  ratiocinii  Mechanici  in  Medicina,  1713. 

t  It  is  well  within  my  own  memor  how  Thomson  and  Tait,  and  Klein  and 
Sylvester  had  to  lay  stress  on  the  mathematical  aspect,  and  urge  the  mathematical 
study,  of  physical  science  itself! 

X  Dr  Johnson  says  that  "to  count  is  a  modern  practice,  the  ancient  method  was 
to  guess";  but  Seneca  was  alive  to  the  difference — "magnum  esse  solem  philosophus 
probabit,  quantus  sit  mathematicus." 

§  Cf.  Pensees,  xxix,  "II  faut  dire,  en  gros,  cela  se  fait  par  figure  et  mouvement, 
car  cela  est  yrai.  Mais  de  dire  quels,  et  composer  la  machine,  cela  est  ridicule, 
car  cela  est  inutile,  et  incertain,  et  penible." 


I]  OF  ADAPTATION  AND  FITNESS  3 

construction,  for  instance  in  the  honeycomb,  he  would  fain  refer  it 
to  psychical  instinct,  or  to  skill  and  ingenuity,  rather  than  to  the 
operation  of  physical  forces  or  mathematical  laws;  when  he  sees  in 
snail,  or  nautilus,  or  tiny  foraminiferal  or  radiolarian  shell  a  close 
approach  to  sphere  or  spiral,  he  is  prone  of  old  habit  to  believe  that 
after  all  it  is  something  more  than  a  spiral  or  a  sphere,  and  that  in 
this  "something  more"  there  lies  what  neither  mathematics  nor 
physics  can  explain.  In  short,  he  is  deeply  reluctant  to  compare 
the  living  with  the  dead,  or  to  explain  by  geometry  or  by  mechanics 
thp  things  which  have  their  part  in  the  mystery  of  life.  Moreover 
he  IS  httle  inclined  to  feel  the  need  of  such  explanations,  or  of  such 
extension  of  his  field  of  thought.  He  is  not  without  some  justifi- 
cation if  he  feels  that  in  admiration  of  nature's  handiwork  he  has 
an  horizon  open  before  his  eyes  as  wide  as  any  man  requires.  He 
has  the  help  of  many  fascinating  theories  within  the  bounds  of  his 
own  science,  which,  though  a  little  lacking  in  precision,  serve  the 
purpose  of  ordering  his  thoughts  and  of  suggesting  new  objects  of 
enquiry.  His  art  of  classification  becomes  an  endless  search  after 
the  blood-relationships  of  things  living  and  the  pedigrees  of  things 
dead  and  gone.  The  facts  of  embryology  record  for  him  (as  Wolff, 
von  Baer  and  Fritz  Miiller  proclaimed)  not  only  the  life-history  of 
the  individual  but  the  ancient  annals  of  its  race.  The  facts  of 
geographical  distribution  or  even  of  the  migration  of  birds  lead  on 
and  on  to  speculations  regarding  lost  continents,  sunken  islands,  or 
bridges  across  ancient  seas.  Every  nesting  bird,  every  ant-hill  or 
spider's  web,  displays  its  psychological  problenis  of  instinct  or  intel- 
ligence. Above  all,  in  things  both  great  and  small,  the  naturajist 
is  rightfully  impressed  and  finally  engrossed  by  the  peculiar  beauty 
which  is  manifested  in  apparent  fitness  or  "adaptation" — the  flower 
for  the  bee,  the  berry  for  the  bird. 

Some  lofty  concepts,  like  space  and  number,  involve  truths  remote 
from  the  category  of  causation;  and  here  we  must  be  content,  as 
Aristotle  says,  if  the  mere  facts  be  known*.  But  natural  history 
deals  with  ephemeral  and  accidental,   not  eternal  nor  universal 

*  ovK  diraiTrjTeop  5  ov8i  ttju  airiav  ofxoiojs,  dW  LKavou  ^v  tlctl  rb  on  deLxdrjvai  /caXcDj 
Eth.  Nic.  1098a,  33.  Teleologist  as  he  was  at  heart,  Aristotle  realised  that  mathematics 
was  on  another  plane  to  teleology:  rds  5^  fxad-qfiariKas  oOdeva  iroieladai  \6you  wepi 
dyaduv  xat  KaKQv.     Met.  996a,  35. 


4  INTRODUCTORY  [ch. 

things ;  their  causes  and  effects  thru^  themselves  on  our  curiosity,  and 
become  the  ultimate  relations  to  which  our  contemplation  extends*. 

Time  out  of  mind  it  has  been  by  way  of  the  "final  cause,"  by  the 
teleological  concept  of  end,  of  purpose  or  of  "design,"  in  one  of  its 
many  forms  (for  its  moods  are  many),  that  men  have  been  chiefly 
wont  to  explain  the  phenomena  of  th«  Hving  world ;  and  it  will  be 
so  while  men  have  eyes  to  see  and  ears  to  hear  withal.  With  Galen, 
as  with  Aristotlef,  it  was  the  physician's  way;  with  John  Ray  J,  as 
with  Aristotle,  it  was  the  naturahst's  way;  with  Kant,  as  with 
Aristotle,  it  was  the  philosopher's  way.  It  was  the  old  Hebrew 
way,  and  has  its  splendid  setting  in  the  story  that  God  made  "every 
plant  of  the  field  before  it  was  in  the  earth,  and  every  herb  of  the 
field  before  it  grew."  It  is  a  common  way,  and  a  great  way;  for  it 
brings  with  it  a  glimpse  of  a  great  vision,  and  it  hes  deep  as  the 
love  of  nature  in  the  hearts  of  men. 

The  argument  of  the  final  cause  is  conspicuous  in  eighteenth- 
century  physics,  half  overshadowing  the  "efficient"  or  physical 
cause  in  the  hands  of  such  men  as  Euler§,  or  Fermat  or  Maupertuis, 
to  whom  Leibniz  1 1  had  passed  it  on.  Half  overshadowed  by  the 
mechanical  concept,  it  runs  through  Claude  Bernard's  Legons  sur  les 
phenomenes  de  la  Vie^,  and  abides  in  much  of  modern  physiology**. 

*  "All  reasonings  concerning  matters  of  fact  seem  to  be  founded  on  the  relation 
of  Cause  and  Effect.  By  means  of  that  relation  alone  we  go  beyond  the  evidence 
of  our  memory  and  senses":   David  Hume,  On  the  Operations  of  the  Understanding. 

t  E.g.  "In  the  works  of  Nature  purpose,  not  accident,  is  the  main  thing":  to  yap 
fir)  TvxofTWs,  d\\'  eveKOL  rtros,  ev  tols  tt]S  (pixreajs  ^pyoLs  ecrl  ko-I  jxaXicra.    PA,  645a,  24. 

X  E.g.  "Quaeri  fortasse  a  nonnullis  potest,  Quis  Papilionum  usus?  Respondeo, 
ad  ornatum  Universi,  et  ut  hominibus  spectaculo  sint."  Joh.  Rail,  Hist.  Insedorum, 
p.  109. 

§  "Quum  enim  Mundi  universi  fabrica  sit  perfectissima,  atque  a  Creators 
sapientissimo  absoluta,  nihil  omnino  in  Mundo  contingit  in  quo  non  maximi 
minimive  ratio  quaepiam  eluceat;  quamobrem  dubium  prorsus  est  nullum  quin 
omnes  Mundi  effectus  ex  causis  finalibus,  ope  Methodi  maximorum  et  minimorum, 
aeque  feliciter  determinari  queant  atque  ex  ipsis  causis  efficientibus."  Methodus 
inveniendi,  etc.,  1744,  p.  24o  {cit.  Mach,  Science  of  Mechanics,  1902,  p.  455). 

!|  Cf.  Opera  (ed.  Erdmann),  p.  106,  "Bien  loin  d'exclure  les  causes  finales... 
c'est  de  la  qu'il  faut  tout  deduire  en  Physique":  in  sharp  contrast  to  Descartes's 
teaching,  "  NuUas  unquam  res  naturales  a  fine,  quem  Deus  aut  Natura  in  iis  faciendis 
sib  jproposuit,  desumemus,  etc."   Princip.  i,  28. 

Tj  Cf.  p.  162.  "La  force  vitale  dirige  des  phenomenes  qu'elle  ne  produit  pas: 
les  agents  physiques  produisent  des  phenomenes  qu'ils  ne  dirigent  pas." 

**  It  is  now  and  then  conceded  with  reluctance.  Thus  Paolo  Enriques,  a  learned 
and  philosophic  naturalist,  writing  '"dell' economia  di  sostanza  nelle  osse  cave" 


I]  OF  THE  FINAL  CAUSE  5 

Inherited  from  Hegel,  it  dominated  Oken's  Naturphilosophie  and 

lingered  among  his  later  disciples,  who  were  wont  to  liken  the  course 

of  organic  evolution  not  to  the  straggling  branches  of  a  tree,  but  to 

the  building  of  a  temple,  divinely  planned,  and  the  crowning  of  it 

with  its  pohshed  minarets*. 

It  is  retained,  somewhat  crudely,  in  modern  embryology,  by  those 

who  see  in  th^  early  processes  of  growth  a  significance  ''rather 

prospective  than  retrospective,"  such  that  the  enlbryonic  phenomena 

must  "be  referred  directly  to  their  usefulness  in  building  up  the 

body  of  the  future  animalf": — which  is  no  more,  and  no  less,  than 

to  say,  with  Aristotle,  that  the  organism  is  the  reAo?,  or  final  cause, 

of  its  own  processes  of  generation  and  development.     It  is  writ 

large  in  that  Entelechyij:  which  Driesch  rediscovered,  and  which  he 

made  known  to  many  who  had  neither  learned  of  it  from  Aristotle, 

nor  studied  it  with  Leibniz,  nor  laughed  at  it  with  Rabelais  and 

Voltaire.     And,  though  it  is  in  a  very  curious  way,   we  are  told 

that  teleology  was   "refounded,   reformed  and  rehabihtated "   by 

Darwin's  concept  of  the  origin  of  species §;    for,  just  as  the  older 

naturalists  held  (as  Addison ||  puts  it)  that  "the  make  of  every  kind 

of  animal  is  different  from  that  of  every  other  kind;   and  ye.  ti^ere 

is  not  the  least  turn  in  the  muscles,  or  twist  in  the  fibres  of  any  one, 

which  does  not  render  them  more  proper  for  that  particular  animal's 

way  of  life  than  any  other  cut  or  texture  of  them  would  have  been  " : 

so,  by  the  theory  of  natural  selection,  "every  variety  of  form  and 

colour  was  urgently  and  absolutely  called  upon  to  produce  its  title 

{Arch.  f.  Entw.  Mech.  xx,  1906),  says  "una  certa  impronta  di  teleologismo  qua 
e  la  e  rimasta,  mio  malgrado,  in  questo  scritto." 

*  Cf.  John  Cleland,  On  terminal  forms  of  life,  Journ.  Anat.  and  Physiol. 
XVIII,  1884. 

t  Conklin,  Embryology  of  Crepidula,  Journ.  of  Morphol.  xin,  p.  203,  1897; 
cf.  F.  R.  Lillie,  Adaptation  in  cleavage.  Wood's  Hole  Biol.  Lectures,  1899,  pp.  43-67. 

X  I  am  inclined  to  trace  back  Driesch's  teaching  of  Entelechy  to  no  less  a  person 
than  Melanchthon.  When  Bacon  {de  Augm.  iv,  3)  states  with  disapproval  that 
the  soul  "has  been  regarded  rather  as  a  function  than  as  a  substance,"  Leslie 
Ellis  points  out  that  he  is  referring  to  Melanchthon's  exposition  of  the  Aristotelian 
doctrine.  For  Melanchthon,  whose  view  of  the  peripatetic  philosophy  had  great 
and  lasting  influence  in  the  Protestant  Universities,  affirmed  that,  according  to 
the  true  view  of  Aristotle's  opinion,  the  soul  is  not  a  substance  but  an  ivreX^xeia,  or 
function.  He  defined  it  as  Sufa/xis  quaedam  ciens  actiones — a  description  all  but 
identical  with  that  of  Claude  Bernard's  '''force  vitale.'' 

§   Ray  Lankester,  art.  Zoology,  Encycl.  Brit.  (9th  edit.),  1888,  p.  806. 

!l   Spectator,  No.  120. 


6  INTRODUCTORY  [ch. 

to  existence  either  as  an  active  useful  agent,  or  as  a  survival"  of 
such  active  usefulness  in  the  past.  But  in  this  last,  and  very 
important  case,  we  have  reached  a  teleology  without  a  rdXos,  as 
men  like  Butler  and  Janet  have  been  prompt  to  shew,  an  "adapta- 
tion" without  "design,"  a  teleology  in  which  the  final  cause  becomes 
little  more,  if  anything,  than  the  mere  expression  or  resultant  of  a 
sifting  out  of  the  good  from  the  bad,  or  of  the  better  from  the  worse, 
in  short  of  a  process  of  mechanism.  The  apparent  manifestations 
of  purpose  or  adaptation  become  part  of  a  mechanical  philosophy, 
"une  forme  methodologique  de  connaissance*,"  according  to  which 
"la  Nature  agit  tou jours  par  les  moyens  les  plus  simplest,"  and 
"chaque  chose  finit  toujours  par  s'accommoder  a  son  miUeu,"  as  in 
the  Epicurean  creed  or  aphorism  that  ^atnTe  finds  a  use  for  every- 
thing J.  In  short,  by  a  road  which  resembles  but  is  not  the  same  as 
Maupertuis's  road,  we  find  our  way  to  the  very  world  in  which  we 
are  living,  and  find  that,  if  it  be  not,  it  is  ever  tending  to  become, 
"the  best  of  all  possible  worlds §." 

But  the  use  of  the  teleological  principle  is  but  one  way,  not  the 
whole  or  the  only  way,  by  which  we  may  seek  to  learn  how  things 
came  to  be,  and  to  take  their  places  in  the  harmonious  complexity 
of  the  world.  To  seek  not  for  end^  but  for  antecedents  is  the  way 
of  the  physicist,  who  finds  "causes"  in  what  he  has  learned  to 
recognise  as  fundamental  properties,  or  inseparable  concomitants, 
or  unchanging  laws,  of  matter  and  of  energy.  In  Aristotle's  parable, 
the  house  is  there  that  men  may  live  in  it ;  but  it  is  also  there  because 
the  builders  have  laid  one  stone  upon  another.  It  is  as  a  mechanism, 
or  a  mechanical  construction,  that  the  physicist  looks  upon  the 
world;  and  Democritus,  first  of  physicists  and  one  of  the  greatest 
of  the  Greeks,  chose  to  refer  all  natural  phenomena  to  mechanism 
and  set  the  final  cause  aside. 

*  So  Newton,  in  the  Preface  to  the  Principia:  "Natura  enim  simplex  est,  et 
rerura  causis  superfluis  non -luxuriat";  "Nature  is  pleased  with  simplicity,  and 
affects  not  the  pomp  of  superfluous  causes."  Modern  physics  finds  the  perfection 
of  mathematical  beauty  in  what  Newton  called  the  perfection  of  simplicity. 

t  Janet,  Les  Causes  Finales,  1876,  p.  350. 

X  "Nil  ideo  quoniam  natumst  in  corpore  ut  uti  Possemus  sed  quod  natumst  id 
procreat  usum."     Lucret.  iv,  834. 

§  The  phrase  is  Leibniz's,  in  his  Theodicee:  and  harks  back  to  Aristotle — If  one 
way  be  better  than  another,  that  you  may  be  sure  is  Nature's  way;  Nic.  Eth. 
10996,  23  et  al. 


I]  OF  EFFICIENT  AND  FINAL  CAUSES  7 

Still,  all  the  whiLe,  like  warp  and  woof,  mechanism  and  teleology 
are  interwoven  together,  and  we  must  not  cleave  to  the  one  nor 
despise  the  other;  for  their  union  is  rooted  in  the  very  nature  of 
totality.  We  may  grow  shy  or  weary  of  looking  to  a  final  cause 
for  an  explanation  of  our  phenomena ;  but  after  we  have  accounted 
for  these  on  the  plainest  principles  of  mechanical  causation  it  may 
be  useful  and  appropriate  to  see  how  the  final  cause  would  tally 
wuth  the  other,  and  lead  towards  the  same  conclusion*.  Maupertuis 
had  Uttle  hking  for  the  final  cause,  and  shewed  some  sympathy  with 
Descartes  in  his  repugnance  to  its  appHcation  to  physical  science. 
But  he  found  at  last,  taking  the  final  and  the  efficient  causes  one  with 
another,  that  "I'harmonie  de  ces  deux  attributs  est  si  parfaite  que 
sans  doute  tous  les  effets  de  la  Nature  se  pourroient  deduire  de 
chacun  pris  separement.  Une  Mecanique  aveugle  et  necessaire  suit 
les  dessins  de  I'lntelHgence  la  plus  eclairee  et  la  plus  hbref."  Boyle 
also,  the  Father  of  Chemistry,  wrote,  in  his  latter  years,  a  Disquisition 
about  the  Final  Causes  of  Natural  Things:  Wherein  it  is  Inquired 
Whether,  And  {if  at  all)  With  what  Cautions,  a  Naturalist  should  admit 
Themi  He  found  "that  all  consideration  of  final  cause  is  not  to  be 
banished  from- Natural  Philosophy...";  but  on  the  other  hand 
''that  the  naturahst  who  would  deserve  that  name  must  not  let 
the  search  and  knowledge  of  final  causes  make  him  neglect  the  in- 
dustrious indagation  of  efficients  J."  In  our  own  day  the  philosopher 
neither  minimises  nor  unduly  magnifies  the  mechanical  aspect  of 
the  Cosmos;  nor  need  the  naturahst  either  exaggerate  •  or  be- 
little the  mechanical  phenomena  which  are  profoundly  associated 
with  Life,  and  inseparable  from  our  understanding  of  Growth  and 
Form.  ^ 

*  "S'il  est  dangereux  de  se  servir  des  causes  finales  a  priori  pour  trouver  les  lois 
des  phenomenes,  11  est  peut-etre  utile  et  il  est  au  moins  curieux  de  faire  voir  com- 
ment le  principe  des  causes  finales  s'accorde  avec  les  lois  des  phenomenes,  pourvu 
qu'on  commence  par  determiner  ces  lois  d'apres  les  principes  de  mecanique  clairs 
et  incontestables."  (D'Aiembert,  Art.  Causes  finales,  Encyclopedie,  ii,  p.  789,  1751.) 
I  SeeJiis  essay  on  the  '''Accord  des  differentes  lois  de  la  Nature." 
X  Cf.  also  Leibniz  {Discours  de  la  Metaphysique:  Lettres  inedites,  ed.  de  Careil, 
1857,  p.  354),  "L'un  et  I'autre  est  bon,  I'un  et  I'autre  peut  etre  utile... et  les 
auteurs  qui  suivent  ces  deux  routes  differentes  ne  devraient  pas  se  maltraiter." 
Or  again  in  the  Monadologie,  "Les  ames  agissent  selon  les  causes  finales. ...  Les 
corps  agissent  selon  les  lois  des  causes  efficientes  ou  des  mouveraents.  Et  les 
deux  regnes,  celui  des  causes  efficientes  et  des  causes  finales  sont  harmonieux 
entre  eux." 


8  INTRODUCTORY  [ch. 

Nevertheless,  when  philosophy  bids  us  hearken  and  obey  the 
lessons  both  of  mechanical  and  of  teleological  interpretation,  the 
precept  is  hard  to  follow:  so  that  oftentimes  it  has  come  to  pass, 
just  as  in  Bacon's  day,  that  a  leaning  to  the  side  of  the  final  cause 
"hath  intercepted  the  severe  and  diligent  enquiry  of  all  real  and 
physical  causes,"  and  has  brought  it  about  that  "the  search  of  the 
physical  cause  hath  been  neglected  and  passed  in  silence."  So  long 
and  so  far  as  "fortuitous  variation*"  and  the  "survival  of  the 
fittest"  remain  engrained  as  fundamental  and  satisfactory  hypo- 
theses in  the  philosophy  of  biology,  so  long  will  these  "satisfactory 
and  specious  causes"  tend  to  stay  "severe  and  diligent  enquiry. . . 
to  the  great  arrest  and  prejudice  of  future  discovery."  Long 
before  the  great  Lord  Keeper  wrote  these  words,  Roger  Bacon  had 
shewn  how  easy  it  is,  and  how  vain,  to  survey  the  operations  of 
Nature  and  idly  refer  her  wondrous  works  to  chance  or  accident, 
or  to  the  immediate  interposition  of  Godf. 

The  difiiculties  which  surround  the  concept  of  ultimate  or  "real" 
caugation,  in  Bacon's  or  Newton's  sense  of  the  word,  the  in- 
superable difficulty  of  giving  any  just  and  tenable  account  of  the 
relation  of  cause  and  effect  from  the  empirical  point  of  view,  need 
scarcely  hinder  us  in  our  physical  enquiry.  As  students  of  mathe- 
matical and  experimental  physics  we  are  content  to  deal  with  those 
antecedents,  or  concomitants,  of  our  phenomena  without  which  the 
phenomenon  does  not  occur — with  causes,  in  short,  which,  aliae  ex 
aliis  aptaetet  necessitate' nexae,  are  no  more,  and  no  less,  than  con- 
ditions sine  qua  non.  Our  purpose  is  still  adequately .  fulfilled : 
inasmuch  as  we  are  still  enabled  to  correlate,  and  to  equate,  our 
particular  phenomena  with  more  and  more  of  the  physical  phenomena 
around,  and  so  to  weave  a  web  of  connection  and  interdependence 
which  shall  serve  our  turn,  though  the  metaphysician  withhold  from 
that  interdependence  the  title  of  causality  J.     We  come  in  touch 

*  The  reader  will  understand  that  I  speak,  not  of  the  "severe  and  diligent 
enquiry"  of  variation  or  of  fortuity,  but  merely  of  the  easy  assumption  that  these 
phenomena  are  a  sufficient  basis  on  which  to  rest,  with  the  all-powerful  help  of 
natural  selection,  a  theory  of  definite  and  progressive  evolution. 

f  Op.  tert.  (ed.  Brewer,  p.  99).  "Ideo  mirabiles  actiones  naturae,  quae  tota 
die  fiunt  in  nobis  et  in  rebus  coram  oculis  nostris,  non  percipimus;  sed  aestimamus 
eas  fieri  vel  per  specialem  operationem  divinam. .  .vel  a  casu  et  fortuna," 

t  Cf.  Fourier's  phrase,  in  his  Theorie  de  la  Chaleur,  with  which  Thomson  and 
Tait  prefaced  their  Treatise  an  Natural  Philosophy:  "Les  causes  primordiales  ne 


I]  OF  ULTIMATE  CAUSATION  9 

with  what  the  schoohnen  called  a  ratio  cognoscendi,  though  the  true 
ratio  efflciendi  is  still  enwrapped  in  many  mysteries.  And  so  handled, 
the  quest  of  physical  causes  merges  with  another  great  Aristotelian 
theme — the  search  for  relations  between  things  apparently  dis- 
connected, and  for  "similitude  in  things  to  common  view  unlike*." 
Newton  did  not  shew  the  cause  of  the  apple  falling,  but  he  shewed 
a  simihtude  ("the  more  to  increase  our  wonder,  with  an  apple") 
between  the  apple  and  the  starsf.  By  doing  so  he  turned  old  facts 
into  new  knowledge ;  and  was  well  content  if  he  could  bring  diverse 
phenomena  under  "two  or  three  Principles  of  Motion"  even  "though 
the  Causes  of  these  Principles  were  not  yet  discovered". 

Moreover,  the  naturalist  and  the  physicist  will  continue  to  speak 
of  "causes",  just  as  of  old,  though  it  may  be  with  some  mental 
reservations :  for,  as  a  French  philosopher  said  in  a  kindred  difficulty : 
"ce  sont  la  des  manieres  de  s'exprimer,  et  si  elles  sont  interdites 
il  faut  renoncer  a  parler  de  ces  choses." 

The  search  for  differences  or  fundamental  contrasts  between  the 
phenomena  of  organic  and  inorganic,  of  animate  and  inanimate, 
things,,  has  occupied  many  men's  minds,  while  the  search  for  com- 
munity of  principles  or  essential  simihtudes  has  been  pursued  by 
few;  and  the  contrasts  are  apt  to  loom  too  large,  great  though  they 
may  be.  M.  Dunan,  discussing  the  Probleme  de  la  Viel,  in  an  essay 
which  M.  Bergson  greatly  commends,  declares  that  "les  lois  physico- 
chimiques  sont  aveugles  et  brutales ;  la  ou  elles  regnent  seules,  au 
lieu  d'un  ordre  et  d'un  concert,  il  ne  pent  y  avoir  qu'incoherence  et 
chaos."  But  the  physicist  proclaims  aloud  that  the  physical 
phenomena  which  meet  us  by  the  way  have  their  forms  not  less 
beautiful  and  scarce  less  varied  than  those  which  move  us  to  admira- 

nous  sont  point  connues;  mais  elles  sont  assujetties  a  des  lois  simples  et  eonstantes, 
que  Ton  peut  decouvrir  par  I'observation,  et  dont  I'etude  est  I'objet  de  la  philosophie 
naturelle." 

*  "Plurimum  amo  analogias,  fidelissimos  meos  magistros,  omnium  Naturae 
arcanorum  conscios,"  said  Kepler;  and  Perrin  speaks  with  admiration,  in  Les 
Atonies,  of  men  like  Galileo  and  Carnot,  who  "possessed  the  power  of  perceiving 
analogies  to  an  extraordinary  degree."  Hure^e  declared,  and  Mill  said  much  the 
same  thing,  that  all  reasoning  whatsoever  depends  on  resemblance  or  analogy, 
and  the  power  to  recognise  it.  Comparative  anatomy  (as  Vicq  d'Azyr  first  called 
it),  or  comparative  physics  (to  use  a  phrase  of  Mach's),  are  particular  instances  of 
a  sustained  search  for  analogy  or  similitude. 

t  As  for  Newton's  apple,  see  De  Morgan,  in  Notes  and  Queries  (2),  vi,  p.  169,  1858. 

I  Revue  Philosophique,  xxxiii,  1892. 


10  INTRODUCTORY  [ch. 

tion  among  living  things.  The  waves  of  the  sea,  the  little  ripples 
on  the  shore,  the  sweeping  curve  of  the  sandy  bay  between  the 
headlands,  the  outline  of  the  hills,  the  shape  of  the  clouds,  all  these 
are  so  many  riddles  of  form,  so  many  problems  of  morphology,  and 
all  of  them  the  physicist  can  more  or  less  easily  read  and  adequately 
solve:  solving  them  by  reference  to  their  antecedent  phenomena, 
in  the  material  system  of  mechanical  forces  to  which  they  belong, 
and  to  which  we  interpret  them  as  being  due.  They  have  also, 
doubtless,  their  immanent  teleological  significance;  but  it  is  on 
another  plane  of  thought  from  the  physicist's  that  we  contemplate 
their  intrinsic  harmony*  and  perfection,  and  "see  that  they  are 
good." 

Nor  is  it  otherwise  with  the  material  forms  of  living  things.  Cell 
and  tissue,  shell  and  bone,  leaf  and  flower,  are  so  many  portions  of 
matter,  and  it  is  in  obedience  to  the  laws  of  physics  that  their 
particles  have  been  moved,  moulded  and  conformedf.  They  are  no 
exception  to  the  rule  that  Qeos  aet  yeajjjLerpeL.  Their  problems  of 
form  are  in  the  first  instance  mathematical  problems,  their  problems 
of  growth  are  essentially  physical  problems,  and  the  morphologist  is, 
ipso  facto,  a  student  of  physical  science.  He  may  learn  from  that 
comprehensive  science,  as  the  physiologists  have  not  failed  to  do, 
the  point  of  view  from  which  her  problems  are  approached,  the 
quantitative  methods  by  which  they  are  attacked,  and  the  whole- 
some restraints  under  which  all  her  work  is  done.  He  may  come 
to  realise  that  there  is  no  branch  of  mathematics,  however  abstract, 
which  may  not  some  day  be  applied  to  phenomena  of  the  real 

*  What  I  understand  by  "holism"  is  what  the  Greeks  called  apfiovia.  This  is 
something  exhibited  not  only  by  a  lyre  in  tune,  but  by  all  the  handiwork  of 
craftsmen,  and  by  all  that  is  " put  together"  by  art  or  nature.  It  is  the  " composite- 
ness  of  any  composite  whole";  and,  like  the  cognate  terms  KpSicns  or  (rvvdeais,  implies 
a  balance  or  attunement.     Cf.  John  Tate,  in  Class.  Review,  Feb.  1939. 

t  This  general  principle  was  clearly  grasped  by  Mr  George  Rainey  many  years 
ago,  and  expressed  in  such  words  as  the  following:  "It  is  illogical  to  suppose  that 
in  the  case  of  vital  organisms  a  distinct  force  exists  to  produce  results  perfectly 
within  the  reach  of  physical  agencies,  especially  as  in  many  instances  no  end  could 
be  attained  were  that  the  case,  but  that  of  opposing  one  force  by  another  capable 
of  effecting  exactly  the  same  purpose."  (On  artificial  calculi,  Q.J. M.S.  {Trans. 
Microsc.  Soc),  vr,  p.  49,  18.58.)  Cf.  also  Helmholtz,  infra  cit.  p.  9.  (Mr  George 
Rainey,  a  man  of  learning  and  originality,  was  demonstrator  of  anatomy  at 
St  Thomas's;  he  followed  that  modest  calling  to  a  great  age,  and  is  remembered 
by  a  few  old  pupils  with  peculiar  affection.) 


I]  OF  EVOLUTION  AND  ENTROPY  11 

world*.  He  may  even  find  a  certain  analogy  between  the  slow, 
reluctant  extension  of  physical  laws  to  vital  phenomena  and  the  slow 
triumphant  demonstration  by  Tycho  Brahe,  Copernicus,  GaHleo  and 
Newton  (all  in  opposition  to  the  Aristotelian  cosmogony),  that  the 
heavens  are  formed  of  like  substance  with  the  earth,  and  that  the 
movements  of  both  are  subject  to  the  selfsame  laws. 

Organic  evolution  has  its  physical  analogue  in  the  universal  law 
that  the  world  tends,  in  all  its  parts  and  particles,  to  pass  from 
certain  less  probable  to  certain  more  probable  configurations  or 
states.  This  is  the  second  law  of  thermodynamics.  It  has  been 
called  the  law  of  evolution  of  the  world'f ;  and  we  call  it,  after  Clausius, 
the  Principle  of  Entropy,  which  is  a  literal  translation  of  Evolution 
into  Greek. 

The  introduction  of  mathematical  concepts  into  natural'  science 
has  seemed  to  many  men  no  mere  stumbling-block,  but  a  very 
parting  of  the  ways.  Bichat  was  a  man  of  genius,  who  did  immense 
service  to  philosophical  anatomy,  but,  like  Pascal,  he  utterly  refused 
to  bring  physics  or  mathematics  into  biology:  "  On  calcule  le  retour 
d'un  comete,  les  resistances  d'un  fluide  parcourant  un  canal  inerte, 
la  Vitesse  d'un  projectile,  etc.;  mais  calculer  avec  BorelU  la  force 
d'un  muscle,  avec  Keil  la  vitesse  du  sang,  avec  Jurine,  Lavoisier  et 
d'autres  la  quantite  d'air  entrant  dans  le  poumon,  c'est  batir  sur  un 
sable  mouvant  un  edifice  sohde  par  lui-meme,  mais  qui  tombe  bientot 
faute  de  base  assureej."  Comte  went  further  still,  and  said  that 
every  attempt  to  introduce  mathematics  into  chemistry  must  be 
deemed  profoundly  irrational,  and  contrary  to  the  whole  spirit  of 
the  science  §.  But  the  great  makers  of  modern  science  have  all  gone 
the  other  way.  Von  Baer,  using  a  bold  metaphor,  thought  that  it 
might  become  possible  "  die  bildenden  Krafte  des  thierischen  Korpers 
.  auf  die  allgemeinen  Krafte  oder  Lebenserscheinun^en  des  Weltganzes 
zuriickzufiihrenll."  Thomas  Young  shewed,  as  BorelU  had  done, 
how  physics  may  subserve  anatomy;  he  learned  from  the  heart  and 
a,rteries  that  "  the  mechanical  motions  which  take  place  in  an  animal's 
body  are  regulated  by  the  same  general  laws  as  the  motions  of 

*  So  said  Lobatchevsky. 

t  Cf.  Chwolson,  Lehrbuch,  iii,  p.  499,  1905;  J.  Perrin,  Traitd  de  chimie  physique, 
I,  p.  142,  1903;  and  Lotka's  Elements  of  Physical  Biology,  1925,  p.  26. 

t  La  Vie  et  la  Mort,  p.  81.  §  Philosophie  Positive,  Bk.  rv. 

]|    Ueber  Entwicklung  der  Thiere:  Beobachtungen  und  Reflexionen,  i,  p.  22,  1828. 


12  INTRODUCTORY  [ch. 

inanimate  bodies*."  And  Theodore  Schwann  said  plainly,  a  hun- 
dred years  ago,  "Ich  wiederhole  iibrigens  dass,  wenn  hier  von  einer 
physikahschen  Erklarung  der  organischen  Erscheinungen  die  Rede 
ist,  darunter  nicht  nothwendig  eine  Erklarung  durch  die  bekannten 
physikalischen  Krafte. .  .zu  verstehen  ist,  sondern  iiberhaupt  eine 
Erklarung  durch  Krafte,  die  nach  strengen  Gesetzen  der  blinden 
Nothwendigkeit  wie  die  physikalischen  Krafte  wirken,  mogen  diese 
Krafte  auch  in  der  anorganischen  Natur  auftreten  oder  nicht f." 

Helmholtz,  in  a  famous  and  influential  lecture,  and  surely  with 
these  very  words  of  Schwann's  in  mind,  laid  it  down  as  the  funda- 
mental principle  of  physiology  that  "there  may  be  other  agents 
acting  in  the  hving  body  than  those  agents  which  act  in  the  inorganic 
world ;  but  these  forces,  so  far  as  they  cause  chemical  and  mechanical 
influence  in  the  body,  must  be  quite  of  the  same  character  as  inorganic 
forces :  in  this,  at  least,  that  their  eff'ects  must  be  ruled  by  necessity, 
and  must  always  be  the  same  when  acting  under  the  same  conditions ; 
and  so  there  cannot  exist  any  arbitrary  choice  in  the  direction  of  their 
actions."  It  follows  further  that,  like  the  other  "physical"  forces, 
they  must  be  subject  to  mathematical  analysis  and  deduction  J. 

So  much  for  the  physico-chemical  problems  of  physiology.  Apart 
from  these,  the  road  of  physico-mathematical  or  dynamical  investi- 
gation in  morphology  has  found  few  to  follow  it;  but  the  pathway 
is  old.  The  way  of  the  old  Ionian  physicians,  of  Anaxagoras  § ,  of 
Empedocles  and  his  disciples  in  the  days  before  Aristotle,  lay  just 
by  that  highway  side.  It  was  Galileo's  and  Borelli's  way;  and 
Harvey's  way,  when  he  discovered  the  circulation  of  the  blood ||. 
It  was  little  trodden  for  long  afterwards,  but  once  in  a  while 
Swammerdam  and  Reaumur  passed  thereby.  And  of  later  years 
Moseley  and  Meyer,  Berthold,  Errera  and  Roux  have  been  among 

*  Croonian  Lecture  on  the  heart  and  arteries,  Phil.  Trans.  1809,  p.  1;  Collected 
Works,  I,  p.  511. 

t  M ikroskopische  Untersuchungen,  1839,  p.  226. 

J  The  conservation  of  forces  applied  to  organic  nature,  Proc.  Royal  Inst. 
April  12,  1861. 

§  Whereby  he  incurred  the  reproach  of  Socrates,  in  the  Phaedo.  See  Clerk 
Maxwell  on  Anaxagoras  as  a  Physicist,  in  Phil.  Mag.  (4),  xlvi,  pp.  453-460,  1873. 

II  Cf.  Harvey's  preface  to  his  Exercitationes  de  Generatione  Animalium,  1651: 
"Quoniam  igitur  in  Generatione  animalium  (ut  etiam  in  caeteris  rebus  omnibus 
de  quibus  aliquid  scire  cupimus),  inquisitio  omnis  a  caussis  petenda  est,  praesertim 
a  materiali  et  efficiente:   visum  est  mihi"  etc. 


I]  OF  NATURAL  PHILOSOPHY  13 

the  little  band  of  travellers.  We  need  not  wonder  if  the  way  be 
hard  to  follow,  and  if  these  wayfarers  have  yet  gathered  little. 
A  harvest  has  been  reaped  by  others,  and  the  gleaning  of  the  grapes 
is  slow. 

It  behoves  us  always  to  remember  that  in  physics  it  has  taken 
great  men  to  discover  simple  things.  They  are  very  great  names 
indeed  which  we  couple  with  the  explanation  of  the  path  of  a  stone, 
the  droop  of  a  chain,  the  tints  of  a  bubble,  the  shadows  in  a  cup. 
It  is  but  the  shghtest  adumbration  of  a  dynamical  morphology  that 
we  can  hope  to  have  until  the  physicist  and  the  mathematician  shall 
have  made  these  problems  of  ours  their  own,  or  till  a  new  Boscovich  shall 
have  written  for  the  naturahst  the  new  TheariaPhilosophiaeNaturalis. 

How  far  even  then  mathematics  will  suffice  to  describe,  and 
physics  to  explain,  the  fabric  of  the  body,  no  man  can  foresee.  It 
may  be  that  all  the  laws  of  energy,  and  all  the  properties  of  matter, 
and  all  the  chemistry  of  all  the  colloids  are  as  powerless  to  explain 
the  body  as  they  are  impotent  to  comprehend  the  soul.  For  my 
part,  I  think  it  is  not  so.  Of  how  it  is  that  the  soul  informs  the 
body,  physical  science  teaches  me  nothing;  and  that  living  matter 
influences  and  is  influenced  by  mind  is  a  mystery  without  a  clu'e. 
Consciousness  is  not  explained  to  my  comprehension  by  all  the 
nerve-paths  and  neurones  of  the  physiologist ;  nor  do  I  ask  of  physics 
how  goodness  shines  in  one  man's  face,  and  evil  betrays  itself  in 
another.  But  of  the  construction  and  growth  and  working  of  the 
body,  as  of  all  else  that  is  of  the  earth  earthy,  physical  science  is, 
in  my  humble  opinion,  our  only  teacher  and  guide. 

Often  and  often  it  happens  that  our  physical  knowledge  is  in- 
adequate to  explain  the  mechanical  working  of  the  organism;  the 
phenomena  are  superlatively .  complex,  the  procedure  is  involved 
and  entangled,  and  the  investigation  has  occupied  but  a  few  short 
lives  of  men.  When  physical  science  falls  short  of  explaining  the 
order  which  reigns  throughout  these  manifold  phenomena — an  order 
more  characteristic  in  its  totality  than  any  of  its  phenomena  in 
themselves — men  hasten:  to  invoke  a  guiding  principle,  an  entelechy, 
or  call  it  what  you  will.  But  all  the  while  no  physical  law,  any 
more  than  gravity  itself,  not  even  among  the  puzzles  of  stereo- 
chemistry or  of  physiological  surface-action  and  osmosis,  is  known 
to  be  transgressed  by  the  bodily  mechanism. 


14  INTRODUCTORY  [ch. 

Some  physicists  declare,  as  Maxwell  did,  that  atoms  or  molecules 
more  compHcated  by  far  than  the  chemist's  hypotheses  demand,  are 
requisite  to  explain  the  phenomena  of  life.  If  what  is  impHed  be 
an  explanation  of  psychical  phenomena,  let  the  point  be  granted  at 
once;  we  may  go  yet  further  and  decHne,  with  Maxwell,  to  believe 
that  anything  of  the  nature  of  physical  complexity,  however  exalted, 
could  ever  suffice.  Other  physicists,  like  Auerbach*,  or  Larmorj, 
or  Joly  J,  assure  us  that  our  laws  of  thermodynamics  do  not  suffice, 
or  are  inappropriate,  to  explain  the  maintenance,  or  (in  Joly's  phrase) 
the  accelerative  absorption,  of  the  bodily  energies,  the  retardation 
of  entropy,  and  the  long  battle  against  the  cold  and  darkness  which 
is  death.  With  these  weighty  problems  I  am  not  for  the  moment 
concerned.  My  sole  purpose  is  to  correlate  with  mathematical  state- 
ment and  physical  law  certain  of  the  simpler  outward  phenomena 
of  organic  growth  and  structure  or  form,  while  all  the  while  regarding 
the  fabric  of  the  organism,  ex  hypothesi,  as  a  material  and  mechanical 
configuration.  This  is  my  purpose  here.  But  I  would  not  for  the 
world  be  thought  to  beheve  that  this  is  the  only  story  which  Life 
and  her  Children  have  tp  tell.  One  does  not  come  by  studying 
living  things  for  a  lifetime  to  suppose  that  physics  and  chemistry 
can  account  for  them  all§. 

Physical  science  and  philosophy  stand  side  by  side,  and  one 
upholds  the  other.  Without  something  of  the  strength  of  physics 
philosophy  would  be  weak ;  and  without  something  of  philosophy's 
wealth  physical  science  would  be  poor.  "Rien  ne  retirera  du  tissu 
de  la  science  les  fils  d'or  que  la  main  du  philosophe  y  a  introduits||." 
But  there  are  fields  where  each,  for  a  while  at  least,  must  work  alone; 
and  where  physical  science  reaches  its  limitations  physical  science 
itself  must  help  us  to  discover.     Meanwhile  the  appropriate  and 

*  Ektropismus,  oder  die  physikalische  Theorie  des  Lebens,  Leipzig,  1810. 

t  Wilde  Lecture,  Nature,  March  12,  1908;  ibid.  Sept.  6,  1900;  Aether  and  Matter, 
p.  288.     Cf.  also  Kelvin,  Fortnightly  Review,  1892,  p.  313. 

X  The  abundance  of  life,  Proc.  Roy.  Dublin  Soc.  vii,  1890;  Scientific  Essays, 
1915,  p.  60  seq. 

§  That  mechanism  has  its  share  in  the  scheme  of  nature  no  philosopher  has 
denied.  Aristotle  (or  whosoever  wrote  the  De  Mundo)  goes  so  far  as  to  assert  that 
in  the  most  mechanical  operations  of  nature  we  behold  some  of  the  divinest 
attributes  of  God. 

II  J.  H,  Fr.  Papillon,  Histoire  de  la  jyhilosophie  moderne  dans  ses  rapports  avec  le 
developpement  des  sciences  de  la  nature,  i,  p.  300,  1870. 


I]  OF  LIFE  ITSELF  15 

legitimate  postulate  of  the  physicist,  in  approaching  the  physical 
problems  of  the  living  body,  is  that  with  these  physical  phenomena 
no  ahen  influence  interferes.  But  the  postulate,  though  it  is  certainly 
legitimate,  and  though  it  is  the  proper  and  necessary  prelude  to 
scientific  enquiry,  may  some  day  be  proven  to  be  untrue;  and  its 
disproof  will  not  be  to  the  physicist's  confusion,  but  will  come  as 
his  reward.  In  dealing  with  forms  which  are  so  concomitant  with 
life  that  they  are  seemingly  controlled  by  life,  it  is  in  no  spirit  of 
arrogant  assertiveness  if  the  physicist  begins  his  argument,  after  the 
fashion  of  a  most  illustrious  exemplar,  with  the  old  formula  of 
scholastic  challenge:   An  Vita  sit?     Dico  quod  non. 

The  terms  Growth  and  Form,  which  make  up  the  title  of  this  book, 
are  to  be  understood,  as  I  need  hardly  say,  in  their  relation  to  the 
study  of  organisms.  We  want  to  see  how,  in  some  cases  at  least, 
the  forms  of  living  things,  and  of  the  parts  of  living  things,  can  be 
explained  by  physical  considerations,  and  to  realise  that  in  general 
no  organic  forms  exist  save  such  as  are  in  conformity  with  physical 
and  mathematical  laws.  And  while  growth  is  a  somewhat  vague 
word  for  a  very  complex  matter,  which  may  depend  on  various 
things,  from  simple  imbibition  of  water  to  the  complicated  results 
of  the  chemistry  of  nutrition,  it  deserves  to  be  studied  in  relation 
to  form :  whether  it  proceed  by  simple  increase  of  size  without  obvious 
alteration  of  form,  or  whether  it  so  proceed  as  to  bring  about  a 
gradual  change  of  form  and  the  slow  development  of  a  more  or  less 
complicated  structure. 

In  the  Newtonian  language*  of  elementary  physics,  force  is 
recognised  by  its  action  in  producing  or  in  changing  motion,  or 
in  preventing  change  of  motion  or  in  maintaining  rest.  When  we 
deal  with  matter  in  the  concrete,  force  does  not,  strictly  speaking, 
enter  into  the  question,  for  force,  unlike  matter,  has  no  independent 
objective  existence.  It  is  energy  in  its  various  forms,  known  or 
unknown,  that  acts  upon  matter.  But  when  we  abstract  our 
thoughts  from  the  material  to  its  form,  or  from  the  thing  moved  to 
its  motions,  when  we  deal  with  the  subjective  conceptions  of  form, 

*  It  is  neither  unnecessary  nor  superfluous  to  explain  that  physics  is  passing 
through  an  empirical  phase  into  a  phase  of  pure  mathematical  reasoning.  But 
when  we  use  physics  to  interpret  and  elucidate  our  biology,  it  is  the  old-fashioned 
empirical  physics  which  we  endeavour,  and  are  alone  able,  to  apply. 


16  INTRODUCTORY  [ch. 

or  movement,  or  the  movements  that  change  of  form  impHes,  then 
Force  is  the  appropriate  term  for  our  conception  of  the  causes  by 
which  these  forms  and  changes  of  form  are  brought  about.  When 
we  use  the  term  force,  we  use  it,  as  the  physicist  always  does,  for 
the  sake  of  brevity,  using  a  symbol  for  the  magnitude  and  direction 
of  an  action  in  reference  to  the  symbol  or  diagram  of  a  material 
thing.  It  is  a  term  as  subjective  and  symbolic  as  form  itself,  and 
SO'  is  used  appropriately  in  connection  therewith. 

The  form,  then,  of  any  portion  of  matter,  whether  it  be  living 
or  dead,  and  the  changes  of  form  which  are  apparent  in  its  movements 
and  in  its  growth,  may  in  all  cases  alike  be  described  as  due  to 
the  action  of  force.  In  short,  the  form  of  an  object  is  a  "diagram 
of  forces,"  in  this  sense,  at  least,  that  from  it  we  can  judge  of  or 
deduce  the  forces  that  are  acting  or  have  acted  upon  it:  in  this 
strict  and  particular  sense,  it  is  a  diagram — in  the  case  of  a  sohd, 
of  the  forces  which  have  been  impressed  upon  it  when  its  conformation 
was  produced,  together  with  those  which  enable  it  to  retain  its 
conformation;  in  the  case  of  a  Hquid  (or  of  a  gas)  of  the  forces  which 
are  for  the  moment  acting  on  it  to  restrain  or  balance  its  own 
inherent  mobility.  In  an  organism,  great  or  small,  it  is  not  merely 
the  nature  of  the  motions  of  the  hving  substance  which  we  must 
interpret  in  terms  of  force  (according  to  kinetics),  but  also  the 
conformation  of  the  organism  itself,  whose  permanence  or  equilibrium 
is  explained  by  the  interaction  or  balance  of  forces,  as  described  in 
statics. 

If  we  look  at  the  hving  cell  of  an  Amoeba  or  a  Spirogyra,  we 
see  a  something  which  exhibits  certain  active  movements,  and  a 
certain  fluctuating,  or  more  or  less  lasting,  form;  and  its  form  at 
a  given  moment,  just  like  its  motions,  is  to  be  investigated  by  the 
help  of  physical  methods,  and  explained  by  the  invocation  of  the 
mathematical  conception  of  force. 

Now  the  state,  including  the  shape  or  form,  of  a  portion  of  matter 
is  the  resultant  of  a  number  of  forces,  which  represent  or  symbolise 
the  manifestations  of  various  kinds  of  energy;  and  it  is  obvious, 
accordingly,  that  a  great  part  of  physical  science  must  be  under- 
stood or  taken  for  granted  as  the  necessary  preliminary  to  the 
discussion  on  which  we  are  engaged.  But  we  may  at  least  try  to 
indicate,  very  briefly,  the  nature  of  the  principal  forces  and  the 


I]  OF  MATTER  AND  ENERGY  17 

principal  properties  of  matter  with  which  our  subject  obhges  us  to 
deal.  Let  us  imagine,  for  instance,  the  case  of  a  so-called  "simple" 
organism,  such  as  Amoeba;  and  if  our  short  list  of  its  physical 
•properties  and  conditions  be  helpful  to  our  further  discussion,  we 
need  not  consider  how  far  it  be  complete  or  adequate  from  the 
wider  physical  point  of  view*. 

This  portion  of  matter,  then,  is  kept  together  by  the  inter- 
molecular  force  of  cohesion;  in  the  movements  of  its  particles 
relatively  to  one  another,  and  in  its  own  movements  relative  to 
adjacent  matter,  it  meets  with  the  opposing  force  of  friction — 
without  the  help  of  which  its  creeping  movements  could  not  be 
performed.  It  is  acted  on  by  gravity,  and  this  force  tends  (though 
slightly,  owing  to  the  Amoeba's  small  mass,  and  to  the  small 
difference  between  its  density  and  that  of  the  surrounding  fluid) 
to  flatten  it  down  upon  the  solid  substance  on  which  it  may  be 
creeping.  Our  Amoeba  tends,  in  the  next  place,  to  be  deformed 
by  any  pressure  from  outside,  even  though  slight,  which  may  be 
applied  to  it,  and  this  circumstance  shews  it  to  consist  of  matter 
in  a  fluid,  or  at  least  semi-fluid,  state:  which  state  is  further 
indicated  when  we  observe  streaming  or  current  motions  in  its 
interior.  Like  other  fluid  bodies,  its  surfacef,  whatsoever  other 
substance — gas,  hquid  or  solid — it  be  in  contact  with,  and  in  varying 
degree  according  to  the  nature  of  that  adjacent  substance,  is  the 
seat  of  molecular  force  exhibiting  itself  as  a  surface-tension,  from 
the  action  of  which  many  important  consequences  follow,  greatly 
affecting  the  form  of  the  fluid  surface. 

While  the  protoplasmj  of  the  Amoeba  reacts  to  the  shghtest 
pressure,  and  tends  to  "flow,"  and  while  we  therefore  speak  of  it 

*  With  the  special  and  impprtant  properties  of  colloidal  matter  we  are,  for 
the  time  being,  not  concerned. 

t  Whether  an  animal  cell  has  a  membrane,  or  only  a  pellicle  or  zona  limitans, 
was  once  deemed  of  great  importance,  and  played  a  big  part  in  the  early  contro- 
versies between  the  cell-theory  of  Schwann  and  the  protoplasma-theory  of  Max 
Schultze  and  others,  Dujardin  came  near  the  truth  when  he  said,  somewhat 
naively,  "en  niant  la  presence  d'un  tegument  propre,  je  ne  pretends  pas  du  tout 
nier  i'existence  d'une  surface." 

%  The  word  protoplasm  is  used  here  in  its  most  general  sense,  as  vaguely  as  when 
Huxley  spoke  of  it  as  the  "physical  basis  of  life."  Its  many  changes  and  shades 
of  meaning  in  early  years  are  discussed  by  Van  Bambeke  in  the  Bull.  Sac.  Beige 
de  Microscopie,  xxn,  pp.  1-16,  1896. 


18  INTRODUCTORY  [ch. 

as  a  fluid*,  it  is  evidently  far  less  mobile  than  such  a  fluid  (for 
instance)  as  water,  but  is  rather  Uke  treacle  in  its  slow  creeping 
movements  as  it  changes  its  shape  in  response  to  force.  Such  fluids 
are  said  to  have  a  high  viscosity,  and  this  viscosity  obviously  acts 
in  the  way  of  resisting  change  of  form,  or  in  other  words  of 
retarding  the  efl'ects  of  any  disturbing  action  of  force.  When  the 
viscous  fluid  is  capable  of  being  drawn  out  into  fine  threads,  a 
property  in  which  we  know  that  some  Amoebae  differ  greatly  from 
others,  we  say  that  the  fluid  is  also  viscid,  or  exhibits  viscidity. 
Again,  not  by  virtue  of  our  Amoeba  being  liquid,  but  at  the  same 
time  in  vastly  greater  measure  than  if  it  were  a  sohd  (though  far  less 
rapidly  than  if  it  were  a  gas),  a  process  of  molecular  diffusion  is 
constantly  going  on  within  its  substance,  by  which  its  particles 
interchange  their  places  within  the  mass,  while  surrounding  fluids, 
gases  and  soUds  in  solution  diffuse  into  and  out  of  it.  In  so  far 
as  the  outer  wall  of  the  cell  is  different  in  character  from  the 
interior,  whether  it  be  a  mere  pelhcle  as  in  Amoeba  or  a  firm 
cell-wall  as  in  Protococcus,  the  diffusion  which  takes  place  throtigh 
this  wall  is  sometimes  distinguished  under  the  term  osmosis. 

Within  the  cell,  chemical  forces  are  at  work,  and  so  also  in  all 
probabihty  (to  judge  by  analogy)  are  electrical  forces;  and  the 
organism  reacts  also  to  forces  from  without,  that  have  their  origin 
in  chemical,  electrical  and  thermal  influences.  The  processes  of 
diffusion  and  of  chemical  activity  within  the  cell  result,  by  the 
drawing  in  of  water,  salts,  and  food-material  with  or  without 
chemical  transformation  into  protoplasm,  in  growth,  and  this  com- 
plex phenomenon  we  shall  usually,  without  discussing  its  nature 
and  origin,  describe  and  picture  as  a  force.  Indeed  we  shall 
manifestly  be  incHned  to  use  the  term  growth  in  two  senses,  just 
indeed  as  we  do  in  the  case  of  attraction  or  gravitation,  on  the  one 
hand  as  a  process,  and  on  the  other  as  a  force. 

In  the  phenomena  of  cell-division,  in  the  attractions  or  repulsions 
of  the  parts  of  the  dividing  nucleus,  and  in  the  "  caryokinetic " 
figures  which  appear  in  connection  with  it,  we  seem  to  see  in 
operation  forces  and  the  effects  of  forces  which  have,  to  say  the 

*  One  of  the  first  statements  which  Dujardin  made  about  protoplasm  (or,  as 
he  called  it,  sarcode)  was  that  it  was  not  a  fluid;  and  he  relied  greatly  on  this  fact 
to  shew  that  it  was  a  living,  or  an  organised,  structure. 


I]  OF  VITAL  PHENOMENA  19 

least  of  it,  a  close  analogy  with  known  physical  phenomena :  and 
to  this  matter  we  shall  presently  return.  But  though  they  resemble 
known  physical  phenomena,  their  nature  is  still  the  subject  of  much 
dubiety  and  discussion,  and  neither  the  forms  produced  nor  the 
forces  at  work  can  yet  be  satisfactorily  and  simply  explained.  We 
may  readily  admit  then,  that,  besides  phenomena  which  are  obviously 
physical  in  their  nature,  there  are  actions  visible  as  well  as  invisible 
taking  place  within  living  cells  which  our  knowledge  does  not  permit 
us  to  ascribe  with  certainty  to  any  known  physical  force;  and  it 
may  or  may  not  be  that  these  phenomena  will  yield  in  time  to  the 
methods  of  physical  investigation.  Whether  they  do  or  no,  it  is 
plain  that  we  have  no  clear  rule  or  guidance  as  to  what  is  "vital" 
and  what  is  not;  the  whole  assemblage  of  so-called  vital  phenomena, 
or  properties  of  the  organism,  cannot  be  clearly  classified  into  those 
that  are  physical  in  origin  and  those  that  are  sui  generis  and  peculiar 
to  living  things.  All  we  can  do  meanwhile  is  to  analyse,  bit  by  bit, 
those  parts  of  the  whole  to  which  the  ordinary  laws  of  the  physical 
forces  more  or  less  obviously  and  clearly  and  indubitably  apply. 

But  even  the  ordinary  laws  of  the  physical  forces  are  by  no  means 
simple  and  plain.  In  the  winding  up  of  a  clock  (so  Kelvin  once 
said),  and  in  the  properties  of  matter  which  it  involves,  there  is 
enough  and  more  than  enough  of  mystery  for  our  limited  under- 
standing: "a  watchspring  is  much  farther  beyond  our  understanding 
than  a  gaseous  nebula."  We  learn  and  learn,  but  never  know  all, 
about  the  smallest,  humblest  thing.  So  said  St  Bonaventure :  "  Si  per 
multos  annos  viveres,  adhuc  naturam  unius  festucae  seu  muscae  seu 
minimae  creaturae  de  mundo  ad  plenum  cognoscere  non  valeres*." 
There  is  a  certain  fascination  in  such  ignorance ;  and  we  learn  (like 
the  Abbe  Galiani)  without  discouragement  that  Science  is  "plutot 
destine  a  etudier  qu'a  connaitre,  a  chercher  qu'a  trouver  la  verite." 

Morphology  is  not  only  a  study  of  material  things  and  of  the  forms 
of  material  things,  but  has  its  dynamical  aspect,  under  which  we 
deal  with  the  interpretation,  in  term^  of  force,  of  the  operations  of 
Energyf .     And  here  it  is  well  worth  while  to  remark  that,  in  deahng 

*  Op.  V,  p.  541 ;  cit.  E.  Gilson. 

t  This  is  a  great  theme.  Boltzmann,  writing  in  1886  on  the  second  law  of 
thermodynamics,  declared  that  available  energy  was  the  main  object  at  stake 
in  the  struggle  for  existence  and  the  evolution  of  the  world.  Cf.  Lotka,  The 
energetics  of  evolution,  Proc.  Nat.  Acad.  Sci.  1922,  p.  147. 


20  INTRODUCTORY  [ch. 

with  the  facts  of  embryology  or  the  phenomena  of  inheritance,  the 
common  language  of  the,  books  seems  to  deal  too  much  with  the 
material  elements  concerned,  as  the  causes  of  development,  of 
variation  or  of  hereditary  transmission.  Matter  as  such  produces 
nothing,  changes  nothing,  does  nothing;  and  however  convenient 
it  may  afterwards  be  to  abbreviate  our  nomenclature  and  our 
descriptions,  we  must  most  carefully  realise  in  the  outset  that  the 
spermatozoon,  the  nucleus,  the  chromosomes  or  the  germ-plasma 
can  never  act  as  matter  alone,  but  only  as  seats  of  energy  and  as 
centres  of  force.  And  this  is  but  an  adaptation  (in  the  light,  or 
rather  in  the  conventional  symboHsm,  of  modern  science)  of  the  old 
saying  of  the  philosopher :  apx^  yo.p  r)  <j>voLs  fxdXXov  rrjs  vXrjg. 

Since  this  book  was  written,  some  five  and  twenty  years  ago, 
certain  great  physico-mathematical  concepts  have  greatly  changed. 
Newtonian  mechanics  and  Newtonian  concepts  of  space  and  time 
are  found  unsuitable,  even  untenable  or  invahd,  for  the  all  but 
infinitely  great  and  the  all  but  infinitely  small.  The  very  idea  of 
physical  causation  is  said  to  be  illusory,  and  the  physics  of  the 
atom  and  the  electron,  and  of  the  quantum  theory,  are  to  be 
elucidated  by  the  laws  of  probability  rather  than  by  the  concept 
of  causation  and  its  effects.  But  the  orders  of  magnitude,  whether 
of  space  or  time,  within  which  these  new  concepts  become  useful, 
or  hold  true,  lie  far  away.  We  distinguish,  and  can  never  help 
distinguishing,  between  the  things  which  are  of  our  own  scale  and 
order,  to  which  our  minds  are  accustomed  and  our  senses  attuned, 
and  those  remote  phenomena  which  ordinary  standards  fail  to 
measure,  in  regions  where  (as  Robert  Louis  Stevenson  said)  there 
is  no  habitable  city  for  the  mind  of  man. 

It  is  no  wonder  if  new  methods,  new  laws,  new  words,  new  modes 
of  thought  are  needed  when  we  make  bold  to  contemplate  a  Universe 
within  which  all  Newton's  is  but  a  speck.  But  the  world  of  the 
Hving,  wide  as  it  may  be,  is  bounded  by  a  famihar  horizon  within 
which  our  thoughts  and  senses  are  at  home,  our  scales  of  time  and 
magnitude  suffice,  and  the  Natural  Philosophy  of  Newton  and 
Gahleo  rests  secure. 

We  start,  like  Aristotle,  with  our  own  stock-in-trade  of  know- 
ledge: dpKTeov  OLTTO  Tcov  rjfjuv  yvajpLjjLojv.     And  only  when  we  are 


I]  OF  NEWTONIAN  PHYSICS  21 

steeped  to  the  marrow  (as  Henri  Poincare  once  said)  in  the  old  laws, 
and  in  no  danger  of  forgetting  them,  may  we  be  allowed  to  learn 
how  they  have  their  remote  but  subtle  limitations,  and  cease  afar 
off  to  be  more  than  approximately  true  *.  Kant's  axiom  of  causahty, 
that  it  is  denknotwendig — indispensable  for  thought — remains  true 
however  physical  science  may  change.  His  later  aphorism,  that  all 
changes  take  place  subject  to  the  law  which  links  cause  and  effect 
together — "alle  Veranderungen  geschehen  nach  dem  Gesetz  der 
Verkniipfung  von  Ursache  und  Wirkung" — is  still  an  axiom  a  priori, 
independent  of  experience:  for  experience  itself  depends  upon  its 
truth  t- 

*  So  Max  Planck  himself  says  somewhere:  "In  my  opinion  the  teaching  of 
mechanics  will  still  have  to  begin  with  Newtonian  force,  just  as  optics  begins  in 
the  sensation  of  colour  and  thermodynamics  with  the  sensation  of  warmth, 
despite  th^  fact  that  a  more  precise  basis  is  substituted  later  on." 

t  "Weil  er  [der  Grundsatz  das  Kausalverhaltnisses]  selbst  der  grund  der  Moglich- 
keit  einer  solchen  Erfahrung  ist":  Kritik  d.  reinen  Vernunft,  ed.  Odicke,  1889,  p.  221. 
Cf.  also  G.  W.  Kellner,  Die  Kausalitat  in  der  Physik,  Ztschr.f.  Physik,  lx,iv,  pp.  568- 
580.  1930. 


CHAPTER  II 

ON  MAGNITUDE 

To  terms  of  magnitude,  and  of  direction,  must  we  refer  all  our 
conceptions  of  Form.  For  the  form  of  an  object  is  defined  when  we 
know  its  magnitude,  actual  or  relative,  in  various  directions;  and 
Growth  involves  the  same  concepts  of  magnitude  and  direction, 
related  to  the  further  concept,  or  "dimension,"  of  Time.  Before 
we  proceed  to  the  consideration  of  specific  form,  it  will  be  well  to 
consider  certain  general  phenomena  of  spatial  magnitude,  or  of  the 
extension  of  a  body  in  the  several  dimensions  of  space. 

We  are  taught  by  elementary  mathematics — and  by  Archimedes 
himself — that  in  similar  figures  the  surface  increases  as  the  square, 
and  the  volume  as  the  cube,  of  the  linear  dimensions.  If  we  take 
the  simple  case  of  a  sphere,  with  radius  r,  the  area  of  its  surface  is 
equal  to  4:7rr^,  and  its  volume  to  ^ttt^  ;  from  which  it  follows  that  the 
ratio  of  its  volume  to  surface,  or  V/S,  is  Jr.  That  is  to  say,  VfS 
varies  as  r;  or,  in  other  words,  the  larger  the  sphere  by  so  much  the 
greater  will  be  its  volume  (or  its  mass,  if  it  be  uniformly  dense 
throughout)  in  comparison  with  its  superficial  area.  And,  taking 
L  to  represent  any  linear  dimension,  we  may  write  the  general 
equations  in  the  form 

Soz  L\  F  oc  L^ 

or  iS  =  kL\  and   V  =  k'L\ 

where  k,  k',  are  "factors  of  proportion," 

V  V       k 

and  ^  cc  L,      or      —  =  j-,  L  =  KL. 

o  ok 

So,  in  Lilliput,  "His  Majesty's  Ministers,  finding  that  Gulhver's 
stature  exceeded  theirs  in  the  proportion  of  twelve  to  one,  concluded 
from  the  similarity  of  their  bodies  that  his  must  contain  at  least 
1728  [or  12^]  of  theirs,  and  must  needs  be  rationed  accordingly*." 

*  Likewise  Gulliver  had  a  whole  Lilliputian  hogshead  for  his  half-pint  of  wine: 
in  the  due  proportion  of  1728  half-pints,  or  108  gallons,  equal  to  one  pipe  or 


CH.  II]  OF  DIMENSIONS  23 

From  these  elementary  principles  a  great  many  consequences 
follow,  all  more  or  less  interesting,  and  some  of  them  of  great 
importance.  In  the  first  place,  though  growth  in  length  (let  us  say) 
and  growth  in  volume  (which  is  usually  tantamount  to  mass  or 
weight)  are  parts  of  one  and  the  same  process  or  phenomenon,  the 
one  attracts  our  attention  by  its  increase  very  much  more  than  the 
other.  For  instance  a  fish,  in  doubhng  its  length,  multiphes  its 
weight  no  less  than  eight  times;  and  it  all  but  doubles  its  weight  in 
growing  from  four  inches  long  to  five. 

In  the  second  place,  we  see  that  an  understanding  of  the  correla- 
tion between  length  and  weight  in  any  particular  species  of  animal, 
in  other  words  a  determination  of  k  in  the  formula  W  =  k.L^, 
enables  us  at  any  time  to  translate  the  one  magnitude  into  the  other, 
and  (so  to  speak)  to  weigh  the  animal  with  a  measuring-rod;  this, 
however,  being  always  subject  to  the  condition  that  the  animal  shall 
in  no  way  have  altered  its  form,  nor  its  specific  gravity.  That  its 
specific  gravity  or  density  should  materially  or  rapidly  alter  is  not 
very  likely;  but  as  long  as  growth  lasts  changes  of  form,  even 
though  inappreciable  to  the  eye,  are  apt  and  hkely  to  occur.  Now 
weighing  is  a  far  easier  and  far  more  accurate  operation  than 
measuring;  and  the  measurements  which  would  reveal  slight  and 
otherwise  imperceptible  changes  in  the  form  of  a  fish — slight  relative 
differences  between  length,  breadth  and  depth,  for  instance — would 
need  to  be  very  dehcate  indeed.  But  if  we  can  make  fairly  accurate 
determinations  of  the  length,  which  is  much  the  easiest  linear 
dimension  to  measure,  and  correlate  it  with  the  weight,  then  the 
value  of  k,  whether  it  varies  or  remains  constant,  will  tell  us  at  once 
whether  there  has  or  has  not  been  a  tendency  to  alteration  in  the 
general  form,  or,  in  other  words,  a  difference  in  the  rates  of  growth 
in  different  directions.  To  this  subject  we  shall  return,  when  we 
come  to  consider  more  particularly  the  phenomenon  of  rate  of  growth. 

double-hogshead.  But  Gilbert  White  of  Selborne  could  not  see  what  was  plain 
to  the  Lilliputians;  for  finding  that  a  certain  little  long-legged  bird,  the  stilt, 
weighed  4J  oz.  and  had  legs  8  in.  long,  he  thought  that  a  flamingo,  weighing  4  lbs., 
should  have  legs  10  ft.  long,  to  be  in  the  same  proportion  as  the  stilt's.  But 
it  is  obvious  to  us  that,  as  the  weights  of  the  two  birds  are  as  1  :  15,  so  the  legs 
(or  other  linear  dimensions)  should  be  as  the  cube-roots  of  these  numbers,  or 
nearly  as  1  :  2^.  And  on  this  scale  the  flamingo's  legs  should  be,  as  they  actually 
are,  about  20  in.  long. 


24  ON  MAGNITUDE  [ch. 

We  are  accustomed  to  think  of  magnitude  as  a  purely  relative 
matter.  We  call  a  thing  big  or  little  with  reference  to  what  it  is 
wont  to  be,  as  when  we  speak  of  a  small  elephant  or  a  large  rat ;  and 
we  are  apt  accordingly  to  suppose  that  size  makes  no  other  or  more 
essential  difference,  and  that  Lilliput  and  Brobdingnag*  are  all 
alike,  according  as  we  look  at  them  through  one  end  of  the  glass 
or  the  other.  Gulliver  himself  declared,  in  Brobdingnag,  that 
"undoubtedly  philosophers  are  in  the  right  when  they  tell  us  that 
nothing  is  great  and  little  otherwise  than  by  comparison":  and 
Oliver  Heaviside  used  to  say,  in  like  manner,  that  there  is  no 
absolute  scale  of  size  in  the  Universe,  for  it  is  boundless  towards 
the  great  and  also  boundless  towards  the  small.  It  is  of  the  very 
essence  of  the  Newtonian  philosophy  that  we  should  be  able  to 
extend  our  concepts  and  deductions  from  the  one  extreme  of  magni- 
tude to  the  other;  and  Sir  John  Herschel  said  that  "the  student 
must  lay  his  account  to  finding  the  distinction  of  great  and  little 
altogether  annihilated  in  nature." 

All  this  is  true  of  number,  and  of  relative  magnitude.  The  Universe 
has  its  endless  gamut  of  great  and  small,  of  near  and  far,  of  many 
and  few.  Nevertheless,  in  physical  science  the  scale  of  absolute 
magnitude  becomes  a  very  real  and  important  thing;  and  a  new 
and  deeper  interest  arises  out  of  the  changing  ratio  of  dimensions 
when  we  come  to  consider  the  inevitable  changes  of  physical  rela- 
tions with  which  it  is  bound  up.  The  effect  of  scale  depends  not  on 
a  thing  in  itself,  but  in  relation  to  its  whole  environment  or  milieu ; 
it  is  in  conformity  with  the  thing's  "place  in  Nature,"  its  field  of 
action  and  reaction  in  the  Universe.  Everywhere  Nature  works 
true  to  scale,  and  everything  has  its  proper  size  accordingly.  Men 
and  trees,  birds  and  fishes,  stars  and  star-systems,  have  their 
appropriate  dimensions,  and  their  more  or  less  narrow  range  of 
absolute  magnitudes.  The  scale  of  human  observation  and  ex- 
perience lies  within  the  narrow  bounds  of  inches,  feet  or  miles,  all 
measured  in  terms  drawn  from  our  own  selves  or  our  own  doings. 
Scales  which  include  light-years,  parsecs.  Angstrom  units,  or  atomic 

*  Swift  paid  close  attention  to  the  arithmetic  of  magnitude,  but  none  to  its 
physical  aspect.  See  De  Morgan,  on  Lilliput,  in  N.  and  Q.  (2),  vi,  pp.  123-125, 
1858.  On  relative  magnitude  see  also  Berkeley,  in  his  Essay  towards  a  New  Theory 
of  Visio7i,  1709. 


II]  THE  EFFECT  OF  SCALE  25 

and  sub-atomic  magnitudes,  belong  to  other  orders  of  things  and 
other  principles  of  cognition. 

A  common  effect  of  scale  is  due  to  the  fact  that,  of  the  physical 
forces,  some  act  either  directly  at  the  surface  of  a  body,  or  otherwise 
in  proportion  to  its  surface  or  area;  while  others,  and  above  all 
gravity,  act  on  all  particles,  internal  and  external  alike,  and  exert 
a  force  which  is  proportional  to  the  mass,  and  so  usually  to  the 
volume  of  the, body. 

A  simple  case  is  that  of  two  similar  weights  hung  by  two  similar 
wires.  The  forces  exerted  by  the  weights  are  proportional  to  their 
masses,  and  these  to  their  volumes,  and  so  to  the  cubes  of  the 
several  Hnear  dimensions,  including  the  diameters  of  the  wires. 
But  the  areas  of  cross-section  of  the  wires  are  as  the  squares  of  the 
said  linear  dimensions;  therefore  the  stresses  in  the  wires  'per  unit 
area  are  not  identical,  but  increase  in  the  ratio  of  the  linear  dimen- 
sions, and  the  larger  the  structure  the  more  severe  the  strain  becomes : 

Force        l^ 

A^  ^  r^  ^  ^' 

and  the  less  the  wires  are  capable  of  supporting  it. 

In  short,  it  often  happens  that  of  the  forces  in  action  in  a  system 
some  vary  as  one  power  and  some  as  another,  of  the  masses,  distances 
or  other  magnitudes  involved;  the  "dimensions"  remain  the  same 
in  our  equations  of  equilibrium,  but  the  relative  values  alter  with 
the  scale.  This  is  known  as  the  "Principle  of  Similitude,"  or  of 
dynamical  similarity,  and  it  and  its  consequences  are  of  great 
importance.  In  a  handful  of  matter  cohesion,  capillarity,  chemical 
affinity,  electric  charge  are  all  potent;  across  the  solar  system 
gravitation*  rules  supreme;  in  the  mysterious  region  of  the  nebulae, 
it  may  haply  be  that  gravitation  grows  negligible  again. 

To  come  back  to  homelier  things,  the  strength  of  an  iron  girder 
obviously  varies  with  the  cross-section  of  its  members,  and  each 
cross-section  varies  as  the  square  of  a  linear  dimension;  but  the 
weight  of  the  whole  structure  varies  as  the  cube  of  its  linear  dimen- 

*  In  the  early  days  of  the  theory  of  gravitation,  it  was  deemed  especially 
remarkable  that  the  action  of  gravity  "is  proportional  to  the  quantity  of  solid 
matter  in  bodies,  and  not  to  their  surfaces  as  is  usual  in  mechanical  causes;  this 
power,  therefore,  seems  to  surpass  mere  mechanism"  (Colin  Maclaurin,  on  Sir 
Isaac  Newton's  Philosophical  Discoveries,  iv,  9). 


26  ON  MAGNITUDE  [ch. 

sions.  It  follows  at  once  that,  if  we  build  two  bridges  geometrically 
similar,  the  larger  is  the  weaker  of  the  two*,  and  is  so  in  the  ratio 
of  their  linear  dimensions.  It  was  elementary  engineering  experience 
such  as  this  that  led  Herbert  Spencer  to  apply  the  principle  of 
simihtude  to  biologyf. 

But  here,  before  w^e  go  further,  let  us  take  careful  note  that 
increased  weakness  is  no  necessary  concomitant  of  increasing  size. 
There  are  exceptions  to  the  rule,  in  those  exceptional  cases  where  we 
have  to  deal  only  with  forces  which  vary  merely  with  the  area  on 
which  they  impinge.  -  If  in  a  big  and  a  httle  ship  two  similar  masts 
carry  two  similar  sails,  the  two  sails  will  be  similarly  strained,  and 
equally  stressed  at  homologous  places,  and  alike  suitable  for  resisting 
the  force  of  the  same  wind.  Two  similar  umbrellas,  however 
differing  in  size,  will  serve  ahke  in  the  same  weather;  and  the 
expanse  (though  not  the  leverage)  of  a  bird's  wing  may  be  enlarged 
with  little  alteration. 

The  principle  of  similitude  had  been  admirably  apphed  in  a  few 
clear  instances  by  Lesage J,  a  celebrated  eighteenth-century  physician, 
in  an  unfinished  and  unpublished  work.  Lesage  argued,  for  example, 
that  the  larger  ratio  of  surface  to  mass  in  a  small  animal  would  lead 
to  excessive  transpiration,  were  the  skin  as  "porous"  as  our  own; 
and  that  we  may  thus  account  for  the  hardened  or  thickened  skins 
of  insects  and  many  other  small  terrestrial  animals.  Again,  since 
the  weight  of  a  fruit  increases  as  the  cube  of  its  linear  dimensions, 
while  the  strength  of  the  stalk  increases  as  the  square,  it  follows 
that  the  stalk  must  needs  grow  out  of  apparent  due  proportion  to 
the  fruit:    or,  alternatively,  that  tall  trees  should  not  bear  large 

*  The  subject  is  treated  from  the  engineer's  point  of  view  by  Prof.  James 
Thomson,  Comparison  of  similar  structures  as  to  elasticity,  strength  and  stability, 
Coll.  Papers,  1912,  pp.  361-372,  and  Trans.  Inst.  Engineers,  Scotland,  1876;  also 
by  Prof.  A.  Barr,  ibid.  1899.  See  also  Rayleigh,  Nature,  April  22,  1915;  Sir  G. 
Greenhill,  On  mechanical  similitude,  Math.  Gaz.  March  1916,  Coll.  Works,  vi, 
p.  300.  For  a  mathematical  account,  sec  (e.g.)  P.  VV.  Bridgeman,  Dimensional 
Analysis  (2nd  ed.),  1931,  or  F.  W.  Lanchester,  The  Theory  of  Dimensions,  1936. 

t  Herbert  Spencer,  The  form  of  the  earth,  etc.,  Phil.  Mag.  xxx,  pp.  194-6, 
1847;   also  Principles  of  Biology,  pt.  ii,  p.  123  seq.,  1864. 

I  See  Pierre  Prevost,  Notices  de  la  vie  et  des  ecrits  de  Lesage,  1805.  George 
Louis  Lesage,  born  at  Geneva  in  1724,  devoted  sixty-three  years  of  a  life  of  eighty 
to  a  mechanical  theory  of  gravitation;  see  W.  Thomson  (Lord  Kelvin),  On  the 
ultramundane  corpuscles  of  Lesage,  Proc.  E.S.E.  vii,  pp.  577-589,  1872;  Phil.  Mag. 
XLV,  pp.  321-345,  1873;  and  Clerk  Maxwell,  art.  "Atom,"  Encyd.  Brit.  (9),  p.' 46. 


II]  THE  PRINCIPLE  OF  SIMILITUDE  27 

fruit  on  slender  branches,  and  that  melons  and  pumpkins  must  lie 
upon  the  ground.  And  yet  again,  that  in  quadrupeds  a  large  head 
must  be  supported  on  a  neck  which  is  either  excessively  thick  and 
strong  like  a  bull's,  or  very  short  like  an  elephant's*. 

But  it  was  Gahleo  who,  wellnigh  three  hundred  years  ago,  had 
first  laid  down  this  general  principle  of  simiUtude;  and  he  did  so 
with  the  utmost  possible  clearness,  and  with  a  great  wealth  of  illustra- 
tion drawn  from  structures  living  and  deadf.  He  said  that  if  we 
tried  building  ships,  palaces  or  temples  of  enormous  size,  yards, 
beams  and  bolts  would  cease  to  hold  together;  nor  can  Nature 
grow  a  tree  nor  construct  an  animal  beyond  a  certain  size,  while 
retaining  the  proportions  and  employing  the  materials  which  suffice 
in  the  case  of  a  smaller  structure  J.  The  thing  will  fall  to  pieces  of 
its  own  weight  unless  we  either  change  its  relative  proportions,  which 
will  at  length  cause  it  to  become  clumsy,  monstrous  and  inefficient, 
or  else  we  must  find  new  material,  harder  and  stronger  than  was 
used  before.  Both  processes  are  famihar  to  us  in  Nature  and  in 
art,  and  practical  apphcations,  undreamed  of  by  Gahleo,  meet  us  at 
every  turn  in  this  modern  age  of  cement  and  steel  §. 

Again,  as  Galileo  was  also  careful  to  explain,  besides  the  questions 
of  pure  stress  and  strain,  of  the  strength  of  muscles  to  hft  an 
increasing  weight  or  of  bones  to  resist  its  crushing  stress,  we  have 
the  important  question  of  bending  ynornents.  This  enters,  more  or 
less,  into  our  whole  range  of  problems ;  it  aifects  the  whole  form  of 
the  skeleton,  and  sets  a  limit  to  the  height  of  a  tall  tree||. 

*  Cf.  W.  Walton,  On  the  debility  of  large  animals  and  trees,  Quart.  Journ. 
of  Math.  IX,  pp.  179-184,  1868;  also  L.  J.  Henderson,  On  volume  in  Biology, 
Proc.  Amer.  Acad.  Sci.  ii,  pp.  654-658,  1916;   etc. 

t  Discorsi  e  Dimostrazioni  matematiche,  intorno  a  due  nuove  scienze  attenenti 
alia  Mecanica  ed  ai  Muovimenti  Locali:  appresso  gli  Elzevirii,  1638;  Opere, 
ed.  Favaro,  viir,  p.  169  seq.     Transl.  by  Henry  Crew  and  A.  de  Salvio,  1914,  p.  130. 

X  So  Werner  remarked  that  Michael  Angelo  and  Bramanti  could  not  have  built 
of  gypsum  at  Paris  on  the  scale  they  built  of  travertin  at  Rome. 

§  The  Chrysler  and  Empire  State  Buildings,  the  latter  1048  ft.  high  to  the  foot 
of  its  200  ft.  "mooring  mast,"  are  the  last  word,  at  present,  in  this  brobdingnagian 
architecture. 

II  It  was  Euler  and  Lagrange  who  first  shewed  (about  1776-1778)  that  a  column 
of  a  certain  height  would  merely  be  compressed,  but  one  of  a  greater  height  would 
be  bent  by  its  own  weight.  See  Euler,  De  altitudine  columnarum  etc..  Acta  Acad. 
Sci.  Imp.  Petropol.  1778,  pp.  163-193;  G,  Greenhill,  Determination  of  the  greatest 
height  to  which  a  tree  of  given  proportions  can  grow,  Cambr.  Phil.  Soc.  Proc.  rv, 
p.  65,  1881,  and  Chree,  ibid,  vu,  1892. 


28  ON  MAGNITUDE  [ch. 

We  learn  in  elementary  mechanics  the  simple  case  of  two  similar 
beams,  supported  at  both  ends  and  carrying  no  other  weight  than 
their  own.  Within  the  limits  of  their  elasticity  they  tend  to  be 
deflected,  or  to  sag  downwards,  in  proportion  to  the  squares  of  their 
linear  dimensions ;  if  a  match-stick  be  two  inches  long  and  a  similar 
beam  six  feet  (or  36  times  as  long),  the  latter  will  sag  under  its  own 
weight  thirteen  hundred  times  as  much  as  the  other.  To  counteract 
this  tendency,  as  the  size  of  an  animal  increases,  the  limbs  tend  to 
become  thicker  and  shorter  and  the  whole  skeleton  bulkier  and 
heavier;  bones  make  up  some  8  per  cent,  of  the  body  of  mouse  or  wren, 
13  or  14  per  cent,  of  goose  or  dog,  and  17  or  18  per  cent,  of  the  body 
of  a  man.  Elephant  and  hippopotamus  have  grown  clumsy  as  well  as 
big,  and  the  elk  is  of  necessity  less  graceful  than  the  gazelle.  It  is  of 
high  interest,  on  the  other  hand,  to  observe  how  little  the  skeletal 
proportions  differ  in  a  httle  porpoise  and  a  great  whale,  even  in  the 
limbs  and  hmb-bones ;  for  the  whole  influence  of  gravity  has  become 
neghgible,  or  nearly  so,  in  both  of  these. 

In  ifhe  problem  of  the  tall  tree  we  have  to  determine  the  point 
at  which  the  tree  will  begin  to  bend  under  its  own  weight  if  it  be 
ever  so  little  displaced  from  the  perpendicular*.  In  such  an 
investigation  we  have  to  make  certain  assumptions — for  instance 
that  the  trunk  tapers  uniformly,  and  that  the  sectional  area  of  the 
branches  varies  according  to  some  definite  law,  or  (as  Ruskin 
assumed)  tends  to  be  constant  in  any  horizontal  plane;  and  the 
mathematical  treatment  is  apt  to  be  somewhat  difficult.  But 
Greenhill  shewed,  on  such  assumptions  as  the  above,  that  a  certain 
British  Columbian  pine-tree,  of  which  the  Kew  flag-staff,  which  is 
221  ft.  high  and  21  inches  in  diameter  at  the  base,  was  made,  could 
not  possibly,  by  theory,  have  grown  to  more  than  about  300  ft.  It 
is  very  curious  that  Galileo  had  suggested  precisely  the  same  height 
(ducento  braccie  alta)  as  the  utmost  limit  of  the  altitude  of  a  tree. 
In  general,  as  Greenhill  shewed,  the  diameter  of  a  tall  homogeneous 
body  must  increase  as  the  power  3/2  of  its  height,  which  accounts 
for  the  slender  proportions  of  young  trees  compared  with  the  squat 

*  In  like  manner  the  wheat-straw  bends  over  under  the  weight  of  the  loaded 
ear,  and  the  cat's  tail  bends  over  when  held  erect — not  because,  they  "possess 
flexibility,"  but  because  they  outstrip  the  dimensions  within  which  stable  equi- 
librium is  possible  in  a  vertical  position.  The  kitten's  tail,  on  the  other  hand, 
stands  up  spiky  and  straight. 


II]  OF  THE  HEIGHT  OF  A  TREE  29 

or  stunted  appearance  of  old  and  large  ones*.  In  short,  as  Goethe 
says  in  Dicktung  und  Wahrheit,  "Es  ist  dafiir  gesorgt  dass  die  Baume 
nicht  in  den  Himmel  wachsen." 

But  the  tapering  pine-tree  is  but  a  special  case  of  a  wider  problem. 
The  oak  does  not  grow  so  tall  as  the  pine-tree,  but  it  carries  a  heavier 
load,  and  its  boll,  broad-based  upon  its  spreading  roots,  shews  a 
different  contour.  Smeaton  took  it  for  the  pattern  of  his  Hghthouse, 
and  Eiffel  built  his  great  tree  of  steel,  a  thousand  feet  high,  to  a 
similar  but  a  stricter  plan.  Here  the  profile  of  tower  or  tree  follows, 
or  tends  to  follow,  a  logarithmic  curve,  giving  equal  strength 
throughout,  according  to  a  principle  which  we  shall  have  occasion 
to  discuss  later  on,  when  we  come  to  treat  of  form  and  mechanical 
efficiency  in  the  skeletons  of  animals.  In  the  tree,  moreover, 
anchoring  roots  form  powerful  wind-struts,  and  are  most  de- 
veloped opposite  to  the  direction  of  the  prevailing  winds;  for  the 
lifetime  of  a  tree  is  affected  by  the  frequency  of  storms,  and  its 
strength  is  related  to  the  wind-pressure  which  it  must  needs  with- 
standf. 

Among  animals  we  see,  without  the  help  of  mathematics  or  of 
physics,  how  small  birds  and  beasts  are  quick  and  agile,  how  slower 
and  sedater  movements  come  with  larger  size,  and  how  exaggerated 
bulk  brings  with  it  a  certain  clumsiness,  a  certain  inefficiency,  an 
element  of  risk  and  hazard,  a  preponderance  of  disadvantage.  The 
case  was  well  put  by  Owen,  in  a  passage  which  has  an  interest  of 
its  own  as  a  premonition,  somewhat  Hke  De.  Candolle's,  of  the 
"struggle  for  existence."  Owen  wrote  as  follows  J:  "  In  proportion 
to  the  bulk  of  a  species  is  the  difficulty  of  the  contest  which,  as  a 
living  organised  whole,  the  individual  of  each  species  has  to  maintain 
against  the  surrounding  agencies  that  are  ever  tending  to  dissolve 
the  vital  bond,  and  subjugate  the  Hving  matter  to  the  ordinary 
chemical  and  physical  forces.  Any  changes,  therefore,  in  such 
external  conditions  as  a  species  may  have  been  original|y  adapted 

*  The  stem  of  the  giant  bamboo  may  attain  a  height  of  60  metres  while  not  more 
than  about  40  cm.  in  diameter  near  its  base,  which  dimensions  fall  not  far  short 
of  the  theoretical  limits;   A.  J.  Ewart,  Phil.  Trans,  cxcviii,  p.  71,  1906. 

t  Cf.  {int.  al.)  T.  Fetch,  On  buttress  tree-roots,  Ann.  R.  Bot.  Garden,  Peradenyia, 
XI,  pp.  277-285,  1930.  Also  au  interesting  paper  by  James  Macdonald,  on  The 
form  of  coniferous  trees.  Forestry,  vi,  1  and  2,  1931/2. 

X  Trans.  Zool.  Soc.  iv,  p.  27,  1850. 


30  ON  MAGNITUDE  [ch. 

to  exist  in,  will  militate  against  that  existence  in  a  degree 
proportionate,  perhaps  in  a  geometrical  ratio,  to  the  bulk  of  the 
species.  If  a  dry  season  be  greatly  prolonged,  the  large  mammal 
will  suffer  from  the  drought  sooner  than  the  small  one;  if  any 
alteration  of  climate  affect  the  quantity  of  vegetable  food,  the 
bulky  Herbivore  will  be  the  first  to  feel  the  effects  of  stinted 
nourishment." 

But  the  principle  of  GaHleo  carries  us  further  and  along  more 
certain  lines.  The  strength  of  a  muscle,  like  that  of  a  rope  or 
girder,  varies  with  its  cross-section;  and  the  resistance  of  a  bone 
to  a  crushing  stress  varies,  again  hke  our  girder,  with  its  cross- 
section.  But  in  a  terrestrial  animal  the  weight  which  tends  to 
crush  its  limbs,  or  which  its  muscles  have  to  move,  varies  as  the 
cube  of  its  hnear  dimensions;  and  so,  to  the  possible  magnitude 
of  an  animal,  living  under  the  direct  action  of  gravity,  there  is  a 
definite  limit  set.  The  elephant,  in  the  dimensions  of  its  limb-bones, 
is  already  shewing  signs  of  a  tendency  to  disproportionate  thickness 
as  compared  with  the  smaller  mammals;  its  movements  are  in 
many  ways  hampered  and  its  agility  diminished:  it  is  already 
tending  towards  the  maximal  Hmit  of  size  which  the  physical  forces 
permit*.  The  spindleshanks  of  gnat  or  daddy-long-legs  have  their 
own  factor  of  safety,  conditional  on  the  creature's  exiguous  bulk 
and  weight;  for  after  their  own  fashion  even  these  small  creatures 
tend  towards  an  inevitable  limitation  of  their  natural  size.  But,  as 
Gahleo  also  saw,  if  the  animal  be  wholly  immersed  in  water  like  the 
whale,  or  if  it  be  partly  so,  as  was  probably  the  case  with  the  giant 
reptiles  of  the  mesozoic  age,  then  the  weight  is  counterpoised  to 
the  extent  of  an  equivalent  volume  of  water,  and  is  completely 
counterpoised  if  the  density  of  the  animal's  body,  with  the  included 
air,  be  identical  (as  a  whale's  very  nearly  is)  with  that  of  the  water 
around^.  Under  these  circumstances  there  is  no  longer  the  same 
physical  barrier  to  the  indefinite  growth  of  the  animal.  Indeed,  in  the 
case  of  the  aquatic  animal,  there  is,  as  Herbert  Spencer  pointed  out, 


*  Cf.  A.  Rauber,  Galileo  iiber  Knochenformen,  Morphol.  Jahrb.  vii,  p.  327,  1882. 

t  Cf.  W.  S.  Wall,  A  New  Sperm  Whale  etc.,  Sydney,  1851,  p.  64:  "As  for 
the  immense  size  of  Cetacea,  it  evidently  proceeds  from  their  buoyancy  in  the 
medium  in  which  they  live,  and  their  being  enabled  thus  to  counteract  the  force  of 

gravity." 


II]  OF  THE  SPEED  OF  A  SHIP  31 

a  distinct  advantage,  in  that  the  larger  it  grows  the  greater  is  its 
speed.  For  its  available  energy  depends  on  the  mass  of  its  muscles, 
while  its  motion  through  the  water  is  opposed,  not  by  gravity,  but 
by  "skin-friction,"  which  increases  only  as  the  square  of  the  linear 
dimensions*:  whence,  other  things  being  equal,  the  bigger  the  ship 
or  the  bigger  the  fish  the  faster  it  tends  to  go,  but  only  in  the  ratio 
of  the  square  root  of  the  increasing  length.  For  the  velocity  (F) 
which  the  fish  attains  depends  on  the  work  (W)  it  can  do  and  the 
resistance  (R)  it  must  overcome.  Now  we^  have  seen  that  the 
dimensions  of  W  are  l^,  and  of  R  are  l'^ ;  and  by  elementary  mechanics 

WocRV^  or   F^oc^. 

73 

Therefore  V^oCr-=l,  and   F  oc  Vl. 

This  is  what  is  known  as  Fronde's  Law,  of  the  correspondence 
of  speeds — a  simple  and  most  elegant  instance  of  "dimensional 
theory!." 

But  there  is  often  another  side  to  these  questions,  which  makes 
them  too  complicated  to  answer  in  a  word.  For  instance,  the  work 
(per  stroke)  of  which  two  similar  engines  are  capable  should  vary  as 
the  cubes  of  their  linear  dimensions,  for  it  varies  on  the  one  hand 
with  the  area  of  the  piston,  and  on  the  other  with  the  length  of  the 
stroke;  so  is  it  likewise  in  the  animal,  where  the  corresponding 
ratio  depends  on  the  cross-section  of  the  muscle,  and  on  the  distance 
through  which  it  contracts.  But  in  two  similar  engines,  the  available 
horse-power  varies  as  the  square  of  the  linear  dimensions,  and  not 
as  the  cube ;  and  this  for  the  reason  that  the  actual  energy  developed 
depends  on  the  heating-surface  of  the  boiler  J.     So  likewise  must 

*  We  are  neglecting  "drag"  or  "head-resistance,"  which,  increasing  as  the  cube 
of  the  speed,  is  a  formidable  obstacle  to  an  unstreamlined  body.  But  the  perfect 
streamlining  of  whale  or  fish  or  bird  lets  the  surrounding  air  or  water  behave  like 
a  perfect  fluid,  gives  rise  to  no  "surface  of  discontinuity,"  and  the  creature  passes 
through  it  without  recoil  or  turbulence.  Froude  reckoned  skin-friction,  or  surface- 
resistance,  as  equal  to  that  of  a  plane  as  long  as  the  vessel's  water-line,  and  of  area 
equal  to  that  of  the  wetted  surface  of  the  vessel. 

t  Though,  as  Lanchester  says,  the  great  designer  "was  not  hampered  by  a 
knowledge  of  the  theory  of  dimensions." 

X  The  analogy  is  not  a  very  strict  or  complete  one.  We  are  not  taking  account, 
for  instance,  of  the  thickness  of  the  boiler-plates. 


32  ON  MAGNITUDE  [ch. 

there  be  a  similar  tendency  among  animals  for  the  rate  of  supply 
o^  kinetic  energy  to  vary  with  the  surface  of  the  lung,  that  is  to  say 
(other  things  being  equal)  with  the  square  of  the  linear  dimensions 
of  the  animal ;  which  means  that,  caeteris  paribus,  the  small  animal 
is  stronger  (having  more  power  per  unit  weight)  than  a  large  one. 
We  may  of  course  (departing  from  the  condition  of  similarity)  increase 
the  heating-surface  of  the  boiler,  by  means  of  an  internal  system  of 
tubes,  without  increasing  its  outward  dimensions,  and  in  this  very 
way  Nature  increases  the  respiratory  surface  of  a  lung  by  a  complex 
system  of  branching  tubes  and  minute  air-cells ;  but  nevertheless  in 
two  similar  and  closely  related  animals,  as  also  in  two  steam-engines 
of  the  same  make,  the  law  is  bound  to  hold  that  the  rate  of  working 
tends  to  vary  with  the  square  of  the  linear  dimensions,  according  to 
Froude's  law  of  steamship  comparison.  In  the  case  of  a  very  large 
ship,  built  for  speed,  the  difficulty  is  got  over  by  increasing  the  size 
and  number  of  the  boilers,  till  the  ratio  between  boiler-room  and 
engine-room  is  far  beyond  what  is  required  in  an  ordinary  small 
vessel*;  but  though  we  find  lung-space  increased  among  animals 
where  greater  rate  of  working  is  required,  as  in  general  among  birds, 
I  do  not  know  that  it  can  be  shewn  to  increase,  as  in  the  "over- 
boilered"  ship,  with  the  size  of  the  animal,  and  in  a  ratio  which 
outstrips  that  of  the  other  bodily  dimensions.  If  it  be  the  case  then, 
that  the  working  mechanism  of  the  muscles  should  be  able  to  exert 
a  force  proportionate  to  the  cube  of  the  linear  bodily  dimensions, 

*  Let  L  be  the  length,  S  the  (wetted)  surface,  T  the  tonnage,  D  the  displacement 
(or  volume)  of  a  ship;  and  let  it  cross  the  Atlantic  at  a  speed  V.  Then,  in  com- 
paring two  ships,  similarly  constructed  but  of  different  magnitudes,  we  know  that 
L=V\  S=L^  =  V\  D  =  T  =  L^=V^;  also  B  (resistance)  =/Sf.  F^^  F«;  H  (horse- 
power) =  i2 .  F  =  F' ;  and  the  coal  (C)  necessary  for  the  voy a,ge— HjV  =  V^.  That 
is  to  say,  in  ordinary  engineering  language,  to  increase  the  speed  across  the  Atlantic 
by  1  per  cent,  the  ship's  length  must  be  increased  2  per  cent.,  her  tonnage  or 
displacement  6  per  cent.,  her  coal- consumption  also  6  per  cent.,  her  horse-power, 
and  therefore  her  boiler-capacity,  7  per  cent.  Her  bunkers,  accordingly,  keep 
pace  with  the  enlargement  of  the  ship,  but  her  boilers  tend  to  increase  out  of 
proportion  to  the  space  available.  Suppose  a  steamer  400  ft.  long,  of  2000  tons, 
2000  H.P.,  and  a  speed  of  14  knots.  The  corresponding  vessel  of  800  ft.  long  should 
develop  a  speed  of  20  knots  (I  :  2  ::  14^  :  20^),  her  tonnage  would  be  16,000,  her 
H.p.  25,000  or  thereby.  Such  a  vessel  would  probably  be  driven  by  four  propellers 
instead  of  one,  each  carrying  8000  h.p.  See  (int.  al.)  W.  J.  Millar,  On  the  most 
economical  speed  to  drive  a  steamer,  Proc.  Edin.  Math.  Soc.  vii,  pp.  27-29,  1889; 
Sir  James  R.  Napier,  On  the  most  profitable  speed  for  a  fully  laden  cargo  steamer 
for  a  given  voyage,  Proc.  Phil.  Soc,  Glasgow,  vi,  pp.  33-38,  1865. 


II]  OF  FROUDE'S  LAW  33 

while  the  respiratory  mechanism  can  only  supply  a  store  of  energy 
at  a  rate  proportional  to  the  square  of  the  said  dimensions,  the 
singular  result  ought  to  follow  that,  in  swimming  for  instance,  the 
larger  fish  ought  to  be  able  to  put  on  a  spurt  of  speed  far  in  excess 
of  the  smaller  one;  but  the  distance  travelled  by  the  year's  end 
should  be  very  much  ahke  for  both  of  them.  And  it  should  also 
follow  that  the  curve  of  fatigue  is  a  steeper  one,  and  the  staying 
power  less,  in  the  smaller  than  ixx  the  larger  individual.  This  is  the 
c^ase  in  long-distance  racing,  where  neither  draws  far  ahead  until 
the  big  winner  puts  on  his  big  spurt  at  the  end;  on  which  is  based 
an  aphorism  of  the  turf,  that  "a  good  big  'un  is  better  than  a  good 
httle  'un."  For  an  analogous  reason  wise  men  know  that  in  the 
'Varsity  boat-race  it  is  prudent  and  judicious  to  bet  on  the  heavier 
crew. 

Consider  again  the  dynamical  problem  of  the  movements  of  the 
body  and  the  hmbs.  The  work  done  (W)  in  moving  a  Hmb,  whose 
weight  is  p,  over  a  distance  s,  is  measured  hy  ps;  p  varies  as  the 
cube  of  the  hnear  dimensions,  and  s,  in  ordinary  locomotion,  varies 
as  the  linear  dimensions,  that  is  to  say  as  the  length  of  Hmb : 

Wocpsocl^  xl^lK 

But  the  work  done  is  limited  by  the  power  available,  and  this 
varies  as  the  mass  of  the  muscles,  or  as  l^;  and  under  this  hmitation 
neither  p  nor  s  increase  as  they  would  otherwise  tend  to  do.  The 
limbs  grow  shorter,  relatively,  as  the  animal  grows  bigger;  and 
spiders,  daddy-long-legs  and  such-hke  long-limbed  creatures  attain 
no  great  size. 

Let  us  consider  more  closely  the  actual  energies  of  the  body. 
A  hundred  years  ago,  in  Strasburg,  a  physiologist  and  a  mathema- 
tician were  studying  the  temperature  of  warm-blooded  animals*. 
The  heat  lost  must,  they  said,  be  proportional  to  the  surface  of  the 
animal :  and  the  gain  must  be  equal  to  the  loss,  since  the  temperature 
of  the  body  keeps  constant.  It  would  seem,  therefore,  that  the 
heat  lost  by  radiation  and  that  gained  by  oxidation  vary  both  alike, 
as  the  surface-area,  or  the  square  of  the  Hnear  dimensions,  of  the 
animal.     But  this  result  is  paradoxical;    for  whereas  the  heat  lost 

*  MM.  Rameaux  et  Sarrus,  Bull.  Acad.  R.  de  Me'decine,  in,  pp.   1094-1100, 
1838-39. 


34  ON  MAGNITUDE  [ch. 

may  well  vary  as  the  surface-area,  that  produced  by  oxidation 
ought  rather  to  vary  as  the  bulk  of  the  animal:  one  should  vary 
as  the  square  and  the  other  as  the  cube  of  the  Hnear  dimensions. 
Therefore  the  ratio  of  loss  to  gain,  hke  that  of  surface  to  volume, 
ought  to  increase  as  the  size  of  the  creature  diminishes.  Another 
physiologist,  Carl  Bergmann*,  took  the  case  a  step  further.  It  was  he, 
by  the  way,  whp  first  said  that  the  real  distinction  was  not  between 
warm-blooded  and  cold-blooded  animals,  but  between  those  of 
constant  and  those  of  variable  temperature:  and  who  coined  the 
terms  homoeothermic  and  poecilothermic  which  we  use  today.  He 
was  driven  to  the  conclusion  that  the  smaller  animal  does  produce 
more  heat  (per  unit  of  mass)  than  the  large  one,  in  order  to  keep 
pace  with  surface-loss;  and  that  this  extra  heat-production  means 
more  energy  spent,  more  food  consumed,  more  work  donef.  Sim- 
plified as  it  thus  was,  the  problem  still  perplexed  the  physiologists 
for  years  after.  The  tissues  of  one  mammal  are  much  like  those  of 
another.  We  can  hardly  imagine  the  muscles  of  a  small  mammal 
to  produce  more  heat  {caeteris  paribus)  than  those  of  a  large ;  and 
we  begin  to  wonder  whether  it  be  not  nervous  excitation,  rather  than 
quahty  of  muscular  tissue,  which  determines  the  rate  of  oxidation 
and  the  output  of  heat.  It  is  evident  in  certain  cases,  and  may  be 
a  general  rule,  that  the  smaller  animals  have  the  bigger  brains; 
"plus  I'animal  est  petit,"  says  M.  Charles  Richet,  "plus  il  a  des 
echanges  chimiques  actifs,  et  plus  son  cerveau  est  volumineuxj." 
That  the  smaller  animal  needs  more  food  is.  certain  and  obvious. 
The  amount  of  food  and  oxygen  consumed  by  a  small  flying  insect 
is  enormous;    and  bees  and  flies  and  hawkmoths  and  humming- 

*  Carl  Bergmann,  Verhaltnisse  der  Warmeokonomie  der  Tiere  zu  ihrer  Grosse, 
Gottinger  Studien,  i,  pp.  594-708,  1847 — a  very  original  paper. 

t  The  metabolic  activity  of  sundry  mammals,  per  24  hours,  has  been  estimated 
as  follows : 

Weight  (kilo.)       Calories  per  kilo. 
Guinea-pig  0-7  223 

Rabbit  2  58 

Man  70  33 

Horse  600  22 

Elephant  4000  13 

Whale  150000  circa  1-7 

J  Ch.  Richet,  Recherches  de  calorimetrie.  Arch,  de  Physiologie  (3),  vi,  pp.  237-291, 
450-497,  1885.  Cf.  also  an  interesting  historical  account  by  M.  Elie  le  Breton, 
Sur  la  notion  de  "masse  protoplasmique  active":  i.  Probl^mes  poses  par  la 
signification  de  la  loi  des  surfaces,  ibid.  1906,  p.  606. 


II]  OF  BERGMANN'S  LAW  35 

birds  live  on  nectar,  the  richest  and  most  concentrated  of  foods*. 
Man  consumes  a  fiftieth  part  of  his  own  weight  of  food  daily,  but 
a  mouse  will  eat  half  its  own  weight  in  a  day;  its  rate  of  Hving  is 
faster,  it  breeds  faster,  and  old  age  comes  to  it  much  sooner  than 
to  man.  A  warm-blooded  animal  much  smaller  than  a  mouse 
becomes  an  impossibihty;  it  could  neither  obtain  nor  yet  digest  the 
food  required  to  maintain  its  constant  temperature,  and  hence  no 
mammals  and  no  birds  are  as  small  as  the  smallest  frogs  or  fishes. 
The  disadvantage  of  small  size  is  all  the  greater  when  loss  of  heat 
is  accelerated  by  conduction  as  in  the  Arctic,  or  by  convection  as 
in  the  sea.  The  far  north  is  a  home  of  large  birds  but  not  of  small ; 
bears  but  not  mice  five  through  an  Arctic  winter;  the  'least  of  the 
dolphins  Hve  in  warm  waters,  and  there  are  no  small  mammals  in 
the  sea.     This  principle  is  sometimes  spoken  of  as  Bergmann's  Law. 

The  whole  subject  of  the  conservation  of  heat  and  the  maintenance 
of  an  all  but  constant  temperature  in  warm-blooded  animals  interests 
the  physicist  and  the  physiologist  ahke.  It  drew  Kelvin's  attention 
many  years  agof,  and  led  him  to  shew,  in  a  curious  paper,  how 
larger  bodies  are  kept  warm  by  clothing  while  smaller  are  only 
cooled  the  more.  If  a  current  be  passed  through  a  thin  wire,  of 
which  part  is  covered  and  part  is  bare,  the  thin  bare  part  may  glow 
with  heat,  while  convection-currents  streaming  round  the  covered 
part  cool  it  off  and  leave  it  in  darkness.  The  hairy  coat  of  very 
small  animals  is  apt  to  look  thin  and  meagre,  but  it  may  serve  them 
better  than  a  shaggier  covering. 

Leaving  aside  the  question  of  the  supply  of  energy,  and  keeping 
to  that  of  the  mechanical  efficiency  of  the  machine,  we  may  find 
endless  biological  illustrations  of  the  principle  of  simihtude.  All 
through  the  physiology  of  locomotion  we  meet  with  it  in  various 
ways :  as,  for  instance,  when  we  see  a  cockchafer  carry  a  plate  many 
times  its  own  weight  upon  its  back,  or  a  flea  jump  many  inches  high. 
''A  dog,"  says  Gahleo,  "could  probably  carry  two  or  three  such 
dogs  upon  his  back;  but  I  believe  that  a  horse  could  not  carry 
even  one  of  his  own  size." 

♦  Cf.  R.  A.  Davies  and  G.  Fraenkel,  The  oxygen- consumption  of  flies  during 
flight,  Jl.  Exp.  Biol.  XVII,  pp.  402-407,  1940. 

t  W.  Thomson,  On  the  efficiency  of  clothing  for  maintaining  temperature, 
Nature,  xxix,  p.  567,  1884. 


36  ON  MAGNITUDE  [ch. 

Such  problems  were  admirably  treated  by  Galileo  and  Borelli, 
but  many  writers  remained  ignorant  of  their  work.  Linnaeus 
remarked  that  if  an  elephant  were  as  strong  in  proportion  as  a 
stag-beetle,  it  would  be  able  to  pull  up  rocks  and  level  mountains; 
and  Kirby  and  Spence  have  a  well-known  passage  directed  to  shew 
that  such  powers  as  have  been  conferred  upon  the  insect  have  been 
withheld  from  the  higher  animals,  for  the  reason  that  had  these 
latter  been  endued  therewith  they  would  have  "caused  the  early 
desolation  of  the  world*." 

Such  problems  as  that  presented  by  the  flea's  jumping  powersf , 
though  essentially  physiological  in  their  nature,  have  their  interest 
for  us  here:  because  a  steady,  progressive  diminution  of  activity 
with  increasing  size  would  tend  to  set  Hmits  to  the  possible  growth 
in  magnitude  of  an  animal  just  as  surely  as  those  factors  which 
tend  to  break  and  crush  the  living  fabric  under  its  own  weight.  In 
the  case  of  a  leap,  we  have  to  do  rather  with  a  sudden  impulse  than 
with  a  continued  strain,  and  this  impulse  should  be  measured  in 
terms  of  the  velocity  imparted.  The  velocity  is  proportional  to 
the  impulse  (x),  and  inversely  proportional  to  the  mass  (M)  moved : 
V  =  xjM.  But,  according  to  what  we  still  speak  of  as  "Borelh's 
law,"  the  impulse  (i.e.  the  work  of  the  impulse)  is  proportional  to 
the  volume  of  the  muscle  by  which  it  is  produced  {,  that  is  to  say 
(in  similarly  constructed  animals)  to  th^  mass  of  the  whole  body; 
for  the  impulse  is  proportional  on  the  one  hand  to  the  cross-section 
of  the  muscle,  and  on  the  other  to  the  distance  through  which  it 

*  Introduction  to  Entomology,  ii,  p.  190,  1826.  Kirby  and  Spence,  like  many  less 
learned  authors,  are  fond  of  popular  illustrations  of  the  "wonders  of  Nature," 
to  the  neglect  of  dynamical  principles.  They  suggest  that  if  a  white  ant  were  as 
big  as  a  man,  its  tunnels  would  be  "magnificent  cylinders  of  more  than  three 
hundred  feet  in  diameter";  and  that  if  a  certain  noisy  Brazilian  insect  were  as 
big  as  a  man,  its  voice  would  be  heard  all  the  world  over,  "so  that  Stentor  becomes 
a  mute  when  compared  with  these  insects!"  It  is  an  easy  consequence  of 
anthropomorphism,  and  hence  a  common  characteristic  of  fairy-tales,  to  neglect 
the  dynamical  and  dwell  on  the  geometrical  aspect  of  similarity. 

•f  The  flea  is  a  very  clever  jumper;  he  jumps  backwards,  is  stream-lined  ac- 
cordingly, and  alights  on  his  two  long  hind-legs.  Cf.  G.  I.  Watson,  in  Nature, 
21  May  1938. 

X  That  is  to  say,  the  available  energy  of  muscle,  in  ft.-lbs.  per  lb.  of  muscle,  is 
the  same  for  all  animals:  a  postulate  which  requires  considerable  qualification 
when  we  come  to  compare  very  different  kinds  of  muscle,  such  as  the  insect's  and 
the  mammal's. 


II]  OF  BORELLI'S  LAW  37 

contracts.  It  follows  from  this  that  the  velpcity  is  constant,  what- 
ever be  the  size  of  the  animal. 

Putting  it  still  more  simply,  the  work  done  in  leaping  is  propor- 
tional to  the  mass  and  to  the  height  to  which  it  is  raised,  W  oc  mH. 
But  the  muscular  power  available  for  this  work  is  proportional  to 
the  mass  of  muscle,  or  (in  similarly  constructed  animals)  to  the  mass 
of  the  animal,  W  oc  m.  It  follows  that  H  is,  or  tends  to  be,  a 
constant.  In  other  words,  all  animals,  provided  always  that  they 
are  similarly  fashioned,  with  their  various  levers  in  Hke  proportion, 
ought  to  jump  not  to  the  same  relative  but  to  the  same  actual 
height*.  The  grasshopper  seems  to  be  as  well  planned  for  jumping 
as  the  flea,  and  the  actual  heights  to  which  they  jump  are  much  of 
a  muchness;  but  the  flea's  jump  is  about  200  times  its  own  height, 
the  grasshopper's  at  most  20-30  times;  and  neither  flea  nor  grass- 
hopper is  a  better  but  rather  a  worse  jumper  than  a  horse  or  a  manf . 

As  a  matter  of  fact,  Borelli  is  careful  to  point  out  that  in  the  act 
of  leaping  the  impulse  is  not  actually  instantaneous,  like  the  blow 
of  a  hammer,  but  takes  some  httle  time,  during  which  the  levers 
are  being  extended  by  which  the  animal  is  being  propelled  forwards ; 
and  this  interval  of  time  will  be  longer  in  the  case  of  the  longer 
levers  of  the  larger  animal.  To  some  extent,  then,  this  principle 
acts  as  a  corrective  to  the  more  general  one,  and  tends  to  leave  a 
certain  balance  of  advantage  in  regard  to  leaping  power  on  the  side 
of  the  larger  animal  J.  But  on  the  other  hand,  the  question  of 
strength  of  materials  comes  in  once  more,  and  the  factors  of  stress 
and  strain  and  bending  moment  make  it  more  and  more  difficult 
for  nature  to  endow  the  larger  animal  with  the  length  of  lever  with 
which  she  has  provided  the  grasshopper  or  the  flea.  To  Kirby  and 
Spence  it  seemed  that  "This  wonderful  strength  of  insects  is 
doubtless  the  result  of  something  peculiar  in  the  structure  and 
arrangement  of  their  muscles,  and  principally  their  extraordinary 

*  Borelli,  Prop.  CLxxvn.  Animalia  minora  et  minus  ponderosa  majores  saltus 
efficiunt  respectu  sui  corporis,  si  caetera  fuerint  paria. 

t  The  high  jump  is  nowadays  a  highly  skilled  performance.  For  the  jumper 
contrives  that  his  centre  of  gravity  goes  under  the  bar,  while  his  body,  bit  by  bit, 
goes  over  it. 

J  See  also  {int.  al.),  John  Bernoulli,  De  Motu  Musculorum,  Basil.,  1694; 
Chabry,  Mecanisme  du  saut,  J.  de  VAnat.  et  de  la  Physiol,  xix,  1883;*  Sur  la 
longueur  des  membres  des  animaux  sauteurs,  ibid,  xxi,  p.  356,  1885;  Le  Hello, 
De  Taction  des  organes  locomoteurs,  etc.,  ibid,  xxix,  pp.  65-93,  1893;  etc. 


38  ON  MAGNITUDE  [ch. 

power  of  contraction."  This  hypothesis,  which  is  so  easily  seen  on 
physical  grounds  to  be  unnecessary,  has  been  amply  disproved  in 
a  series  of  excellent  papers  by  Felix  Plateau*. 

From  the  impulse  of  the  preceding  case  we  may  pass  to  the  momentum 
created  (or  destroyed)  under  similar  circumstances  by  a  given  force  acting 
for  a  given  time :  mv  =  Ft. 

We  know  that    .  m  oc  P,    and    t  =  Ifv, 

so  that  lH  =  Fllv,    or    v^  =  F/l\ 

But  whatsoever  force  be  available,  the  animal  may  only  exert  so  much  of 
it  as  is  in  proportion  to  the  strength  of  his  own  limbs,  that  is  to  say  to  the 
cross-section  of  bone,  sinew  and  muscle ;  and  all  of  these  cross-sections  are 
proportional  to  P,  the  square  of  the  linear  dimensions.  The  maximal  force, 
-f'max,  which  the  animal  dare  exert  is  proportional,  then,  to  l^;  therefore 

Fraa^JP  =  coustant. 

And  the  maximal  speed  which  the  animal  can  safely  reach,  namely 
F'max  =  ^max/^,  IS  also  constaut,  or  independent  {ceteris  paribus)  of  the  dimensions 
of  the  animal. 

A  spurt  or  effort  may  be  well  within  the  capacity  of  the  animal  but  far 
beyond  the  margin  of  safety,  as  trainer  and  athlete  well  know.  This  margin 
is  a  narrow  one,  whether  for  athlete  or  racehorse;  both  run  a  constant  risk 
of  overstrain,  under  which  they  may  "pull"  a  muscle,  lacerate  a  tendon,  or 
even  "break  down"  a  bonet. 

It  is  fortunate  for  their  safety  that  animals  do  not  jump  to  heights  pro- 
portional to  their  own.  For  conceive  an  animal  (of  mass  m)  to  jump  to 
a  certain  altitude,  such  that  it  reaches  the  ground  with  a  velocity  v;  then 
if  c  be  the  crushing  strain  at  any  point  of  the  sectional  area  (A)  of  the  limbs, 
the  limiting  condition  is  that  mv  =  cA. 

If  the  animal  vary  in  magnitude  without  change  in  the  height  to  which 
it  jumps  (or  in  the  velocity  with  which  it  descends),  then 

m       P  , 

coc  2  oc  ^"2,     orZ. 

The  crushing  strain  varies  directly  with  the  linear  dimensions  of  the  animal ; 
and  this,  a  dynamical  case,  is  identical  with  the  usual  statical  limitalfion  of 
magnitude. 

*  Recherches  sur  la  force  absolue  des  muscles  des  Inverlebres,  Bull.  Acad.  R, 
de  Belgique  (3),  vi,  vii,  1^83-84:  see  also  ibid.  (2),  xx,  1865;  xxii,  1866;  Ann. 
Mag.  N.H.  xvii,  p.  139,  1866;  xix,  p.  95,  1867.  Cf.  M.  Radau,  Sur  la  force 
musculaire  des  insectes,  Revue  des  deux  Mondes,  lxiv,  p.  770,  1866.  The  subject 
had  been  well  treated  by  Straus-Diirckheim,  in  his  Considerations  generates  sur 
Vanatomie  comparee  des  animauz  articuUs,  1828. 

t  Cf.  The  dynamics  of  sprint -running,  by  A.  V.  Hill  and  others,  Proc.  R.S.  (B), 
cii,  pp.  29-42,  1927;  or  Muscular  Movement  in  Man,  by  A.  V.  Hill,  New  York, 
1927,  ch.  VI,  p.  41. 


II]  OF  LOCOMOTION  39 

But  if  the  animal,  with  increasing  size  or  stature,  jump  to  a  correspondingly 
increasing  height,  the  case  becomes  much  more  serious.  For  the  final  velocity 
of  descent  varies  as  the  square  root  of  the  altitude  reached,  and  therefore  as 
the  square  root  of  tte  linear  dimensions  of  the  animal.     And  since,  as  before, 

13 

cccmvcc  j^  Vy 

c  oc  75 .  V ;,    or  c  oc  Z». 

If  a  creature's  jump  were  in  proportion  to  its  height,  the  crushing  strains 
would  so  increase  that  its  dimensions  would  be  limited  thereby  in  a  much  higher 
degree  than  was  indicated  by  statical  considerations.  An  animal  may  grow 
to  a  size  where  lit  is  unstable  dynamically,  though  still  on  the  safe  side 
statically — a  size  where  it  moves  with  difficulty  though  it  rests  secure.  It  is 
by  reason  of  dynamical  rather  than  of  statical  relations  that  an  elephant 
is  of  graver  deportment  than  a. mouse. 

An  apparently  simple  problem,  much  less  simple  than  it  looks,  lies 
in  the  act  of  walking,  where  there  will  evidently  be  great  economy  of 
work  if  the  leg  swing  with  the  help  of  gravity,  that  is  to  say,  at  a 
peTidulum-rate.  The  conical  shape  and  jointing  of  the  limb,  the  time 
spent  with  the  foot  upon  the  ground,  these  and  other  mechanical 
differences  complicate  the  case,  and  make  the  rate  hard  to  define  or 
calculate.  Nevertheless,  we  may  convince  ourselves  by  counting  our 
steps,  that  the  leg  does  actually  tend  to  swing,  as  a  pendulum  does, 
at  a  certain  definite  rate*.  So  on  the  same  principle,  but  to  the 
slower  beat  of  a  longer  pendulum,  the  scythe  swings  smoothly  in 
the  mower's  hands. 

To  walk  quicker,  we  "step  out";  we  cause  the  leg-pendulum  to 
describe  a  greater  arc,  but  it  does  not  swing  or  vibrate  faster  until 
we  shorten  the  pendulum  and  begin  to  run.  Now  let  two  similar 
individuals,  A  and  B,  walk  in  a  similar  fashion,  that  is  to  say  with 
a  similar  an^le  of  swing  (Fig.  1).  The  arc  through  which  the  leg 
swings,  or  the  amplitude  of  each  step,  will  then  vary  as  the  length 
of  leg  (say  as  a/6),  and  so  as  the  height  or  other  linear  dimension  (/) 
of  the  manf.     But  the  time  of  swing  varies  inversely  as'the  square 

*  The  assertion  that  the  hmb  tends  to  swing  in  pendulum-time  was  first  made 
by  the  brothers  Weber  {Mechanik  der  menschl.  Gehwerkzeuge,  Gottingen,  1836). 
Some  later  writers  have  criticised  the  statement  (e.g.  Fischer,  Die  Kinematik  dea 
Beinschwingens  etc.,  AhJi.  math.  phys.  Kl.  k.  Sachs.  Ges.  xxv-xxvin,  1899-1903), 
but  for  all  that,  with  proper  and  large  qualifications,  it  remains  substantially  true. 

t  So  the  stride  of  a  Brobdingnagian  was  10  yards  long,  or  just  twelve  times  the 
2  ft.  6  in.,  which  make  the  average  stride  or  half-pace  of  a  man. 


40 


ON  MAGNITUDE 


[CH. 


root  of  the  pendulum-length,  or  Va/Vb.     Therefore  the  velocity, 
which  is  measured  by  amphtude/time,  or  a/b  x  Vb/Va,  will  also  vary 

as  the  square  root  of  the  hnear  dimen- 
sions; which  is  Froude's  law  over  again. 
The  smaller  man,  or  smaller  animal, 
goes  slower  than  the  larger,  but  only  in 
the  ratio  of  the  square  roots  of  their 
linear  dimensions ;  whereas,  if  the  limbs 
moved  alike,  irrespective  of  the  size  of 
the  animal — if  the  limbs  of  the  mouse 
swung  no  faster  than  those  of  the  horse 
— then  the  mouse  would  be  as  slow  in 
its  gait  or  slower  than  the  tortoise. 
M.  DeHsle*  saw  a  fly  walk  three  inches 
in  half-a-second ;  this  was  good  steady 
walking.  When  we  walk  five  miles  an 
hour  we  go  about  88  inches  in  a  second, 
or  88/6  =14-7  times  the  pace  of  M.  DeHsle 's  fly.  We  should  walk 
at  just  about  the  fly's  pace  if  our  stature  were  1/(14-7)2,  or  1/216 
of  our  present  height — say  72/216  inches,  or  one-third  of  an  inch 
high.  Let  us  note  in  passing  that  the  number  of  legs  does  not 
matter,  any  more  than  the  number  of  wheels  to  a  coach;  the 
centipede  runs  none  the  faster  for  all  his  hundred  legs. 

But  the  leg  comprises  a  complicated  system  of  levers,  by  whose 
various  exercise  we  obtain  very  different  results.  For  instance,  by 
being  careful  to  rise  upon  our  instep  we  increase  the  length  or 
amplitude  of  our  stride,  and  improve  our  speed  very  materially; 
and  it  is  curious  to  see  how  Nature  lengthens  this  metatarsal  joint, 
or  instep-lever,  in  horse f  and  hare  and  greyhound,  in  ostrich  and 
in  kangaroo,  and  in  every  speedy  animal.  Furthermore,  in  running 
we  bend  and  so  shorten  the  leg,  in  order  to  accommodate  it  to  a 
quicker  rate  of  pendulum-swing  J.    In  short  the  jointed  structure 

*  Quoted  in  Mr  John  Bishop's  interesting  article  in  Todd's  Cyclopaedia,  iii, 
p.  443. 

t  The  "cannon-bones"  are  not  only  relatively  longer  but  may  even  be  actually 
longer  in  a  little  racehorse  than  a  great  carthorse. 

t  There  is  probably  another  factor  involved  here:  for  in  bending  and  thus 
shortening  the  leg,  we  bring  its  centre  of  gravity  nearer  to  the  pivot,  that  is  to 
say  to  the  joint,  and  so  the  muscle  tends  to  move  it  the  ,more  quickly.     After  all, 


II]  OF  RATE  OF  WALKING  41 

of  the  leg  permits  us  to  use  it  as  the  shortest  possible  lever  while 
it  is  swinging,  and  as  the  longest  possible  lever  when  it  is  exerting 
its  propulsive  force. 

The  bird's  case  is  of  peculiar  interest.  In  running,  walking  or 
swimming,  we  consider  the  speed  which  an  animal  can  attain,  and 
the  increase  of  speed  which  increasing  size  permits  of.  But  in  flight 
there  is  a  certain  necessary  speed— a  speed  (relative  to  the  air)  which 
the  bird  must  attain  in  order  to  maintain  itself  aloft,  and  which  nfiust 
increase  as  its  size  increases.  It  is  highly  probable,  as  Lanchester 
remarks,  that  Lilienthal  met  his  untimely  death  (in  August  1896) 
not  so  much  from  any  intrinsic  fault  in  the  design  or  construction 
of  his  machine,  but  simply  because  his  engine  fell  somewhat 
short  of  the  power  required  to  give  the  speed  necessary  for  its 
stability. 

Twenty-five  years  ago,  when  this  book  was  written,  the  bird,  or 
the  aeroplane,  was  thought  of  as  a  machine  whose  sloping  wings, 
held  at  a  given  angle  and  driven  horizontally  forward,  deflect  the 
air  downwards  and  derive  support  from  the  upward  reaction.  In 
other  words,  the  bird  was  supposed  to  communicate  to  a  mass  of 
air  a  downward  momentum  equivalent  (in  unit  time)  to  its  own 
weight,  and  to  do  so  by  direct  and  continuous  impact.  The  down- 
ward momentum  is  then  proportional  to  the  mass  of  air  thrust 
downwards,  and  to  the  rate  at  which  it  is  so  thrust  or  driven:  the 
mass  being  proportional  to  the  wing-area  and  to  the  speed  of  the 
bird,  and  the  rate  being  again  proportional  to  the  flying  speed;  so 
that  the  momentum  varies  as  the  square  of  the  bird's  linear  dimen- 
sions and  also  as  the  square  of  its  speed.  But  in  order  to  balance 
its  weight,  this  momentum  must  also  be  proportional  to  the 
cube  of  the  bird's  linear  dimensions;  therefore  the  bird's  necessary 
speed,  such  as  enables  it  to  maintain  level  flight,  must  be  pro- 
portional to  the  square  root  of  its  linear  dimensions,  and  the  whole 
work  done  must  be  proportional  to  the  power  3 J  of  the  said  linear 
dimensions. 

The  case  stands,  so  far,  as  follows :  m,  the  mass  of  air  deflected 
downwards;  M,  the  momentum  so  communicated;  W,  the  work 
done — all  in  unit  time;   w,  the  weight,  and  F,  the  velocity  of  the 

we  know  that  the  pendulum  theory  is  not  the  whole  story,  but  only  an  important 
first  approximation  to  a  complex  phenomenon. 


42  ON  MAGNITUDE  [ch. 

bird;  I,  a  linear  dimension,  the  form  of  the  bird  being  supposed 
constant.       j^j  _  ^^  ^  ^3^  but  M  =  mV ,  and  m  =  W , 

Therefore  M  -  ^272  _  ^3^ 

and  therefore  F  =  VI 

and  Tf  =  MF  =  R 

The  gist  of  the  matter  is,  or  seems  to  be,  that  the  work  which 
can  he  done  varies  with  the  available  weight  of  muscle,  that  is  to  say, 
with  the  mass  of  the  bird ;  but  the  work  which  has  to  be  done  varies 
with  mass  and  distance;  so  the  larger  the  bird  grows,  the  greater 
the  disadvantage  under  which  all  its  work  is  done*.  The  dispropor- 
tion does  not  seem  very  great  at  first  sight,  but  it  is  quite  enough 
to  tell.  It  is  as  much  as  to  say  that,  every  time  we  double  the 
linear  dimensions  of  the  bird,  the  difficulty  of  flight,  >  or  the  work 
which  must  needs  be  done  in  order  to  fly,  is  increased  in  the  ratio 
of  2^  to  2^*,  or  1  :  V2,  or  say  1:14.  If  we  take  the  ostrich  to  exceed 
the  sparrow  in  linear  dimensions  as  25  :  1,  which  seems  well  within 
the  mark,  the  ratio  would  be  that  between  25^*  and  25^,  or  between 
5'  and  5^;  in  other  words,  flight  would  be  five  times  more  difficult 
for  the  larger  than  for  the  smaller  bird. 

But  this  whole  explanation  is  doubly  inadequate.  For  one  thing, 
it  takes  no  account  of  gliding  flight,  in  which  energy  is  drawn  from 
the  wind,  and  neither  muscular  power  nor  engine  power  are  em- 
ployed; and  we  see  that  the  larger  birds,  vulture,  albatross  or 
solan-goose,  depend  on  gliding  more  and  more.  Secondly,  the  old 
simple  account  of  the  impact  of  the  wing  upon  the  air,  and  the 
manner  in  which  a  downward  momentum  is  communicated  and 
support  obtained,  is  now  known  to  be  both  inadequate  and 
erroneous.  For  the  science  of  flight,  or  aerodynamics,  has  grown 
out  of  the  older  science  of  hydrodynamics;  both  deal  with  the 
special  properties  of  a  fluid,  whether  water  or  air;  and  in  our  case, 
to  be  content  to  think  of  the  air  as  a  body  of  mass  m,  to  which  a 
velocity  v  is  imparted,  is  to  neglect  all  its  fluid  properties.    How  the 

*  This  is  the  result  arrived  at  by  Helmholtz,  Ueber  ein  Theorem  geometrisch- 
ahnliche  Bewegungen  fliissiger  Korper  betrefFend,  nebst  Anwendung  auf  das 
Problem  Luftballons  zu  lenken,  Monatsber.  Akad.  Berlin,  1873,  pp.  501-514.  It  was 
criticised  and  challenged  (somewhat  rashly)  by  K.  Miillenhof,  Die  Grosse  der  Flug- 
flachen  etc.,  PfiUger's  Archiv,  xxxv,  p.  407;   xxxvi,  p.  548,  1885. 


II]  OF  FLIGHT  43 

fish  or  the  dolphin  swims,  and  how  the  bird  flies,  are  up  to  a  certain 
point  analogous  problems ;  and  stream-lining  plays  an  essential  part 
in  both.  But  the  bird  is  much  heavier  than  the  air,  and  the  fish 
has  much  the  same  density  as  the  water,  so  that  the  problem  of 
keeping  afloat  or  aloft  is  negligible  in  the  one,  and  all-important  in 
the  other.  Furthermore,  the  one  fluid  is  highly  compressible,  and 
the  other  (to  all  intents  and  purposes)  incompressible ;  and  it  is  this 
very  difference  which  the  bird,  or  the  aeroplane,  takes  special 
advantage  of,  and  which  helps,  or  even  enables,  it  to  fly. 

It  remains  as  true  as  ever  that  a  bird,  in  order  to  counteract 
gravity,  must  cause  air  to  move  downward  and  obtains  an  upward 
reaction  thereby.  But  the  air  displaced  downward  beneath  the 
wing  accounts  for  a  small  and  varying  part,  perhaps  a  third  perhaps 
a  good  deal  less,  of  the  whole  force  derived;  and  the  rest  is  generated 
above  the  wing,  in  a  less  simple  way.  For,  as  the  air  streams  past 
the  slightly  sloping  wing,  as  smoothly  as  the  stream-lined  form 
and  polished  surface  permit,  it  swirls  round  the  front  or  "leading" 
edge*,  and  then  streams  swiftly  over  the  upper  surface  of  the  wing; 
while  it  passes  comparatively  slowly,  checked  by  the  opposing  slope 
of  the  wing,  across  the  lower  side.  And  this  is  as  much  as  to  say 
that  it  tends  to  be  compressed  below  and  rarefied  above;  in  other 
words,  that  a  partial  vacuum  is  formed  above  the  wing  and  follows 
it  wherever  it  goes,  so  long  as  the  stream-lining  of  the  wing  and  its 
angle  of  incidence  are  suitable,  and  so  long  as  the  bird  travels  fast 
enough  through  the  air. 

The  bird's  weight  is  exerting  a  downward  force  upon  the  air,  in 
one  way  just  as  in  the  other;  and  we  can  imagine  a  barometer 
delicate  enough  to  shew  and  measure  it  as  the  bird  flies  overhead. 
But  to  calculate  that  force  we  should  have  to  consider  a  multitude 
of  component  elements;  we  should  have  to  deal  with  the  stream- 
lined tubes  of  flow  above  and  below,  and  the  eddies  round  the  fore- 
edge  of  the  wing  and  elsewhere;  and  the  calculation  which  was  too 
simple  before  now  becomes  insuperably  difficult.  But  the  principle 
of  necessary  speed  remains  as  true  as  ever.     The  bigger  the  bird 

*  The  arched  form,  or  "dipping  front  edge"  of  the  wing,  and  its  use  in  causing 
a  vacuum  above,  were  first  recognised  by  Mr  H.  F.  Phillips,  who  put  the  idea  into 
a  patent  in  1884.  The  facts  were  discovered  independently,  and  soon  afterwards, 
both  by  Lilienthal  and  Lanchester. 


44  ON  MAGNITUDE  [ch. 

becomes,  the  more  swiftly  must  the  air  stream  over  the  wing 
to  give  rise  to  the  rarefaction  or  negative  pressure  which  is  more 
and  more  required;  and  the  harder  must  it  be  to  fly,  so  long  as 
work  has  to  be  done  by  the  muscles  of  the  bird.  The  general 
principle  is  the  same  as  before,  though  the  quantitative  relation 
does  not  work  out  as  easily  as  it  did.  As  a  matter  of  fact,  there 
is  probably  little  difference  in  the  end;  and  in  aeronautics,  the 
"total  resultant  force"  which  the  bird  employs  for  its  support  is 
said,  e^npirically,  to  vary  as  the  square  of  the  air-speed:  which  is 
then  a  result  analogous  to  Froude's  law,  and  is  just  what  we  arrived 
at  before  in  the  simpler  and  less  accurate  setting  of  the  case. 

But  a  comparison  between  the  larger  and  the  smaller  bird,  like 
all  other  comparisons,  applies  only  so  long  as  the  other  factors  in 
the  case  remain  the  same ;  and  these  vary  so  much  in  the  complicated 
action  of  flight  that  it  is  hard  indeed  to  compare  one  bird  with 
another.  For  not  only  is  the  bird  continually  changing  the  incidence 
of  its  wing,  but  it  alters  the  lie  of  every  single  important  feather; 
and  all  the  ways  and  means  of  flight  vary  so  enormously,  in  big 
wings  and  small,  and  Nature  exhibits  so  many  refinements  and 
"  improvements"  in  the  mechanism  required,  that  a  comparison  based 
on  size  alone  becomes  imaginary,  and  is  little  worth  the  making. 

The  above  considerations  are  of  great  practical  importance  in 
aeronautics,  for  they  shew  how  a  provision  of  increasing  speed  must  ac- 
company every  enlargement  of  our  aeroplanes.  Speaking  generally, 
the  necessary  or  minimal  speed  of  an  aeroplane  varies  as  the  square 
root  of  its  Unear  dimensions;  if  (ceteris  paribus)  we  make  it  four 
times  as  long,  it  must,  in  order  to  remain  aloft,  fly  twice  as  fast  as 
before*.  If  a  given  machine  weighing,  say,  500  lb.  be  stable  at 
40  miles  an  hour,  then  a  geometrically  similar  one  which  weighs, 
say,  a  couple  of  tons  has  its  speed  determined  as  follows : 

W:w::L^:l^::S:l. 

Therefore  L:l::2:l. 

But  V^:v^::L:l 

Therefore  V:v::V2:l  =  1-414  : 1. 

*  G.  H.  Bryan,  Stability  in  Aviation,  1911;  F.  W.  Lanchester,  Aerodynamics, 
1909;  cf.  (int.  al.)  George  Greenhill,  The  Dynamics  of  Mechanical  Flight,  1912; 
F.  W.  Headley,  The  Flight  of  Birds,  and  recent  works. 


II]  OF  NECESSARY  SPEED  45 

That  is  to  say,  the  larger  machine  must  be  capable  of  a  speed  of 
40  X  1-414,  or  about  56|,  miles  per  hour. 

An  arrow  is  a  somewhat  rudimentary  flying-machine;  but  it  is 
capable,  to  a  certain  extent  and  at  a  high  velocity,  of  acquiring 
"stability,"  and  hence  of  actual  flight  after  the  fashion  of  an  aero- 
plane; the  duration  and  consequent  range  of  its  trajectory  are 
vastly  superior  to  those  of  a  bullet  of  the  same  initial  velocity. 
Coming  back  to  our  birds,  and  again  comparing  the  ostrich  with 
the  sparrow,  we  find  we  know  little  or  nothing  about  the  actual 
speed  of  the  latter ;  but  the  minimal  speed  of  the  swift  is  estimated 
at  100  ft.  per  second,  or  even  more — say  70  miles  an  hour.  We 
shall  be  on  the  safe  side,  and  perhaps  not  far  wrong,  to  take  20  miles 
an  hour  as  the  sparrow's  minimal  speed;  and  it  would  then  follow 
that  the  ostrich,  of  25  times  the  sparrow's  linear  dimensions,  would 
have  to  fly  (if  it  flew  at  all)  with  a  minimum  velocity  of  5  x  20, 
or  100  miles  an  hour*. 

The  same  principle  of  necessary  speed,  or  the  inevitable  relation 
between  the  dimensions  of  a  flying  object  and  the  minimum  velocity 
at  which  its  flight  is  stable,  accounts  for  a  considerable  number  of 

*  Birds  have  an  ordinary  and  a  forced  speed.  Meinertzhagen  puts  the  ordinary 
flight  of  the  swift  at  68  m.p.h.,  which  talhes  with  the  old  estimate  of  Athanasius 
Kircher  (Physiologia,  ed.  1680,  p.  65)  of  100  ft.  per  second  for  the  swallow.  Abel 
Chapman  {Retrospect,  1928,  ch.  xiv)  puts  the  gliding  or  swooping  flight  of  the  swift 
at  over  150  m.p.h.,  and  that  of  the  griffon  vulture  at  180  m.p.h.;  but  these  skilled 
fliers  doubtless  far  exceed  the  necessary  minimal  speeds  which  we  are  speaking  of. 
An  airman  flying  at  70  m.p.h.  has  seen  a  golden  eagle  fly  past  him  easily;  but 
even  this  speed  is  exceptional.  Several  observers  agree  in  giving  50  m.p.h.  for 
grouse  and  woodcock,  and  30  m.p.h.  for  starling,  chaffinch,  quail  and  crow.  A 
migrating  flock  of  lapwing  travelled  at  41  m.p.h.,  ten  or  twelve  miles  more  than 
the  usual  speed  of  the  single  bird.  Lanchester,  on  theoretical  considerations, 
estimates  the  speed  of  the  herring  gull  at  26  m.p.h.,  and  of  the  albatross  at  about 
34  miles.  A  tern,  a  very  skilful  flier,  was  seen  to  fly  as  slowly  as  15  m.p.h. 
A  hornet  or  a  large  dragonfly  may  reach  14  or  18  m.p.h.;  but  for  most  insects 
2-4  metres  per  sec,  say  4-9  m.p.h.,  is  a  common  speed  (cf,  A.  Magnan,  Vol. 
des  Insectes,  1834,  p.  72).  The  larger  diptera  are  very  swift,  but  their  speed  is  much 
exaggerated.  A  deerfly  (Cephenomyia)  has  been  said  to  fly  at  400  yards  per  second, 
or  say  800  m.p.h.,  an  impossible  velocity  (Irving  Langmuir,  Science,  March  11,  1938). 
It  would  mean  a  pressure  on  the  fly's  head  of  half  an  atmosphere,  probably  enough 
to  crush  the  fly;  to  maintain  it  would  take  half  a  horsepower;  and  this  would  need 
a  food- consumption  of  1^  times  the  fly's  weight  per  second\  25  m.p.h.  is  a  more 
reasonable  estimate.  The  naturalist  should  not  forget,  though  it  does  not  touch 
our  present  argument,  that  the  aeroplane  is  built  to  the  pattern  of  a  beetle  rather 
than  of  a  bird;  for  the  elytra 'are  not  wings  but  planes.  Cf.  int.  ah,  P.  Amans, 
Geometrie. .  .des  ailes  rigides,  C.R.  Assoc.  Frang.  pour  Vavancem.  des  Sc.  1901. 


46  ON  MAGNITUDE  [ch, 

observed  phenomena.  It  tells  us  why  the  larger  birds  have  a 
marked  difficulty  in  rising  from  the  ground,  that  is  to  say,  in 
acquiring  to  begin  with  the  horizontal  velocity  necessary  for  their 
support;  and  why  accordingly,  as  Mouillard*  and  others  have 
observed,  the  heavier  birds,  even  those  weighing  no  more  than  a 
pound  or  two,  can  be  effectually  caged  in  small  enclosures  open 
to  the  sky.  It  explains  why,  as  Mr  Abel  Chapman  says,  "all 
ponderous  birds,  wild  swans  and  geese,  great  bustard  and  caper- 
cailzie, even  blackcock,  fly  faster  than  they  appear  to  do,"  while 
"light-built  types  with  a  big  wing-areaf,  such  as  herons  and  harriers, 
possess  no  turn  of  speed  at  all."  For  the  fact  is  that  the  heavy 
birds  must  fly  quickly,  or  not  at  all.  It  tells  us  why  very  small 
birds,  especially  those  as  small  as  humming-birds,  and  a  fortiori  the 
still  smaller  insects,  are  capable  of  "stationary  flight,"  a  very  slight 
and  scarcely  perceptible  velocity  relatively  to  the  air  being  sufficient 
for  their  support  and  stabihty.  And  again,  since  it  is  in  all 
these  cases  velocity  relatively  to  the  air  which  we  are  speaking 
of,  we  comprehend  the  reason  why  one  may  always  tell  which 
way  the  wind  blows  by  watching  the  direction  in  which  a  bird  starts 
to  fly. 

The  wing  of  a  bird  or  insect,  like  the  tail  of  a  fish  or  the  blade 
of  an  oar,  gives  rise  at  each  impulsion  to  a  swirl  or  vortex,  which 
tends  (so  to  speak)  to  chng  to  it  and  travel  along  with  it;'  and  the 
resistance  which  wing  or  oar  encounter  comes  much  more  from 
these  vortices  than  from  the  viscosity  of  the  fluid.  J  We  learn  as  a 
corollary  to  this,  that  vortices  form  only  at  the  edge  of  oar  or  wing — 
it  is  only  the  length  and  not  the  breadth  of  these  which  matters. 
A  long  narrow  oar  outpaces  a  broad  one,  and  the  efficiency  of  the 
long,  narrow  wing  of  albatross,  swift  or  hawkmoth  is  so  far  accounted 
for.  From  the  length  of  the  wing  we  can  calculate  approximately 
its  rate  of  swing,  and  more  conjecturally  the  dimensions  of  each 
vortex,  and  finally  the  resistance  or  Ufting  power  of  the  stroke; 
and  the  result  shews  once  again  the  advantages  of  the  small-scale 

*  Mouillard,  L' empire  de  Vair;  essai  d'ornithologie  appliqu4e  a  Vaviationf  1881; 
transl.  in  Annual  Report  of  the  Smithsonian  Institution^  1892. 

t  On  wing-area  in  relation  to  weight  of  bird  see  Lendenfeld  in  Naturw.  Wochenschr. 
Nov.  1904,  transl.  in  Smithsonian  Inst.  Rep.  1904;  also  E.  H.  Hankin,  Animal 
Flight,  1913;  etc. 

X  Cf.  V.  Bjerknes,  Hydrodynamique  physique,  n,  p.  293,  1934. 


II]  OF  MODES  OF  FLIGHT  47 

mechanism,  and  the  disadvantage  under  which  the  larger  machine 
or  larger  creature  hes. 


Length 

Speed  of 

Radius 

Force  of 

Specific 

Weight 

of  wing     Beats 

wing-tip 

of 

wing-beat 

j  orce, 

gm. 

m.        per  sec 
(From  V. 

m./s. 
Bjerknes) 

vortex* 

gra. 

FjW 

Stork 

3500 

0-91              2 

5-7 

1-5 

1480 

2:5 

Gull 

1000 

0-60             3 

5-7 

1-0 

640 

2:3 

Pigeon 

350 

0-30             6 

5-7 

0-5 

160 

1:2 

Sparrow 

30 

Oil            13 

4-5 

0-18 

13 

2:5 

Bee 

0-07 

0-01          200 

6-3 

002 

0-2 

3i:l 

Fly 

001 

0-007        190 

4-2 

001 

004 

4:1 

*  Conjectural. 

A  bird  may  exert  a  force  at  each  stroke  of  its  wing  equal  to 
one-half,  let  us  say  for  safety  one-quarter,  of  its  own  weight,  more 
or  less ;  but  a  bee  or  a  fly  does  twice  or  thrice  the  equivalent  of  its 
own  weight,  at  a  low  estimate.  If  stork,  gull  or  pigeon  can  thus 
carry  only  one-fifth,  one-third,  one- quarter  of  their  weight  by  the 
beating  of  their  wings,  it  follows  that  all  the  rest  must  be  borne  by 
sailing-flight  between  the  wing-beats.  But  an  insect's  wings  lift  it 
easily  and  with  something  to  spare;  hence  saihng-flight,  and  with 
it  the  whole  principle  of  necessary  speed,  does  not  concern  the  lesser 
insects,  nor  the  smallest  birds,  at  all;  for  a  humming-bird  can 
"stand  still"  in  the  air,  like  a  hover-fly,  and  dart  backwards  as 
well  as  forwards,  if  it  please. 

There  is  a  little  group  of  Fairy-flies  (Mymaridae),  far  below  the 
size  of  any  small  famiUar  insects;  their  eggs  are  laid  and  larvae 
reared  within  the  tiny  eggs  of  larger  insects ;  their  bodies  may  be  no 
more  than  J  mm.  long,  and  their  outspread  wings  2  mm.  from  tip 
to  tip  (Fig.  2).  It  is  a  pecuharity  of  some  of  these  that  their  Httle 
wings  are  made  of  a  few  hairs  or  bristles,  instead  of  the  continuous 
membrane  of  a  wing.  How  these  act  on  the  minute  quantity  of  air 
involved  we  can  only  conjecture.  It  would  seem  that  that  small 
quantity  reacts  as  a  viscous  fluid  to  the  beat  of  the  wing ;  but  there 
are  doubtless  other  unobserved  anomahes  in  the  mechanism  and 
the  mode  of  flight  of  these  pigmy  creaturesf . 

The  ostrich  has  apparently  reached  a  magnitude,  and  the  moa 
certainly  did  so,  at  which  flight  by  muscular  action,  according  to 

t  It  is  obvious  that  in  a  still  smaller  order  of  magnitude  tfie  Brownian  movement 
would  suffice  to  make^  flight  impossible. 


48 


ON  MAGNITUDE 


[CH. 


the  normal  anatomy  of  a  bird,  becomes  physiologically  impossible. 
The  same  reasoning  applies  to  the  case  of  man.  It  would  be  very 
difficult,  and  probably  absolutely  impossible,  for  a  bird  to  flap  its 
way  through  the  air  were  it  of  the  bigness  of  a  man;  but  Borelli, 
in  discussing  the  matter,  laid  even  greater  stress  on  the  fact  that 
a  man's  pectoral  muscles  are  so  much  less  in  proportion  than  those 
of  a  bird,  that  however  we  might  fit  ourselves  out  with  wings,  we 
could  never  expect  to  flap  them  by  any  power  of  our  own  weak 
muscles.  Borelli  had  learned  this  lesson  thoroughly,  and  in  one  of 
his  chapters  he  deals  with  the  proposition :  Est  impossibile  ut  homines 
jyropriis  viribus  artificiose  volare  possinV^,     But  gliding  flight,  where 


a  '        b 

2.   Fairy-flies  (Mymaridae) :  after  F.  Enock.    x  20. 


wind-force  and  gravitational  energy  take  the  place  of  muscular 
power,  is  another  story,  and  its  limitations  are  of  another  kind. 
Nature  has  many  modes  and  mechanisms  of  flight,  in  birds  of  one 
kind  and  another,  in  bats  and  beetles,  butterflies,  dragonflies  and 
what  not ;  and  gliding  seems  to  be  the  common  way  of  birds,  and 
the  flapping  flight  {remigio  alarum)  of  sparrow  and  of  crow  to  be 
the  exception  rather  than  the  rule.  But  it  were  truer  to  say  that 
gliding  and  soaring,  by  which  energy  is  captured  from  the  wind,  are 
modes  of  flight  little  needed  by  the  small  birds,  but  more  and  more 
essential  to  the  large.  BorelH  had  proved  so  convincingly  that 
we  could  never  hope  to  fly  propriis  viribus,  that  all  through  the 
eighteenth  century  men  tried  no  more  to  fly  at  all.  It  was  in  trying 
to  glide  that  the  pioneers  of  aviation,  Cayley,  Wenham  and  Mouillard, 

*  Giovanni  Alfonso  Borelli,  De  Motu  Animalium,  i,  Prop,  cciv,  p.  243,  edit. 
1685.  The  part  on  The  Flight  of  Birds  is  issued  by  the  Royal  Aeronautical  Society 
as  No.  6  of  its  Aeronautical  Classics. 


II]  OF  GLIDING  FLIGHT  49 

Langley,  Lilienthal  and  the  Wrights — all  careful  students  of  birds — 
renewed  the  attempt*;  and  only  after  the  Wrights  had  learned  to 
ghde  did  they  seek  to  add  power  to  their  glider.  Flight,  as  the 
Wrights  declared,  is  a  matter  of  practice  and  of  skill,  and  skill  in 
gliding  has  now  reached  a  point  which  more  than  justifies  all 
Leonardo  da  Vinci's  attempts  to  fly.  Birds  shew  infinite  skill  and 
instinctive  knowledge  in  the  use  they  make  of  the  horizontal  accelera- 
tion of  the  wind,  and  the  advantage  they  take  of  ascending  currents 
in  the  air.  Over  the  hot  sands  of  the  Sahara,  where  every  here 
and  there  hot  air  is  going  up  and  cooler  coming  down,  birds  keep 
as  best  they  can  to  the  one,  or  ghde  quickly  through  the  other; 
so  we  may  watch  a  big  dragonfly  planing  slowly  down  a  few  feet 
above  the  heated  soil,  and  only  every  five  minutes  or  so  regaining 
height  with  a  vigorous  stroke  of  his  wings.  The  albatross  uses  the 
upward  current  on  the  lee-side  of  a  great  ocean- wave ;  so,  on  a  lesser 
scale,  does  the  flying- fish;  and  the  seagull  flies  in  curves,  taking 
every  advantage  of  the  varying  wind- velocities  at  different  levels 
over  the  sea.  An  Indian  vulture  flaps  his  way  up  for  a  few  laborious 
yards,  then  catching  an  upward  current  soars  in  easy  spirals  to 
2000  feet ;  here  he  may  stay,  effortless,  all  day  long,  and  come  down 
at  sunset.  Nor  is  the  modern  sail-plane  much  less  efiicient  than  a 
soaring  bird ;  for  a  skilful  pilot  in  the  tropics  should  be  able  to  roam 
all  day  long  at  willf . 

A  bird's  sensitiveness  to  air-pressure  is  indicated  in  other  ways 
besides.  Heavy  birds,  hke  duck  and  partridge,  fly  low  and  ap- 
parently take  advantage  of  air-pressure  reflected  from  the  ground. 
Water-hen  and  dipper  follow  the  windings  of  the  stream  as  they  fly 
up  or  down;  a  bee-hne  would  give  them  a  shorter  course,  but  not 
so  smooth  a  journey.  Some  small  birds — wagtails,  woodpeckers  and 
a  few  others — fly,  so  to  speak,  by  leaps  and  bounds ;  they  fly  briskly 

*  Sir  George  Cayley  (1774-1857),  father  of  British  aeronautics,  was  the  first  to 
perceive  the  capabilities  of  rigid  planes,  and  to  experiment  on  gliding  flight.  He 
anticipated  all  the  essential  principles  of  the  modern  aeroplane,  and  his  first  paper 
"On  Aerial  Navigation"  appeared  in  Nicholson's  Journal  for  November  1809. 
F.  H.  Wenham  (1824-1908)  studied  the  flight  of  birds  and  estimated  the  necessary 
proportion  of  surface  to  weight  and  speed;  he  held  that  "the  whole  secret  of 
success  in  flight  depends  upon  a  proper  concave  form  of  the  supporting  surface." 
See  his  paper  "On  Aerial  Locomotion"  in  the  Report  of  the  Aeronautical  Society 
1866. 

t  Sir  Gilbert  Walker,  in  Nature,  Oct.  2,  1937. 


50  ON  MAGNITUDE  [ch. 

for  a  few  moments,  then  close  their  wings  and  shoot  along*.  The 
flying-fishes  do  much  the  same,  save  that  they  keep  their  wings 
outspread.  The  best  of  them  "taxi"  along  with  only  their  tails  in 
the  water,  the  tail  vibrating  with  great  rapidity,  and  the  speed 
attained  lasts  the  fish  on  its  long  glide  through  the  airf. 

Flying  may  have  begun,  as  in  Man's  case  it  did,  with  short  spells 
of  gUding  flight,  helped  by  gravity,  and  far  short  of  sustained  or 
continuous  locomotion.  The  short  wings  and  long  tail  of  Archae- 
opteryx  would  be  efficient  as  a  slow-speed  ghder;  and  we  may  still 
see  a  Touraco  glide  down  from  his  perch  looking  not  much  unlike 
Archaeopteryx  in  the  proportions  of  his  wings  and  tail.  The  small 
bodies,  scanty  muscles  and  narrow  but  vastly  elongated  wings  of 
a  Pterodactyl  go  far  beyond  the  hmits  of  mechanical  efficiency  for 
ordinary  flapping  flight;  but  for  ghding  they  approach  perfection  J. 
Sooner  or  later  Nature  does  everything  which  is  physically  possible ; 
and  to  glide  with  skill  and  safety  through  the  air  is  a  possibihty 
which  she  did  not  overlook. 

Apart  from  all  differences  in  the  action  of  the  limbs — apart  from 
differences  in  mechanical  construction  or  in  the  manner  in  which 
the  mechanism  is  used — we  have  now  arrived  at  a  curiously  simple 
and  uniform  result.  For  in  all  the  three  forms  of  locomotion  which 
we  have  attempted  to  study,  alike  in  swimming  and  in  walking,  and 
even  in  the  more,  complex  problem  of  flight,  the  general  result, 
obtained  under  very  different  conditions  and  arrived  at  by  different 
modes  of  reasoning,  shews  in  every  case  that  speed  tends  to  vary  as 
the  square  root  of  the  linear  dimensions  of  the  animal. 

While  the  rate  of  progress  tends  to  increase  slowly  with  increasing 
size  (according  to  Froude's  law),  and  the  rhythm  or  pendulum-rate 
of  the  limbs  to  increase  rapidly  with  decreasing  size  (according  to 
Galileo's   law),    some   such   increase   of  velocity   with   decreasing 

*  Why  large  birds  cannot  do  the  same  is  discussed  by  Lanchester,  op.  cit. 
Appendix  iv., 

t  Cf.  Carl  L.  Hubbs,  On  the  flight  of. .  .the  Cypselurinae,  and  remarks  on  the 
evolution  of  the  flight  of  fishes.  Papers  of  the  Michigan  Acad,  of  Sci.  xvii,  pp.  575- 
611,  1933.  See  also  E.  H.  Hankin,  P.Z.S.  1920,  pp.  467-474;  and  C.  M.  Breeder, 
On  the  structural  specialisation  of  flying  fishes  from  the  standpoint  of  aero- 
dynamics, Copeia,  1930,  pp.  114-121. 

X  The  old  conjecture  that  their  flight  was  helped  or  rendered  possible  by  a  denser 
atmosphere  than  ours  is  thus  no  longer  called  for. 


II]  OF  THE  FORCE  OF  GRAVITY  51 

magnitude  is  true  of  all  the  rhythmic  actions  of  the  body,  though 
for  reasons  not  always  easy  to  explain.  The  elephant's  heart  beats 
slower  than  ours*,  the  dog's  quicker;  the  rabbit's  goes  pit-a-pat; 
the  mouse's  and  the  sparrow's  are  too  quick  to  count.  But  the  very 
"rate  of  Hving"  (measured  by  the  0  consumed  and  COg  produced) 
slows  down  as  size  increases;  and  a  rat  lives  so  much  faster  than 
a  man  that  the  years  of  its  life  are  three,  instead  of  threescore  and 
ten. 

From  all  the  foregoing  discussion  we  learn  that,  as  Crookes  once 
upon  a  time  remarked  I,  the  forms  as  well  as  the  actions  of  our 
bodies  are  entirely  conditioned  (save  for  certain  exceptions  in  the 
case  of  aquatic  animals)  by  the  strength  of  gravity  upon  this  globe; 
or,  as  Sir  Charles  Bell  had  put  it  some  sixty  years  before,  the  very 
animals  which  move  upon  the  surface  of  the  earth  are  proportioned 
to  its  magnitude.  Were  the  force  of  gravity  to  be  doubled  our 
bipedal  form  would  be  a  failure,  and  the  majority  of  terrestrial 
animals  would  resemble  short-legged  saurians,  or  else  serpents. 
Birds  and  insects  would  suffer  Ukewise,  though  with  some  com- 
pensation in  the  increased  density  of  the  air.  On  the  other  hand, 
if  gravity  were  halved,  we  should  get  a  lighter,  slenderer,  more  active 
type,  needing  less  energy,  less  heat,  less  heart,  less  lungs,  less  blood. 
Gravity  not  only  controls  the  actions  but  also  influences  the  forms 
of  all  save  the  least  of  organisms.  The  tree  under  its  burden  of 
leaves  or  fruit  has  changed  its  every  curve  and  outline  since  its 
boughs  were  bare,  and  a  mantle  of  snow  will  alter  its  configuration 
again.  Sagging  wrinkles,  hanging  breasts  and  many  another  sign 
of  age  are  part  of  gravitation's  slow  relentless  handiwork. 

There  are  other  physical  factors  besides  gravity  which  help  to 
limit  the  size  to  which  an  animal  may  grow  and  to  define  the  con- 
ditions under  which  it  may  Hve.  The  small  insects  skating  on  a 
pool  have  their  movements  controlled  and  their  freedom  hmited  by 
the  surface-tension  between  water  and  air,  and  the  measure  of  that 
tension  determines  the  magnitude  which  they  may  attain.  A  man 
coming  wet  from  his  bath  carries  a  few  ounces  of  water,  and  is 
perhaps  1  per  cent,  heavier  than  before;  but  a  wet  fly  weighs  twice 
as  much  as  a  dry  one,  and  becomes  a  helpless  thing.  *  A  small 

*  Say  28  to  30  beats  to  the  minute. 

t  Proc.  Psychical  Soc.  xn,  p.  338-355,  1^97. 


52  ON  MAGNITUDE  [ch. 

insect  finds  itself  imprisoned  in  a  drop  of  water,  and  a  fly  with 
two  feet  in  one  drop  finds  it  hard  to  extricate  them. 

The  mechanical  construction  of  insect  or  crustacean  is  highly 
efficient  up  to  a  certain  size,  but  even  crab  and  lobster  never  exceed 
certain  moderate  dimensions,  perfect  within  these  narrow  bounds  as 
their  construction  seems  to  be.  Their  body  lies  within  a  hollow 
shell,  the  stresses  within  which  increase  much  faster  than  the  mere 
scale  of  size ;  every  hollow  structure,  every  dome  or  cyhnder,  grows 
weaker  as  it  grows  larger,  and  a  tin  canister  is  easy  to  make  but  a 
great  boiler  is  a  complicated  affair.  The  boiler  has  to  be  strengthened 
by  "stiffening  rings"  or  ridges,  and  so  has  the  lobster's  shell;  but 
there  is  a  limit  even  to  this  method  of  counteracting  the  weakening 
effect  of  size.  An  ordinary  girder-bridge  may  be  made»efficient  up 
to  a  span  of  200  feet  or  so ;  but  it  is  physically  incapable  of  spanning 
the  Firth  of  Forth.  The  great  Japanese  spider-crab,  Macrocheira, 
has  a  span  of  some  12  feet  across;  but  Nature  meets  the  difficulty 
and  solves  the  problem  by  keeping  the  body  small,  and  building  up 
the  long  and  slender  legs  out  of  short  lengths  of  narrow  tubes. 
A  hollow  shell  is  admirable  for  small  animals,  but  Nature  does  not 
and  cannot  make  use  of  it  for  the  large. 

In  the  case  of  insects,  other  causes  help  to  keep  them  of  small 
dimensions.  In  their  peculiar  respiratory  system  blood  does  not 
carry  oxygen  to  the  tissues,  but  innumerable  fine  tubules  or  tracheae 
lead  air  into  the  interstices  of  the  body.  If  we  imagine  them  growing 
even  to  the  size  of  crab  or  lobster,  a  vast  complication  of  tracheal 
tubules  would  be  necessary,  within  which  friction  would  increase 
and  diffusion  be  retarded,  and  which  would  soon  be  an  inefficient 
and  inappropriate  mechanism. 

The  vibration  of  vocal  chords  and  auditory  drums  has  this  in 
common  with  the  pendulum-hke  motion  of  a  hmb  that  its  rate 
also  tends  to  vary  inversely  as  the  square  root  of  the  linear  dimen- 
sions. We  know  by  common  experience  of  fiddle,  drum  or  organ, 
that  pitch  rises,  or  the  frequency  of  vibration  increases,  as  the 
dimensions  of  pipe  or  membrane  or  string  diminish;  and  in  like 
manner  we  expect  to  hear  a  bass  note  from  the  great  beasts  and  a 
piping  treble  from  the  small.  The  rate  of  vibration  {N)  of  a  stretched 
string  depends  on  its  tension  and  its  density;  these  beins^  equal,  it 
varies  inversely  as  its  own  length  and  as  its  diameter,     i^or  similar 


II]  OF  EYES  AND  EARS  53 

strings,  N  oc  1//^,  and  for  a  circular  membrane,  of  radius  r  and 
thickness  e,  N  oc  ll(r^  Ve). 

But  the  deHcate  drums  or  tympana  of  various  animals  seem  to 
vary  much  less  in  thickness  than  in  diameter,  and  we  may  be  content 
to  write,  once  more,  N  oc  l/r^. 

Suppose  one  animal  to  be  fifty  times  less  than  another,  vocal 
chords  and  all:  the  one's  voice  will  be  pitched  2500  times  as  many 
beats,  or  some  ten  or  eleven  octaves,  above  the  other's;  and  the 
same  comparison,  or  the  same  contrast,  will  apply  to  the  tympanic 
membranes  by  which  the  vibrations  are  received.  But  our  own 
perception  of  musical  notes  only  reaches  to  4000  vibrations  per 
second,  or  thereby;  a  squeaking  mouse  or  bat  is  heard  by  few,  and 
to  vibrations  of  10,000  per  second  we  are  all  of  us  stone-deaf. 
Structure  apart,  mere  size  is  enough  to  give  the  lesser  birds  and 
beasts  a  music  quite  different  to  our  own:  the  humming-bird,  for 
aught  we  know,  may  be  singing  all  day  long.  A  minute  insect  may 
utter  and  receive  vibrations  of  prodigious  rapidity;  even  its  little 
wings  may  beat  hundreds  of  times  a  second*.  Far  more  things 
happen  to  it  in  a  second  than  to  us ;  a  thousandth  part  of  a  second 
is  no  longer  neghgible,  and  time  itself  seems  to  run  a  different  course 
to  ours. 

The  eye  and  its  retinal  elements  have  ranges  of  magnitude  and 
Hmitations  of  magnitude  of  their  own.  A  big  dog's  eye  is  hardly 
bigger  than  a  little  dog's;  a  squirrel's  is  much  larger,  propor- 
tionately, than  an  elephant's;  and  a  robin's  is  but  little  less  than 
a  pigeon's  or  a  crow's.  For  the  rods  and  cones  do  not  vary  with 
the  size  of  the  animal,  but  have  their  dimensions  optically  limited 
by  the  interference-patterns  of  the  waves  of  light,  which  set  bounds 
to  the  production  of  clear  retinal  images.  True,  the  larger  animal 
may  want  a  larger  field  of  view ;  but  this  makeg  little  difference,  for 
but  a  small  area  of  the  retina  is  ever  needed  or  used.  The  eye,  in 
short,  can  never  be  very  small  and  need  never  be  very  big;  it  has 
its  own  conditions  and  limitations  apart  from  the  size  of  the  animal. 
But  the  insect's  eye  tells  another  story.  If  a  fly  had  an  eye  like 
ours,  the  pupil  would  be  so  small  that  diffraction  would  render  a 
clear  image  impossible.     The  only  alternative  is  to  unite  a  number 

*  The  wing-beats  are  said  to  be  as  follows:  dragonfly  28  per  sec,  bee  190, 
housefly  330;   cf.  Erhard,  Verh.  d.  d.  zool.  Gesellsch.  1913,  p.  206. 


54  ON  MAGNITUDE  [ch. 

of  small  and  optically  isolated  simple  eyes  into  a  compound  eye, 
and  in  the  insect  Nature  adopts  this  alternative  possibiHty*. 

Our  range  of  vision  is  limited  to  a  bare  octave  of  "luminous" 
waves,  which  is  a  considerable  part  of  the  whole  range  of  Hght-heat 
rays  emitted  by  the  sun;  the  sun's  rays  extend  into  the  ultra-violet 
for  another  half-octave  or  more,  but  the  rays  to  which  our  eyes  are 
sensitive  are  just  those  which  pass  with  the  least  absorption  through 
a  watery  medium.  Some  ancient  vertebrate  may  have  learned  to 
see  in  an  ocean  which  let  a  certain  part  of  the  sun's  whole  radiation 
through,  which  part  is  our  part  still ;  or  perhaps  the  watery  media 
of  the  eye  itself  account  sufficiently  for  the  selective  filtration.  In 
either  case,  the  dimensions  of  the  retinal  elements  are  so  closely 
related  to  the  wave-lengths  of  light  (or  to  their  interference  patterns) 
that  we  have  good  reason  to  look  upon  the  retina  as  perfect  of  its 
kind,  within  the  hmits  which  the  properties  of  hght  itself  impose; 
and  this  perfection  is  further  illustrated  by  the  fact  that  a  few 
light-quanta,  perhaps  a  single  one,  suffice  to  produce  a  sensation  "j". 
The  hard  eyes  of  insects  are  sensitive  over  a  wider  range.  The  bee 
has  two  visual  optima,  one  coincident  with  our  own,  the  other  and 
principal  one  high  up  in  the  ultra-violet  J.  And  with  the  latt^er  the 
bee  is  able  to  see  that  ultra-violet  which  is  so  well  reflected  by  many 
flowers  that  flower-photographs  have  been  taken  through  a  filter 
which  passes  these  but  transmits  no  other  rays§. 

When  we  talk  of  hght,  and  of  magnitudes  whose  order  is  that  of 
a  wave-length  of  hght,  the  subtle  phenomenon  of  colour  is  near  at 
hand.  The  hues  of  hving  things  are  due  to  sundry  causes;  where 
they  come  from  chemical  pigmentation  they  are  outside  our  theme, 
but  oftentimes  there  is  no  pigment  at  all,  save  perhaps  as  a  screen 
or  background,  and  the  tints  are  those  proper  to  a  scale  of  wave- 
lengths or  range  of  magnitude.  In  birds  these  "optical  colours" 
are  of  two  chief  kinds.     One  kind  include  certain  vivid  blues,  the 

*  Cf.  C.  J.  van  der  Horst,  The  optics  of  the  insect  eye,  Acta  Zoolog.  1933, 
p.  108. 

t  Cf.  Niels  Bohr,  in  Nature,  April  1,  1933,  p.  457.  Also  J.  Joly,  Proc.  R.S.  (B), 
xcn,  p.  222,  1921. 

f  L.  M.  Bertholf,  Reactions  of  the  honey-bee  to  light,  Journ.  of  Agric.  Res. 
XLm,  p.  379;   xliv,  p.  763,  1931. 

§  A.  Kuhn,  Ueber  den  Farbensinn  der  Bienen,  Ztschr.  d.  vergl.  Physiol,  v, 
pp.  762-800,  1927;  cf.  F.  K.  Richtmeyer,  Reflection  of  ultra-violet  by  flowers, 
J  num.  Optical  Soc.  Amer.  vii,  pp.  151-168,  1923;   etc. 


II]  OF  LIGHT  AND  COLOUR  55 

blue  of  a  blue  jay,  an  Indian  roller  or  a  macaw;  to  the  other  belong 
the  iridescent  hues  of  mother-of-pearl,  of  the  humming-bird,  the 
peacock  and  the  dove:  for  the  dove's  grey  breast  shews  many 
colours  yet  contains  but  one — colores  inesse  plures  nee  esse  plus  uno, 
as  Cicero  said.  The  jay's  blue  feather  shews  a  layer  of  enamel-like 
cells  beneath  a  thin  horny  cuticle,  and  the  cell-walls  are  spongy 
with  innumerable  tiny  air-filled  pores.  These  are  about  0-3 /it  in 
diameter,  in  some  birds  even  a  little  less,  and  so  are  not  far  from 
the  hmits  of  microscopic  vision.  A  deeper  layer  carries  dark-brown 
pigment,  but  there  is  no  blue  pigment  at  all;  if  the  feather  be  dipped 
in  a  fluid  of  refractive  index  equal  to  its  own,  the  blue  utterly 
disappears,  to  reappear  when  the  feather  dries.  This  blue  is  like 
the  colour  of  the  sky;  it  is  ''Tyndall's  blue,"  such  as  is  displayed 
by  turbid  media,  cloudy  with  dust-motes  or  tiny  bubbles  of  a  size 
comparable  to  the  wave-lengths  of  the  blue  end  of  the  spectrum. 
The  longer  waves  of  red  or  yellow  pass  through,  the  shorter  violet 
rays  are  reflected  or  scattered;  the  intensity  of  the  blue  depends 
on  the  size  and  concentration  of  the  particles,  while  the  dark  pigment- 
screen  enhances  the  effect. 

Rainbow  hues  are  more  subtle  and  more  complicated ;  but  in  the 
peacock  and  the  humming-bird  we  know  for  certain*  that  the 
colours  are  those  of  Newton's  rings,  and  are  produced  by  thin  plates 
or  films  covering  the  barbules  of  the  feather.  The  colours  are  such 
as  are  shewn  by  films  about  J  [x  thick,  more  or  less ;  they  change 
towards  the  blue  end  of  the  spectrum  as  thei  hght  falls  more  and 
more  obliquely;  or  towards  the  red  end  if  you  soak  the  feather 
and  cause  the  thin  plates  to  swell.  The  barbules  of  the  peacock's 
feather  are  broad  and  flat,  smooth  and  shiny,  and  their  cuticular 
layer  sphts  into  three  very  thin  transparent  films,  hardly  more  than 
1  jjL  thick,  all  three  together.  The  gorgeous  tints  of  the  humming- 
birds have  had  their  places  in  Newton's  scale  defined,  and  the 
changes  which  they  exhibit  at  varying  incidence  have  been  predicted 

*  Rayleigh,  Phil.  Mag.  (6),  xxxvii,  p.  98,  1919.  For  a  review  of  the  whole 
subject,  and  a  discussion  of  its  many  difficulties,  see  H.  Onslow,  On  a  periodic 
structure  in  many  insect  scales,  etc.,  Phil.  Trans.  (B),  ccxi,  pp.  1-74,  1921; 
also  C.  W.  Mason,  Journ.  Physic.  Chemistry,  xxvii,  xxx,  xxxi,  1923-25-27; 
F.  Suffert,  Zeitschr.  f.  Morph.  u.  Oekol.  d.  Tiere,  i,  pp.  171-306,  1924  (scales  of 
butterflies);  also  B.  Reusch  and  Th.  Elsasser  in  Journ.  f.  Ornithologie,  lxxiti, 
1925;   etc. 


56  ON  MAGNITUDE  [ch. 

and  explained.  The  thickness  of  each  film  lies  on  the  very  limit  of 
microscopic  vision,  and  the  least  change  or  irregularity  in  this 
minute  dimension  would  throw  the  whole  display  of  colour  out  of 
gear.  No  phenomenon  of  organic  magnitude  is  more  striking  than 
this  constancy  of  size;  none  more  remarkable  than  that  these  fine 
lamellae  should  have  their  tenuity  so  sharply  defined,  so  uniform 
in  feather  after  feather,  so  identical  in  all  the  individuals  of  a  species, 
so  constant  from  one  generation  to  another. 

A  simpler  phenomenon,  and  one  which  is  visible  throughout  the 
whole  field  of  morphology,  is  the  tendency  (referable  doubtless  in 
each  case  to  some  definite  physical  cause)  for  mere  bodily  surface 
to  keep  pace  with  volume,  through  some  alteration  of  its  form.  The 
development  of  villi  on  the  lining  of  the  intestine  (which  increase 
its  surface  much  as  we  enlarge  the  effective  surface  of  a  bath-towel), 
the  various  valvular  folds  of  the  intestinal  lining,  including  the 
remarkable  "spiral  valve"  of  the  shark's  gut,  the  lobulation  of  the 
kidney  in  large  animals*,  the  vast  increase  of  respiratory  surface  in 
the  air-sacs  and  alveoli  of  the  lung,  the  development  of  gills  in  the 
larger  Crustacea  and  worms  though  the  general  surface  of  the  body 
suffices  for  respiration  in  the  smaller  species — all  these  and  many 
more  are  cases  in  which  a  more  or  less  constant  ratio  tends  to  be 
maintained  between  mass  and  surface,  which  ratio  would  have  been 
more  and  more  departed  from  with  increasing  size,  had  it  not  been 
for  such  alteration  of  surface-form  f.  A  leafy  wood,  a  grassy  sward, 
a  piece  of  sponge,  a  reef  of  coral,  are  all  instances  of  a  hke  pheno- 
menon. In  fact,  a  deal  of  evolution  is  involved  in  keeping  due 
balance  between  surface  and  mass  as  growth  goes  on. 

In  the  case  of  very  small  animals,  and  of  individual  cells,  the 
principle  becomes  especially  important,  in  consequence  of  the 
molecular  forces  whose  resultant  action  is  limited  to  the  superficial 
layer.  In  the  cases  just  mentioned,  action  is  facilitated  by  increase 
of  surface :  diffusion,  for  instance,  of  nutrient  liquids  or  respiratory 
gases  is  rendered  more  rapid  by  the  greater  area  of  surface;    but 

*  Cf.  R.  Anthony,  C.R.  clxix,  p.  1174,  1919,  etc.  Cf.  also  A.  Putter,  Studien 
uber  physiologische  Ahnlichkeit,  Pfluger's  Archiv,  clxviii,  pp.  209-246,  1917. 

t  For  various  calculations  of  the  increase  of  surface  due  to  histological  and 
anatomical  subdivision,  see  E.  Babak,  Ueber  die  Oberflachenentwickelung  bei 
Organismen,  Biol.  Centralbl.  xxx,  pp.  225-239,  257-267,  1910. 


II]  OF  SURFACE  AND  VOLUME  57 

there  are  other  cases  in  which  the  ratio  of  surface  to  mass  may 
change  the  whole  condition  of  the  system.  Iron  rusts  when  exposed 
to  moist  air,  but  it  rusts  ever  so  much  faster,  and  is  soon  eaten  away, 
if  the  iron  be  first  reduced  to  a  heap  of  small  filings ;  this  is  a  mere 
difference  of  degree.  But  the  spherical  surface  of  the  rain-drop 
and  the  spherical  surface  of  the  ocean  (though  both  happen  to  be 
ahke  in  mathematical  form)  are  two  totally  different  phenomena, 
the  one  due  to  surface-energy,  and  the  other  to  that  form  of  mass- 
energy  which  we  ascribe  to  gravity.  The  contrast  is  still  more 
clearly  seen  in  the  case  of  waves:  for  the  little  ripple,  whose  form 
and  manner  of  propagation  are  governed  by  surface-tension,  is 
found  to  travel  with  a  velocity  which  is  inversely  as  the  square 
root  of  its  length;  while  the  ordinary  big  waves,  controlled  by 
gravitation,  have  a  velocity  directly  proportional  to  the  square  root 
of  their  wave-length.  In  hke  manner  we  shall  find  that  the  form 
of  all  very  small  organisms  is  independent  of  gravity,  and  largely 
if  not  mainly  due  to  the  force  of  surface-tension:  either  as  the 
direct  result  of  the  continued  action  of  surface-tension  on  the 
semi-fluid  body,  or  else  as  the  result  of  its  action  at  a  prior  stage 
of  development,  in  bringing  about  a  form  which  subsequent  chemical 
changes  have  rendered  rigid  and  lasting.  In  either  case,  we  shall 
find  a  great  tendency  in  small  organisms  to  assume  either  the 
spherical  form  or  other  simple  forms  related  to  ordinary  inanimate 
surface-tension  phenomena,  which  forms  do  not  recur  in  the 
external  morphology  of  large  animals. 

Now  this  is  a  very  important  matter,  and  is  a  notable  illustration 
of  that  principle  of  simihtude  which  we  have  already  discussed  in 
regard  to  several  of  its  manifestations.  We  are  coming  to  a  con- 
clusion which  will  affect  the  whole  course  of  our  argument  throughout 
this  book,  namely  that  there  is  an  essential  difference  in  kind 
between  the  phenomena  of  form  in  the  larger  and  the  smaller 
organisms.  I  have  called  this  book  a  study  of  Growth  and  Fonn, 
because  in  the  most  familiar  illustrations  of  organic  form,  as  in  our 
own  bodies  for  example,  these  two  factors  are  inseparably  asso- 
ciated, and  because  we  are  here  justified  in  thinking  of  form  as  the 
direct  resultant  and  consequence  of  growth:  of  growth,  whose 
varying  rate  in  one  direction  or  another  has  produced,  by  its  gradual 
and  unequal  increments,  the  successive  stages  of  development  and 


58  ON  MAGNITUDE  [ch. 

the  final  configuration  of  the  whole  material  structure.  But  it  is 
by  no  means  true  that  form  and  growth  are  in  this  direct  and  simple 
fashion  correlative  or  complementary  in  the  case  of  minute  portions 
of  living  matter.  For  in  the  smaller  organisms,  and  in  the  indi- 
vidual cells  of  the  larger,  we  have  reached  an  order  of  magnitude 
in  which  the  intermolecular  forces  strive  under  favourable  conditions 
with,  and  at  length  altogether  outweigh,  the  force  of  gravity,  and 
also  those  other  forces  leading  to  movements  of  convection  which 
are  the  prevailing  factors  in  the  larger  material  aggregate. 

However,  we  shall  require  to  deal  more  fully  with  this  matter  in 
our  discussion  of  the  rate  of  growth,  and  we  may  leave  it  mean- 
while, in  order  to  deal  with  other  matters  more  or  less  directly 
concerned  with  the  magnitude  of  the  cell. 

The  hving  cell  is  a  very  complex  field  of  energy,  and  of  energy 
of  many  kinds,  of  which  surface-energy  is  not  the  least.  Now  the 
whole  surface-energy  of  the  cell  is  by  no  means  restricted  to  its 
outer  surface;  for  the  cell  is  a  very  heterogeneous  structure,  and  all 
its  protoplasmic  alveoli  and  other  visible  (as  well  as  invisible)  hetero- 
geneities make  up  a  great  system  of  internal  surfaces,  at  every  part 
of  which  one  "phase"  comes  in  contact  with  another  "phase,"  and 
surface-energy  is  manifested  accordingly.  But  still,  the  external 
surface  is  a  definite  portion  of  the  system,  with  a  definite  "phase" 
of  its  own,  and  however  little  we  may  know  of  the  distribution  of 
the  total  energy  of  the  system,  it  is  at  least  plain  that  the  conditions 
which  favour  equihbrium  will  be  greatly  altered  by  the  changed 
ratio  of  external  surface  to  mass  which  a  mere  change  of  magnitude 
produces  in  the  cell.  In  short,  the  phenomenon  of  division  of  the 
growing  cell,  however  it  be  brought  about,  will  be  precisely  what 
is  wanted  to  keep  fairly  constant  the  ratio  between  surface  and 
mass,  and  to  retain  or  restore  the  balance  between  surface-energy 
and  the  other  forces  of  the  system*.  But  when  a  germ-cell  divides 
or  "segments"  into  two,  it  does  not  increase  in  mass;  at  least  if 
there  be  some  shght  alleged  tendency  for  the  egg  to  increase  in 

*  Certain  cells  of  the  cucumber  were  found  to  divide  when  they  had  grown  to 
a  volume  half  as  large  again  as  that  of  the  "resting  cells."  Thus  the  volumes 
of  resting,  dividing  and  daughter  cells  were  as  1:1-5:  0-75;  and  their  surfaces, 
being  as  the  power  2/3  of  these  figures,  were,  roughly,  as  1:1-3:  0-8.  The  ratio 
of  SjV  was  then  as  1  :  0-9  :  1-1,  or  much  nearer  equality.  Cf.  F.  T.  Lewis,  Anat. 
Record,  xlvii,  pp.  59-99,  1930. 


II]  OF  THE  SIZE  OF  DROPS  59 

mass  or  volume  during  segmentation  it  is  very  slight  indeed, 
generally  imperceptible,  and  wholly  denied  by  some*.  The  growth 
or  development  of  the  egg  from  a  one-celled  stage  to  stages  of  two 
or  many  cells  is  thus  a  somewhat  pecuhar  kind  of  growth;  it  is 
growth  limited  to  change  of  form  and  increase  of  surface,  unaccom- 
panied by  growth  in  volume  or  in  mass.  In  the  case  of  a  soap-bubble, 
by  the  way,  if  it  divide  into  two  bubbles  the  volume  is  actually 
diminished,  while  the  surface-area  is  greatly  increased! ;  the  diminution 
being  due  to  a  cause  which  we  shall  have  to  study  later,  namely  to 
the  increased  pressure  due  to  the  greater  curvature  of  the  smaller 
bubbles. 

An  immediate  and  remarkable  result  of  the  principles  just 
described  is  a  tendency  on  the  part  of  all  cells,  according  to  their 
kind,  to  vary  but  little  about  a  certain  mean  size,  and  to  have  in 
fact  certain  absolute  limitations  of  magnitude.  The  diameter  of  a 
large  parenchymatous  cell  is  perhaps  tenfold  that  of  a  httle  one; 
but  the  tallest  phanerogams  are  ten  thousand  times  the  height  of 
the  least.  In  short,  Nature  has  her  materials  of  predeterminate 
dimensions,  and  keeps  to  the  same  bricks  whether  she  build  a  great 
house  or  a  small.  Even  ordinary  drops  tend  towards  a  certain 
fixed  size,  which  size  is  a  function  of  the  surface-tension,  and  may 
be  used  (as  Quincke  used  it)  as  a  measure  thereof.  In  a  shower  of 
rain  the  principle  is  curiously  illustrated,  as  Wilding  Roller  and 
V.  Bjerknes  tell  us.  The  drops  are  of  graded  sizes,  each  twice  as  big 
as  another,  beginning  with  the  minute  and  uniform  droplets  of  an 
impalpable  mist.  They  rotate  as  they  fall,  and  if  two  rotate  in 
contrary  directions  they  draw  together  and  presently  coalesce;  but 
this  only  happens  when  two  drops  are  faUing  side  by  side,  and  since 
the  rate  of  fall  depends  on  the  size  it  always  is  a  pair  of  coequal 
drops  which  so  meet,  approach  and  join  together.  A  supreme 
instance  of  constancy  or  quasi-constancy  of  size,  remote  from  but 
yet  analogous  to  the  size-hmitation  of  a  rain-drop  or  a  cell,  is  the 
fact  that  the  stars  of  heaven  (however  else  one  diifereth  from 
another),  and  even  the  nebulae  themselves,  are  all  wellnigh  co-equal 
in  mass.     Gravity  draws  matter  together,  condensing  it  into  a  world 

*  Though  the  entire  egg  is  not  increasing  in  mass,  that  is  not  to  say  that  its  living 
protoplasm  is  not  increasing  all  the  while  at  the  expense  of  the  reserve  material, 
t  Cf.  P.  G.  Tait,  Proc.  E.S.E.  v,  1866  and  vi,  1868. 


60  ON  MAGNITUDE  [ch. 

or  into  a  star;  but  ethereal  pressure  is  an  opponent  force  leading 
to  disruption,  negligible  on  the  small  scale  but  potent  on  the  large. 
High  up  in  the  scale  of  magnitude,  from  about  10^^  to  10^^  grams 
of  matter,  these  two  great  cosmic  forces  balance  one  another;  and 
all  the  magnitudes  of  all  the  stars  he  within  or  hard  by  these  narrow 
limits. 

In  the  hving  cell,  Sachs  pointed  out  (in  1895)  that  there  is  a 
tendency  for  each  nucleus  to  gather  around  itself  a  certain  definite 
amount  of  protoplasm*.  Drieschf,  a  httle  later,  found  it  possible, 
by  artificial  subdivision  of  the  egg,  to  rear  dwarf  sea-urchin  larvae, 
one-half,  one-quarter  or  even  one-eighth  of  their  usual  size;  which 
dwarf  larvae  were  composed  of  only  a  half,  a  quarter  or  an  eighth 
of  the  normal  number  of  cells.  These  observations  have  been  often 
repeated  and  amply  confirmed :  and  Loeb  found  the  sea-urchin  eggs 
capable  of  reduction  to  a  certain  size,  but  no  further. 

In  the  development  of  Crepidula  (an  American  "shpper-Hmpet," 
now  much  at  home  on  our  oyster-beds),  ConklinJ  has  succeeded  in 
rearing  dwarf  and  giant  individuals,  of  which  the  latter  may  be 
five-and-twenty  times  as  big  as  the  former.  But  the  individual 
cells,  of  skin,  gut,  liver,  muscle  and  other  tissues,  are  just  the  same 
size  in  one  as  in  the  other,  in  dwarf  and  in  giant  §.     In  like  manner 

*  Physiologische  Notizen  (9),  p.  425,  1895.  Cf.  Amelung,  Flora,  1893;  Stras- 
biirger,  Ueber  die  Wirkungssphare  der  Kerne  und  die  Zellgrosse,  Histol.  Beitr.  (5), 
pp.  95-129,  1893;  R.  Hertwig,  Ueber  Korrelation  von  Zell-  und  Kerngrosse 
(Kernplasraarelation),  Biol.  Centralbl.  xvm,  pp.  49-62,  108-119,  1903;  G.  Levi 
and  T.  Terni,  Le  variazioni  dell'  indice  plasmatico-nucleare  durante  1'  intercinesi, 
Arch.  Hal.  di  Anat.  x,  p.  545,  1911;  also  E.  le  Breton  and  G.  Schaeffer,  Variations 
hiuchimiqiies  du  rapport  nucleo-plasmatique,  Strasburg,  1923. 

t  Arch.f.  Entw.  Mech.  iv,  1898,  pp.  75,  247. 

X  E.  G.  Conklin,  Cell-size  and  nuclear  size,  Journ.  Exp.  Zool.  xii,  pp.  1-98, 
1912;  Body-size  and  cell-size,  Journ.  of  Morphol.  xxiii,  pp.  159-188,  1912.  Cf. 
M.  Popoff,  Ueber  die  Zellgrosse,  Arch.f.  Zellforschung,  iii,  1909. 

§  Thus  the  fibres  of  the  crystalline  lens  are  of  the  same  size  in  large  and  small 
dogs,  Rabl,  Z.f.  w.  Z.  lxvii,  1899.  Cf.  {int.  al.)  Pearson,  On  the  size  of  the  blood- 
corpuscles  in  Rana,  Biometrika,  vi,  p.  403,  1909.  Dr  Thomas  Young  caught  sight 
of  the  phenomenon  early  in  last  century:  "The  solid  particles  of  the  blood  do  not 
by  any  means  vary  in  magnitude  in  the  same  ratio  with  the  bulk  of  the  animal," 
Natural  Philosophy,  ed.  1845,  p.  466;  and  Leeuwenhoek  and  Stephen  Hales 
were  aware  of  it  nearly  two  hundred  years  before.  Leeuwenhoek  indeed  had 
a  very  good  idea  of  the  size  of  a  human  blood-corpuscle,  and  was  in  the  habit  of 
using  its  diameter — about  1/3000  of  an  inch — as  a  standard  of  comparison.  But 
though  the  blood-corpuscles  shew  no  relation  of  magnitude  to  the  size  of  the 
animal,  they  are  related  without  doubt  to  its  activity;    for  the  corpuscles  in  the 


OF  THE  SIZE  OF  CELLS 


61 


the  leaf-cells  are  found  to  be  of  the  same  size  in  an  ordinary  water- 
lily,  in  the  great  Victoria  regia,  and  in  the  still  hiiger  leaf,  nearly 
3  metres  long,  of  Euryale  ferox  in  Japan*.  Driesch  has  laid  par- 
ticular stress  upon  this  principle  of  a  "fixed  cell-size,"  which  has, 
however,  its  own  limitations  and  exceptions.  Among  these  excep- 
tions, or  apparent  exceptions,  are  the  giant  frond-like  cell  of  a 
Caulerpa  or  the  great  undivided  plasmodium  of  a  Myxomycete. 
The  flattening  of  the  one  and  the  branching  of  the  other  serve  (or 
help)  to  increase  the  ratio  of  surface  to  content,  the  nuclei  tend  to 
multiply,  and  streaming  currents  keep  the  interior  and  exterior  of 
the  mass  in  touch  with  one  another. 


j^ 


Rabbit 


Man  Dog 

Fig.  3.     Motor  ganglion-cells,  from  the  cervical  spinal  cord. 
From  Minot,  after  Irving  Hardesty. 

We  get  a  good  and  even  a  famiUar  illustration  of  the  principle 
of  size-hmitation  in  comparing  the  brain-cells  or  ganghon-cells, 
whether  of  the  lower  or  of  the  higher  animals  f .  In  Fig.  3  we  shew 
certain  identical  nerve-cells  from  various  mammals,  from  mouse  to 
elephant,  all  drawn  to  the  same  scale  of  magnification ;  and  we  see 
that  they  are  all  of  much  the  same  order  of  magnitude.  The  nerve- 
cell  of  the  elephant  is  about  twice  that  of  the  mouse  in  linear 

sluggish  Amphibia  are  much  the  largest  known  to  us,  while  the  smallest  are  found 
among  the  deer  and  other  agile  and  speedy  animals  (cf.  Gulliver,  P.Z.S.  1875, 
p.  474,  etc.).  This  correlation  is  explained  by  the  surface  condensation  or 
adsorption  of  oxygen  in  the  blood-corpuscles,  a  process  greatly  facilitated  and 
intensified  by  the  increase  of  surface  due  to  their  minuteness. 

*  Okada  and  Yomosuke,  in  Sci.  Rep.  Tohoku  Univ.  iii,  pp.  271-278,  1928. 

t  Cf.  P.  Enriques,  La  forma  eome  funzione  della  grandezza :  Ricerche  sui  gangli 
nervosi  degli  invertebrati,  Arch.  f.  Entw.  Mech.  xxv,  p.  655,  1907-8. 


62  ON  MAGNITUDE  [ch. 

dimensions,  and  therefore  about  eight  times  greater  in  volume  or 
in  mass.  But  making  due  allowance  for  difference  of  shape,  the 
linear  dimensions  of  the  elephant  are  to  those  of  the  mouse  as  not 
less  than  one  to  fifty ;  and  the  bulk  of  the  larger  animal  is  something 
like  125,000  times  that  of  the  less.  It  follows,  if  the  size  of  the 
nerve-cells  are  as  eight  to  one,  that,  in  corresponding  parts  of  the 
nervous  system,  there  are  more  than  15,000  times  as  many  individual 
cells  in  one  animal  as  in  the  other.  In  short  we  may  (with  Enriques) 
lay  it  down  as  a  general  law  that  among  animals,  large  or  small,  the 
gangUon-cells  vary  in  size  within  narrow  limits;  and  that,  amidst 
all  the  great  variety  of  structure  observed  in  the  nervous  system 
of  different  classes  of  animals,  it  is  always  found  that  the  smaller 
species  have  simpler  gangha  than  the  larger,  that  is  to  say  ganglia 
containing  a  smaller  number  of  cellular  elements*.  The  bepiing  of 
such  facts  as  this  upon  the  cell-theory  in  general  is  not  to  be  dis- 
regarded; and  the  warning  is  especially  clear  against  exaggerated 
attempts  to  correlate  physiological  processes  with  the  visible 
mechanism  of  associated  cells,  rather  than  with  the  system  of 
energies,  or  the  field  of  force,  which  is  associated  with  them.  For 
the  life  of  the  body  is  more  than  the  swm  of  the  properties  of  the 
cells  of  which  it  is  composed:  as  Goethe  said,  "Das  Lebendige  ist 
zwar  in  Elemente  zerlegt,*  aber  man  kann  es  aus  diesen  nicht  wieder 
zusammenstellen  und  beleben." 

Among  certain  microscopic  organisms  such  as  the  Rotifera  (which 
have  the  least  average  size  and  the  narrowest  range  of  size  of  all 
the  Metazoa),  we  are  still  more  palpably  struck  by  the  small  number 
of  cells  which  go  to  constitute  a  usually  complex  organ,  such  as 
kidney,  stomach  or  ovary ;  we  can  sometimes  number  them  in  a  few 

*  While  the  difference  in  cell-volume  is  vastly  less  than  that  between  the 
volumes,  and  very  much  less  also  than  that  between  the  surfaces,  of  the  respective 
animals,  yet  there  is  a  certain  difference ;  and  this  it  has  been  attempted  to  correlate 
with  the  need  for  each  cell  in  the  many-celled  ganglion  of  the  larger  animal  to 
possess  a  more  complex  "exchange-system"  of  branches,  for  intercommunication 
with  its  more  numerous  neighbours.  Another  explanation  is  based  on  the  fact 
that,  while  such  cells  as  continue  to  divide  throughout  life  tend  to  uniformity  of 
size  in  all  mammals,  those  which  do  not  do  so,  and  in  particular  the  ganglion  cells, 
continue  to  grow,  and  their  size  becomes,  therefore,  a  function  of  the  duration  of 
life.  Cf.  G.  Levi,  Studii  sulla  grandezza  delle  cellule.  Arch.  Jtal.  di  Anat.  e  di 
Embrioloq.  v,  p.  291,  1906;  cf.  also  A.  Berezowski,  Studien  liber  die  Zellgrosse, 
Arch.  f.  Zellforsch.  v,  pp.  375-384,  1910. 


II]  OF  THE  LEAST  OF  ORGANISMS  63 

units,  in  place  of  the  many  thousands  which  make  up  such  an  organ 
in  larger,  if  not  alwayc  higher,  animals.  We  have  already  spoken 
of  the  Fairy-flies,  a  few  score  of  which  would  hardly  weigh  down 
one  of  the  larger  rotifers,  and  a  hundred  thousand  would  weigh  less 
than  one  honey-bee.  Their  form  is  complex  and  their  httle  bodies 
exquisitely  beautiful;  but  I  feel  sure  that  their  cells  are  few,  and 
their  organs  of  great  histological  simplicity.  These  considerations 
help,  I  think,  to  shew  that,  however  important  and  advantageous 
the  subdivision  of  the  tissues  into  cells  may  be  from  the  construc- 
tional, or  from  the  dynamic,  point  of  view,  the  phenomenon  has 
less  fundamental  importance  than  was  once,  and  is  often  still, 
assigned  to  it. 

Just  as  Sachs  shewed  there  was  a  hmit  to  the  amount  of  cytoplasm 
which  could  gather  round  a  nucleus,  so  Boveri  has  demonstrated 
that  the  nucleus  itself  has  its  own  hmitations  of  size,  and  that,  in 
cell-division  after  fertilisation,  each  new  nucleus  has  the  same  size 
as  its  parent  nucleus*;  we  may  nowadays  transfer  the  statement 
to  the  chromosomes.  It  may  be  that  a  bacterium  lacks  a  nucleus 
for  the  simple  reason  that  it  is  too  small  to  hold  one,  and  that  the 
same  is  true  of  such  small  plants  as  the  Cyanophyceae,  or  blue-green 
algae.  Even  a  chromatophore  with  its  "pyrenoids"  seems  to  be 
impossible  below  a  certain  sizej. 

Always  then,  there  are  reasons,  partly  physiological  but  in  large 
part  purely  physical,  which  define  or  regulate  the  magnitude  of  the 
organism  or  the  cell.  And  as  we  have  already  found  definite 
Hmitations  to  the  increase  in  magnitude  of  an  organism,  let  us  now 
enquire  whether  there  be  not  also  a  lower  limit  below  which  the 
very  existence  of  an  organism  becomes  impossible. 

*  Boveri,  Zellen.studien,  V:  Ueber  die  Abhangigkeit  der  Kerngrosse  und 
Zellenzahl  von  der  Chromosomenzahl  der  Ausgangszellen.  Jena,  1905.  Cf.  also 
{int.  al.)  H.  Voss,  Kerngrossenverhaltnisse  in  der  Leber  etc.,  Ztschr.f.  Zellforschung, 
VII,  pp.  187-200,  1928. 

t  The  size  of  the  nucleus  may  be  affected,  even  determined,  by  the  number  of 
chromosomes  it  contains.  There  are  giant  race*  of  Oenothera,  Primula  and  Solanum 
whose  cell-nuclei  contain  twice  the  normal  number  of  chromosomes,  and  a  dwarf 
race  of  a  little  freshwater  crustacean,  Cyclops,  has  half  the  usual  number.  The 
cytoplasm  in  turn  varies  with  the  amount  of  nuclear  matter,  the  whole  cell  is 
unusually  large  or  unusually  small;  and  in  these  exceptional  cases  we  see  a  direct 
relation  between  the  size  of  the  organism  and  the  size  of  the  cell.  Cf.  {int.  al.) 
R.  P.  Gregory,  Proc.  Camb.  Phil  Soc.  xv,  pp.  239-246,  1909;  F.  Keeble,  Journ. 
of  Genetics,  ii,  pp.  163-188,  1912.  • 


64  ON  MAGNITUDE  [ch. 

A  bacillus  of  ordinary  size  is,  say,  1  /x  in  length.  The  length  (or 
height)  of  a  man  is  about  a  million  and  three-quarter  times  as  great, 
i.e.  1-75  metres,  or  1-75  x  10^ /x;  and  the  mass  of  the  man  is  in  the 
neighbourhood  of  5  x  10^®  (five  million,  million,  milHon)  times 
greater  than  that  of  the  bacillus.  If  we  ask  whether  there  may  not 
exist  organisms  as  much  less  than  the  bacillus  as  the  bacillus  is  less 
than  the  man,  it  is  easy  to  reply  that  this  is  quite  impossible,  for  we 
are  rapidly  approaching  a  point  where  the  question  of  molecular 
dimensions,  and  of  the  ultimate  divisibility  of  matter,  obtrudes 
itself  as  a  crucial  factor  in  the  case.  Clerk  Maxwell  dealt  with  this 
matter  seventy  years  ago,  in  his  celebrated  article  Atom"^.  KoUi 
(or  Colley),  a  Russian  chemist,  declared  in  1893  that  the  head  of  a 
spermatozoon  could  hold  no  more  than  a  few  protein  molecules ;  and 
Errera,  ten  years  later,  discussed  the  same  topic  with  great  ingenuity  f. 
But  it  needs  no  elaborate  calculation  to  convince  us  that  the  smaller 
bacteria  or  micrococci  nearly  approach  the  smallest  magnitudes 
which  we  can  conceive  to  have  an  organised  structure.  A  few  small 
bacteria  are  the  smallest  of  visible  organisms,  and  a  minute  species 
associated  with  influenza,  B.  pneumosinter,  is  said  to  be  the  least 
of  them  all.  Its  size  is  of  the  order  of  0-1 /x,  or  rather  less;  and 
here  we  are  in  close  touch  with  the  utmost  limits  of  microscopic 
vision,  for  the  wave-lengths  of  visible  light  run  only  from  about 
400  to  700 m/x.  The  largest  of  the  bacteria,  B.  megatherium,  larger 
than  the  well-known  B.  anthracis  of  splenic  fever,  has  much  the 
same  proportion  to  the  least  as  an  elephant  to  a  guinea-pig  {. 

Size  of  body  is  no  mere  accident.  Man,  respiring  as  he  does, 
cannot  be  as  small  as  an  insect,  nor  vice  versa;  only  now  and  then, 
as  in  the  Goliath  beetle,  do  the  sizes  of  mouse  and  beetle  meet  and 
overlap.  The  descending  scale  of  mammals  stops  short  at  a  weight 
of  about  5  grams,  that  of  beetles  at  a  length  of  about  half  a  milli- 
metre, and  every  group  of  animals  has  its  upper  and  its  lower 
limitations  of  size.  So,  not  far  from  the  lower  limit  of  our  vision, 
does  the  long  series  of  bacteria  come  to  an  end.  There  remain  still 
smaller  particles  which  the  ultra-microscope  in  part  reveals;    and 

*  Encyclopaedia  Britannica,  9th  edition,  1875. 

f  Leo  Errera,  Siir  la  limite  de  la  petitesse  des  organismes,  BvlL  Soc.  Roy.  des 
Sc.  me'd.  et  nat.  de  Bruxelles,  1903;   Recueil  d'osnvres  {Physiologie  gen^rale),  p.  325. 

I  Cf.  A.  E.  Boycott,  The  transition  from  live  to  dead,  Proc.  R.  Soc.  of  Medicine, 
XXII  {Pathology),  pp.  55-69,  1928. 


II]  OF  MOLECULAR  MAGNITUDES  65 

here  or  hereabouts  are  said  to  come  the  so-called  viruses  or  "filter- 
passers,"  brought  within  our  ken  by  the  maladies,  such  as  hydro- 
phobia, or  foot-and-mouth  disease,  or  the  mosaic  diseases  of  tobacco 
and  potato,  to  which  they  give  rise.  These  minute  particles,  of  the 
order  of  one-tenth  the  diameter  of  our  smallest  bacteria,  have  no 
diffusible  contents,  no  included  water — whereby  they  differ  from 
every  living  thing.  They  appear  to  be  inert  colloidal  (or  even 
crystalloid)  aggregates  of  a  nucleo-protein,  of  perhaps  ten  times  the 
diameter  of  an  ordinary  protein-molecule,  and  not  much  larger  than 
the  giant  molecules  of  haemoglobin  or  haemocyanin  *. 

Bejerinck  called  such  a  virus  a  contagium  vivum;  "infective 
nucleo-protein"  is  a  newer  name.  We  have  stepped  down,  by  a 
single  step,  from  Hving  to  non-Hving  things,  from  bacterial  dimen- 
sions to  the  molecular  magnitudes  of  protein  chemistry.  And  we 
begin  to  suspect  that  the  virus-diseases  are  not  due  to  an  "organism, 
capable  of  physiological  reproduction  and  multiphcation,  but  to  a 
mere  specific  chemical  substance,  capable  of  catalysing  pre-existing 
materials  and  thereby  producing  more  and  more  molecules  hke 
itself.  The  spread  of  the  virus  in  a  plant  would  then  be  a  mere 
autocatalysis,  not  involving  the  transport  of  matter,  but  only  a 
progressive  change  of  state  in  substances  already  there  f." 

But,  after  all,  a  simple  tabulation  is  all  we  need  to  shew  how 
nearly  the  least  of  organisms  approach  to  molecular  magnitudes. 
The  same  table  will  suffice  to  shew  how  each  main  group  of  animals 
has  its  mean  and  characteristic  size,  and  a  range  on  either  side, 
sometimes  greater  and  sometimes  less. 

Our  table  of  magnitudes  is  no  mere  catalogue  of  isolated  facts, 
but  goes  deep  into  the  relation  between  the  creature  and  its  world. 
A  certain  range,  and  a-  narrow  one,  contains  mouse  and  elephant, 
and  all  whose  business  it  is  to  walk  and  run ;   this  is  our  own  world, 

*  Cf.  Svedberg,  Journ.  Am.  Chem.  Soc.  XLviir,  p.  30,  1926.  According  to  the 
Foot-and-Mouth  Disease  Research  Committee  {oth  Report,  1937),  the  foot-and- 
mouth  virus  has  a  diameter,  determined  by  graded  filters,  of  8-12m/i;  while 
Kenneth  Smith  and  W.  D.  MacClement  {Proc.  R.S.  (B),  cxxv,  p.  296,  1938)  calculate 
for  certain  others  a  diameter  of  no  more  than  4m^,  or  less  than  a  molecule  of 
haemocyanin. 

t  H.  H.  Dixon,  Croonian  lecture  on  the  transport  of  substances  in  plants, 
Proc.  R.S.  (B),  vol.  cxxv,  pp.  22,  23,  1938. 


66 


ON  MAGNITUDE 


[CH. 


with  whose  dimensions  our  hves,  our  hmbs,  our  senses  are  in  tune. 
The  great  whales  grow  out  of  this  range  by  throwing  the  burden 
of  their  bulk  upon  the  waiters;  the  dinosaurs  wallowed  in  the  swamp, 
and  the  hippopotamus,  the  sea-elephant  and  Steller's  great  sea-cow 
pass  or  passed  their  hves  in  the  rivers  or  the  sea.     The  things  which 


Linear  dimensions  of  organisms,  and  other  objects 


cm. 

(10,000  km.) 

10' 

A  quadrant  of  the  earth's  circumference 

(1000  km.) 

10« 
105 
10* 

Orkney  to  Land's  End 
Mount  Everest 

(km.) 

103 

102 

Giant  trees:   Sequoia 
Large  whale 
Basking  shark 

101 

Elephant;   ostrich;   man 

(metre) 

10" 
10-1 

Dog;  rat;  eagle 

Small  birds  and  mammals;   large  insects 

(cm.) 

10-2 

Small  insects;   minute  fish 

(mm.) 

10-3 

Minute  insects 

10-* 

Protozoa;   pollen -grains 

-  Cells 

10-5 

Large  bacteria;   human  blood-corpuscles 

(micron,  /z,) 

io-« 

10-' 

Minute  bacteria 

Limit  of  microscopic  vision 

io-« 

Viruses,  or  filter-passers                 r.  n    j        *•  i 
Giant  albuminoids,  casein,  etc.f    ^^^^^^  P^^^^^^^^ 
Starch-molecule 

(m/ii) 

io-» 

Water-molecule 

(Angstrom  unit) 

10-10 

fly  are  smaller  than  the  things  which  walk  and  run ;  the  flying  birds 
are  never  as  large  as  the  larger  mammals,  the  lesser  birds  and 
mammals  are  much  of  a  muchness,  but  insects  come  down  a  step 
in  the  scale  and  more.  The  lessening  influence  of  gravity  facilitates 
flight,  but  makes  it  less  easy  to  walk  and  run;  first  claws,  then 
hooks  and  suckers  and  glandular  hairs  help  to  secure  a  foothold, 


II]  OF  SCALES  OF  MAGNITUDE  67 

until  to  creep  upon  wall  or  ceiling  becomes  as  easy  as  to  walk  upon 
the  ground.  Fishes,  by  evading  gravity,  increase  their  range  of 
magnitude  both  above  and  below  that  of  terrestrial  animals.  Smaller 
than  all  these,  passing  out  of  our  range  of  vision  and  going  down  to 
the  least  dimensions  of  hving  things,  are  protozoa,  rotifers,  spores, 
pollen-grains*  and  bacteria.  All  save  the  largest  of  these  float 
rather  than  swim;  they  are  buoyed  up  by  air  or  water,  and  fall 
(as  Stokes's  law  explains)  with  exceeding  slowness. 

There  is  a  certain  narrow  range  of  magnitudes  where  (as  we  have 
partly  said)  gravity  and  surface  tension  become  comparable  forces, 
nicely  balanced  with  one  another.  Here  a  population  of  small 
plants  and  animals  not  only  dwell  in  the  surface  waters  but  are 
bound  to  the  surface  film  itself — the  whirligig  beetles  and  pond- 
skaters,  the  larvae  of  gnat  and  mosquito,  the  duckweeds  (Lemna), 
the  tiny  Wolffia,  and  Azolla;  even  in  mid-ocean,  one  small  insect 
(Halobates)  retains  this  singular  habitat.  It  would  be  a  long  story 
to  tell  the  various  ways  in  which  surface-tension  is  thus  taken  full 
advantage  of.  Gravitation  not  only  Hmits  the  magnitude  but 
controls  the  form  of  things.  With  the  help  of  gravity  the  quadruped 
has  its  back  and  its  belly,  and  its  limbs  upon  the  ground ;  its  freedom 
of  motion  in  a  plane  perpendicular  to  gravitational  force ;  its  sense 
of  fore-and-aft,  its  head  and  tail,  its  bilateral  symmetry.  Gravitation 
influences  both  our  bodies  and  our  minds.  We  owe  to  it  our  sense 
of  the  vertical,  our  knowledge  of  up-and-down;  our  conception  of 
the  horizontal  plane  on  which  we  stand,  and  our  discovery  of  two 
axes  therein,  related  to  the  vertical  as  to  one  another;  it  was  gravity 
which  taught  us  to  think  of  three-dimensional  space.  Our  archi- 
tecture is  controlled  by  gravity,  but  gravity  has  less  influence  over 
the  architecture  of  the  bee ;  a  bee  might  be  excused,  might  even  be 
commended,  if  it  referred  space  to  four  dimensions  instead  of  three !  t 
The  plant  has  its  root  and  its  stem",    but  about  this  vertical  or 

*  Pollen-grains,  like  protozoa,  have  a  considerable  range  of  magnitude.  The 
largest,  such  as  those  of  the  pumpkin,  are  about  200/x  in  diameter;  these  have  to 
be  carried  by  insects,  for  they  are  above  the  level  of  Stokes's  law,  and  no  longer 
float  upon  the  air.  The  smallest  pollen-grains,  such  as  those  of  the  forget-me-not, 
are  about  4^  /x  in  diameter  (Wodehouse). 

f  Corresponding,  that  is  to  say,  to  the  four  axes  which,  meeting  in  a  pfoint,  make 
co-equal  angles  (the  so-called  tetrahedral  angles)  one  with  another,  as  do  the  basal 
angles  of  the  honeycomb.     (See  below,  chap,  vu.) 


68  ON  MAGNITUDE  [ch. 

gravitational  axis  its  radiate  symmetry  remains,  undisturbed  by 
directional  polarity,  save  for  the  sun.  Among  animals,  radiate 
symmetry  is  confined  to  creatures  of  no  great  size ;  and  some  form 
or  degree  of  spherical  symmetry  becomes  the  rule  in  the  small  world 
of  the  protozoon — unless  gravity  resume  its  sway  through  the  added 
burden  of  a  shell.  The  creatures  which  swim,  walk  or  run,  fly, 
creep  or  float  are,  so  to  speak,  inhabitants  and  natural  proprietors 
of  as  many  distinct  and  all  but  separate  worlds.  Humming-bird 
and  hawkmoth  may,  once  in  a  way,  be  co-tenants  of  the  same 
world;  but  for  the  most  part  the  mammal,  the  bird,  the  fish,  the 
insect  and  the  small  life  of  the  sea,  not  only  have  their  zoological 
distinctions,  but  each  has  a  physical  universe  of  its  own.  The 
world  of  bacteria  is  yet  another  world  again,  and  so  is  the  world  of 
colloids ;  but  through  these  small  Lilliputs  we  pass  outside  the  range 
of  hving  things. 

What  we  call  mechanical  principles  apply  to  the  magnitudes 
among  which  we  are  at  home;  but  lesser  worlds  are  governed  by 
other  and  appropriate  physical  laws,  of  capillarity,  adsorption  and 
electric  charge.  There  are  other  worlds  at  the  far  other  end  of  the 
scale,  in  the  uttermost  depths  of  space,  whose  vast  magnitudes  lie 
within  a  narrow  range.  When  the  globular  star-clusters  are  plotted 
on  a  curve,  apparent  diameter  against  estimated  distance,  the 
curve  is  a  fair  approximation  to  a  rectangular  hyperbola;  which 
means  that,  to  the  same  rough  approximation,  the  actual  diameter 
is  identical  in  them  all*. 

It  is  a  remarkable  thing,  worth  pausing  to  reflect  on,  that  we  can 
pass  so  easily  and  in  a  dozen  fines  from  molecular  magnitudes  f  to 
the  dimensions  of  a  Sequoia  or  a  whale.  Addition  and  subtraction, 
the  old  arithmetic  of  the  Egyptians,  are  not  powerful  enough  for 
such  an  operation;  but  the  story  of  the  grains  of  wheat  upon  the 
chessboard  shewed  the  way,  and  Archimedes  and  Napier  elaborated 

*  See  Harlow  Shapley  and  A.  B.  Sayer,  The  angular  diameters  of  globular 
clusters,  Proc.  Nat.  Acad,  of  Sci.  xxi,  pp.  593-597,  1935.  The  same  is  approxi- 
mately true  of  the  spiral  nebulae  also. 

t  We  may  call  (after  Siedentopf  and  Zsigmondi)  the  smallest  visible  particles 
microns,  such  for  instance  as  small  bacteria,  or  the  fine  particles  of  gum-mastich 
in  suspension,  measuring  0-5  to  1-0/x;  sub-microns  are  those  revealed  by  the  ultra- 
microscope,  such  as  particles  of  colloid  gold  (2-15m/Lt),  or  starch-moleculea  (5m/x); 
amicrons,  under  Im^,  are  not  perceptible  by  either  method.  A  water-molecule 
measures,  probably,  about  0-1  m/i. 


II]  OF  THIN  FILMS  69 

the  arithmetic  of  multipHcation.  So  passing  up  and  down  by  easy- 
steps,  as  Archimedes  did  when  he  numbered  the  sands  of  the  sea, 
we  compare  the  magnitudes  of  the  great  beasts  and  the  small,  of 
che  atoms  of  which  they  are  made,  and  of  the  world  in  which  they 
dwell*. 

While  considerations  based  on  the  chemical  composition  of  the 
organism  have  taught  us  that  there  must  be  a  definite  lower  hmit 
to  its  magnitude,  other  considerations  of  a  purely  physical  kind  lead 
us  to  the  same  conclusion.  For  our  discussion  of  the  principle  of 
similitude  has  already  taught  us  that  long  before  we  reach  these 
all  but  infinitesimal  magnitudes  the  dwindling  organism  will  have 
experienced  great  changes  in  all  its  physical  relations,  and  must  at 
length  arrive  at  conditions  surely  incompatible  with  life,  or  what  we 
understand  as  life,  in  its  ordinary  development  and  manifestation. 

We  are  told,  for  instance,  that  the  powerful  force  of  surface-tension, 
or  capillarity,  begins  to  act  within  a  range  of  about  1/500,000  of  an 
inch,  or  say  0-05 />t.  A  soap  film,  or  a  film  of  oil  on  water,  may  be 
attenuated  to  far  less  magnitudes  than  this;  the  black  spots  on  a 
soap  bubble  are  known,  by  various  concordant  methods  of  measure- 
ment, to  be  only  about  6x  10~'  cm.,  or  about  6m/x  thick,  and  Lord 
Rayleigh  and  M.  Devaux  have  obtained  films  of  oil  of  2mjLt,  or  even 
1  m/x  in  thickness.  But  while  it  is  possible  for  a  fluid  film  to  exist 
of  these  molecular  dimensions,  it  is  certain  that  long  before  we 
reach  these  magnitudes  there  arise  conditions  of  which  we  have 
little  knowledge,  and  which  it  is  not  easy  to  imagine.  A  bacillus 
lives  in  a  world,  or  on  the  borders  of  a  world,  far  other  than  our 
own,  and  preconceptions  drawn  from  our  experience  are  not  valid 
there.  Even  among  inorganic,  non-living  bodies,  there  comes  a 
certain  grade  of  minuteness  at  which  the  ordinary  properties  become 
modified.  For  instance,  while  under  ordinary  circumstances  crystal- 
lisation starts  in  a  solution  about  a  minute  sohd  fragment  or  crystal 

*  Observe  that,  following  a  common  custom,  we  have  only  used  a  logarithmic 
scale  for  the  round  numbers  representing  powers  of  ten,  leaving  the  interspaces 
between  these  to  be  filled  up,  if  at  all,  by  ordinary  numbers.  There  is  nothing 
to  prevent  us  from  using  fractional  indices,  if  we  please,  throughout,  and  calling 
a  blood-corpuscle,  for  instance,  10~^"^  cm.  in  diameter,  a  man  lO^'^^  cm.  high,  or 
Sibbald's  Rorqual  lOi"*^  metres  long.  This  method,  implicit  in  that  of  Napier  of 
Merchiston,  was  first  set  forth  by  Wallis,  in  his  Arithmetica  infinitorum. 


70  ON  MAGNITUDE  [ch. 

of  the  salt,  Ostwald  has  shewn  that  we  may  have  particles  so  minute 
that  they  fail  to  serve  as  a  nucleus  for  crystalUsation — which  is  as 
much  as  to  say  that  they  are  too  small  to  have  the  form  and  pro- 
perties of  a  "crystal."  And  again,  in  his  thin  oil-films,  Lord 
Rayleigh  noted  the  striking  change  of  physical  properties  which 
ensues  when  the  film  becomes  attenuated  to  one,  or  something  less 
than  one,  close-packed  layer  of  molecules,  and  when,  in  short,  it  no 
longer  has  the  properties  of  matter  in  mass. 

These  attenuated  films  are  now  known  to  be  "monomolecular,"  the 
long-chain  molecules  of  the  fatty  acids  standing  close-packed,  like  the  cells 
of  a  honeycomb,  and  the  film  being  just  as  thick  as  the  molecules  are  long. 
A  recent  determination  makes  the  several  molecules  of  oleic,  palmitic  and 
stearic  acids  measure  10-4,  14-1  and  15- 1  cm.  in  length,  and  in  breadth  7-4, 
6-0  and  5-5  cm.,  all  by  10~^:  in  good  agreement  with  Lord  Rayleigh  and 
Devaux's  lowest  estimates  (F.  J.  Hill,  Phil.  Mag.  1929,  pp.  940-946).  But 
it  has  since  been  shewn  that  in  aliphatic  substances  the  long-chain  molecules 
are  not  erect,  but  inclined  to  the  plane  of  the  film ;  that  the  zig-zag  constitution 
of  the  molecules  permits  them  to  interlock,  so  giving  the  film  increased 
stability ;  and  that  the  interlock  may  be  by  means  of  a  first  or  second  zig-zag, 
the  measured  area  of  the  film  corresponding  precisely  to  these  two  dimorphic 
arrangements.  (Cf.  C.  G.  Lyons  and  E.  K.  Rideal,  Proc.  R.S.  (A),  cxxvin, 
pp.  468-473,  1930.)  The  film  may  be  lifted  on  to  a  polished  surface  of  metal, 
or  even  on  a  sheet  of  paper,  and  one  monomolecular  layer  so  added  to  another; 
even  the  complex  protein  molecule  can  be  unfolded  to  form  a  film  one  amino- 
acid  molecule  thick.  The  whole  subject  of  monomolecular  layers,  the  nature 
of  the  film,  whether  condensed,  expanded  or  gaseous,  its  astonishing  sensitive- 
ness to  the  least  impurities,  apd  the  manner  of  spreading  of  the  one  liquid 
over  the  other,  has  become  of  great  interest  and  importance  through  the  work 
of  Irving  Langmuir,  Devaux,  N.  K.  Adam  and  others,  and  throws  new  light 
on  the  whole  subject  of  molecular  magnitudes*. 

The  surface-tension  of  a  drop  (as  Laplace"  conceived  it)  is  the 
cumulative  effect,  the  statistical  average,  of  countless  molecular 
attractions,  but  we  are  now  entering  on  dimensions  where  the 
molecules  are  fewf.  The  free  surface-energy  of  a  body  begins  to 
vary  with  the  radius,  when  that  radius  is  of  an  order  comparable 
to  inter-molecular  distances;  and  the  whole  expression  for  such 
energy  tends  to  vanish  away  when  the  radius  of  the  drop  or  particle 
is  less  than  O-Olfx,  or  lOm^it.     The  quahties  and  properties  of  our 

*  Cf.  (int.  al.)  Adam,  Physics  and  Chemistry  of  Surfaces,  1930;  Irving  Langmuir, 
Proc.  R.S.  (A),  CLXX,  1939. 

t  See  a  very  interesting  paper  by  Fred  Vies,  Introduction  a  la  physique  bac- 
terienne,  Revue  Sclent.  11  juin  1921.  Cf.  also  N.  Rashevsky,  Zur  Theorie  d. 
spontanen  Teilung  von  mikroskopischen  Tropfen,  Ztschr.f.  Physik,  xlvi,  p.  578, 1928. 


II]  OF  MINUTE   MAGNITUDES  71 

particle  suffer  an  abrupt  change  here;  what  then  can  we  attribute, 
in  the  way  of  properties,  to  a  corpuscle  or  organism  as  small  or 
smaller  than,  say,  0-05  or  0-03 /x?  It  must,  in  all  probability,  be  a 
homogeneous  structureless  body,  composed  of  a  very  small  number 
of  albumenoid  or  other  molecules.  Its  vital  properties  and  functions 
must  be  extremely  limited;  its  specific  outward  characters,  even  if 
we  could  see  it,  must  be  nil;  its  osmotic  pressure  and  exchanges 
must  be  anomalous,  and  under  molecular  bombardment  they  may 
be  rudely  disturbed;  its  properties  can  be  Httle  more  than  those  of 
an  ion-laden  corpuscle,  enabling  it  to  perform  this  or  that  specific 
chemical  reaction,  to  effect  this  or  that  disturbing  influence,  or 
produce  this  or  that  pathogenic  effect.  Had  it  sensation,  its  ex- 
periences would  be  strange  indeed;  for  if  it  could  feel,  it  would  regard 
a  fall  in  temperature  as  a  movement  of  the  molecules  around,  and 
if  it  could  see  it  would  be  surrounded  with  light  of  many  shifting 
colours,  like  a  room  filled  with  rainbows. 

The  dimensions  of  a  cilium  are  of  such  an  order  that  its  substance 
is  mostly,  if  not  all,  under  the  pecuHar  conditions  of  a  surface-layer, 
and  surface-energy  is  bound  to  play  a  leading  part  in  cihary  action. 
A  cilium  or  flagellum  is  (as  it  seems  to  me)  a  portion  of  matter  in 
a  state  sui  generis,  with  properties  of  its  own,  just  as  the  film  and  the 
jet  have  theirs.  And  just  as  Savart  and  Plateau  have  told  us  about 
jets  and  films,  so  will  the  physicist  some  day  explain  the  properties 
of  the  cilium  and  flagellum.  It  is  certain  that  we  shall  never 
understand  these  remarkable  structures  so  long  as  we  magnify 
them  to  another  scale,  and  forget  that  new  and  pecuhar  physical 
properties  are  associated  with  the  scale  to  which  they  belong*. 

As  Clerk  Maxwell  put  it,  "molecular  science  sets  us  face  to  face 
with  physiological  theories.  It  forbids  the  physiologist  to  imagine 
that  structural  details  of  infinitely  small  dimensions  (such  as  Leibniz 
assumed,  one  within  another,  ad  infinitum)  can  furnish  an  explana- 
tion of  the  infinite  variety  which  exists  in  the  properties  and  functions 
of  the  most  minute  organisms."  And  for  this  reason  Maxwell 
reprobates,  with  not  undue  severity,  those  advocates  of  pangenesis 

*  The  cilia  on  the  gills  of  bivalve  molluscs  are  of  exceptional  size,  measuring 
from  say  20  to  120^  long.  They  are  thin  triangular  plates,  rather  than  filaments; 
they  are  from  4  to  lO/x  broad  at  the  base,  but  less  than  1/x  thick.  Cf.  D.  Atkins. 
Q.J. M.S.,  1938,  and  other  papers. 


72  ON  MAGNITUDE  [ch. 

and  similar  theories  of  heredity,  who  "would  place  a  whole  world 
of  wonders  within  a  body  so  small  and  so  devoid  of  visible  structure 
as  a  germ."  But  indeed  it  scarcely  needed  Maxwell's  criticism  to 
shew  forth  the  immense  physical  difficulties  of  Darwin's  theory  of 
pangenesis:  which,  after  all,  is  as  old  as  Democritus,  and  is  no  other 
than  that  Promethean  particula  undique  desecta  of  which  we  have 
read,  and  at  which  we  have  smiled,  in  our  Horace. 

There  are  many  other  ways  in  which,  when  we  make  a  long 
excursion  into  space,  we  find  our  ordinary  rules  of  physical  behaviour 
upset.  A  very  familiar  case,  analysed  by  Stokes,  is  that  the 
viscosity  of  the  surrounding  medium  has  a  relatively  powerful  effect 
upon  bodies  below  a  certain  size.  A  droplet  of  water,  a  thousandth 
of  an  inch  (25  ju,)  in  diameter,  cannot  fall  in  still  air  quicker  than 
about  an  inch  and  a  half  per  second;  as  its  size  decreases,  its 
resistance  varies  as  the  radius,  not  (as  with  larger  bodies)  as  the 
surface;  and  its  "critical"  or  terminal  velocity  varies  as  the 
square  of  the  radius,  or  as  the  surface  of  the  drop.  A  minute 
drop  in  a  misty  cloud  may  be  one-tenth  that  size,  and  will  fall  a 
hundred  times  slower,  say  an  inch  a  minute;  and  one  again  a  tenth 
of  this  diameter  (say  0-25 />t,  or  about  twice  as  big  as  a  small  micro- 
coccus) will  scarcely  fall  an  inch  in  two  hours*.  Not  only  do 
dust-particles,  spores  f  and  bacteria  fall,  by  reason  of  this  principle, 
very  slowly  through  the  air,  but  all  minute  bodies  meet  with  great 
proportionate  resistance  to  their  movements  through  a  fluid.  In 
salt  water  they  have  the  added  influence  of  a  larger  coefficient  of 
friction  than  in  fresh  J ;  and  even  such  comparatively  large  organisms 
as  the  diatoms  and  the  foraminifera,  laden  though  they  are  with  a 
heavy  shell  of  flint  or  lime,  seem  to  be  poised  in  the  waters  of  the 
ocean,  and  fall  with  exceeding  slowness. 

*  The  resistance  depends  on  the  radius  of  the  particle,  the  viscosity,  and  the 
rate  of  fall  ( V) ;  the  eifective  weight  by  which  this  resistance  is  to  be  overcome 
depends  on  gravity,  on  the  density  of  the  particle  compared  with  that  of  the 
medium,  and  on  the  mass,  which  varies  as  r^.  Resistance  =A;rF,  and  effective 
weight  =  )fcV;  when  these  two  equal  one  another  we  have  the  critical  or  terminal 
velocity,  and  Vccr^. 

t  A.  H.  R.  BuUer  found  the  spores  of  a  fungus  (CoUybia),  measuring  5x3/i, 
to  fall  at  the  rate  of  half  a  millimetre  per  second,  or  rather  more  than  an  inch 
a  minute;   Studies  on  Fungi,  1909. 

X  Cf.  W.  Krause,  Biol.  Centralbl.  i,  p.  578,  1881;  Fliigel,  Meteorol  Ztschr.  1881, 
p.  321. 


II]  OF  STOKES'S  LAW  73 

When  we  talk  of  one  thing  touching  another,  there  may  yet  be 
a  distance  between,  not  only  measurable  but  even  large  compared 
with  the  magnitudes  we  have  been  considering.  Two  polished 
plates  of  glass  or  steel  resting  on  one  another  are  still  about  4/x 
apart — the  average  size  of  the  smallest  dust;  and  when  all  dust- 
particles  are  sedulously  excluded,  the  one  plate  sinks  slowly  down 
to  within  O-S/jl  of  the  other,  an  apparent  separation  to  be  accounted 
for  by  minute  irregularities  of  the  polished  surfaces*. 

The  Brownian  movement  has  also  to  be  reckoned  with — that 
remarkable  phenomenon  studied  more  than  a  century  ago  by  Robert 
Brown  f,  Humboldt's /ac?7e  princeps  botanicorum,  and  discoverer  of 
the  nucleus  of  the  cell  J.  It  is  the  chief  of  those  fundamental 
phenomena  which  the  biologists  have  contributed,  or  helped  to 
contribute,  to  the  science  of  physics. 

The  quivering  motion,  accompanied  by  rotation  and  even  by 
translation,  manifested  by  the  fine  granular  particle  issuing  from  a 
crushed  pollen-grain,  and  which  Brown  proved  to  have  no  vital 
significance  but  to  be  manifested  by  all  minute  particles  whatsoever, 
was  for  many  years  unexplained.  Thirty  years  and  more  after  Brown 
wrote,  it  was  said  to  be  "due,  either  directly  to  some  calorical 
changes  continually  taking  place  in  the  fluid,  or  to  some  obscure 
chemical  action  between  the  solid  particles  and  the  fluid  which  is 
indirectly   promoted   by   heat§."     Soon    after   these    words    were 


*  Cf.  Hardy  and  Nottage,  Proc.  R.S.  (A),  cxxviii,  p.  209,  1928;  Baston  and 
Bowden,  ibid,  cxxxiv,  p.  404,  1931. 

t  A  Brief  Description  of  Microscopical  Observations.  .  .on  the  Particles  contained 
in  the  Pollen  of  Plants;  and  on  the  General  Existence  of  Active  Molecules  in  Organic 
and  Inorganic  Bodies,  London,  1828.  See  also  Edinb.  Netv  Philosoph.  Journ.  v, 
p.  358,  1828;  Edinb.  Journ.  of  Science,  i,  p.  314,  1829;  Ann.  Sc.  Nat.  xiv,  pp.  341- 
362,  1828;  etc.  The  Brownian  movement  was  hailed  by  some  as  supporting 
Leibniz's  theory  of  Monads,  a  theory  once  so  deeply  rooted  and  so  widely  believed 
that  even  under  Schwann's  cell-theory  Johannes  Miiller  and  Henle  spoke  of 
the  cells  as  "organische  Monaden";  cf.  Emit  du  Bois  Reymond,  Leibnizische 
Gedanken  in  der  neueren  Naturwissenschaft,  Monatsber.  d.  k.  Akad.  Wiss.,  Berlin, 
1870. 

J  The  "nucleus"  was  first  seen  in  the  epidermis  of  Orchids;  but  "this  areola, 
or  nucleus  of  the  cell  as  perhaps  it  might  be  termed,  is  not  confined  to  the 
epidermis,"  etc.  See  his  paper  on  Fecundation  in  Orchideae  and  Asclepiadae, 
Trans.  Linn.  Soc.  xvi,  1829-33,  also  Proc.  Linn.  Soc.  March  30,  1832. 

§  Carpenter,  The  Microscope,  edit.  1862,  p.  185. 


74  ON  MAGNITUDE  [ch. 

written  it  was  ascribed  by  Christian  Wiener  *  to  molecular  move- 
ments within  the  fluid,  and  was  hailed  as  visible  proof  of  the 
atomistic  (or  molecular)  constitution  of  the  same.  We  now  know 
that  it  is  indeed  due  to  the  impact  or  bombardment  of  molecules 
upon  a  body  so  small  that  these  impacts  do  not  average  out,  for 
the  moment,  to  approximate  equality  on  all  sides  f.  The  movement 
becomes  manifest  with  particles  of  somewhere  about  20 /x,  and  is 
better  displayed  by  those  of  about  10 /x,  and  especially  well  by 
certain  colloid  suspensions  or  emulsions  whose  particles  are  just 
below  1/Lt  in  diameter  {.  The  bombardment  causes  our  particles  to 
behave  just  hke  molecules  of  unusual  size,  and  this  behaviour  is 
manifested  in  several  ways§.  Firstly,  we  have  the  quivering 
movement  of  the  particles;  secondly,  their  movement  backwards 
and  forwards,  in  short,  straight  disjointed  paths;  thirdly,  the 
particles  rotate,  and  do  so  the  more  rapidly  the  smaller  they  are: 
and  by  theory,  confirmed  by  observation,  it  is  found  that  particles 
of  IjjL  in  diameter  rotate  on  an  average  through  100°  a  second, 
while  particles  of  13/x  turn  through  only  14°  a  minute.  Lastly,  the 
very  curious  result  appears,  that  in  a  layer  of  fluid  the  particles  are 
not  evenly  distributed,  nor  do  they  ever  fall  under  the  influence  of 
gravity  to  the  bottom.  For  here  gravity  and  the  Brownian  move- 
ment are  rival  powers,  striving  for  equilibrium;  just  as  gravity  is 
opposed  in  the  atmosphere  by  the  proper  motion  of  the  gaseous 
molecules.  And  just  as  equihbrium  is  attained  in  the  atmosphere 
when  the  molecules  are  so  distributed  that  the  density  (and  therefore 
the  number  of  molecules  per  unit  volume)  falls  oif  in  geometrical 

*  In  Poggendorffs  Annalen,  cxviii,  pp.  79-94,  1863.  For  an  account  of  this 
remarkable  man,  see  Naturmssensehaften,  xv,  1927;  cf.  also  Sigraund  Exner, 
Ueber  Brown's  Molecularbewegung,  Sitzungsher.  kk.  Akad.  Wien,  lvi,  p.  116,  1867. 

t  Perrin,  Les  preuves  de  la  realite  moleculaire,  Ann.  de  Physique,  xvii,  p.  549, 
1905;  XIX,  p.  571,  1906.  The  actual  molecular  collisions  are  unimaginably 
frequent;   we  see  only  the  residual  fluctuations. 

J  Wiener  was  struck  by  the  fact  that  the  phenomenon  becomes  conspicuous 
just  when  the  size  of  the  particles  becomes  comparable  to  that  of  a  wave-length 
of  light. 

§  For  a  full,  but  still  elementary,  account,  see  J.  Perrin,  Les  Atomes;  cf.  also 
Th.  Svedberg,  Die  Existenz  der  Molekiile,  1912;  R.  A.  Millikan,  The  Electron, 
1917,  etc.  The  modern  literature  of  the  Brownian  movement  (by  Einstein,  Perrin, 
de  Broglie,  Smoluchowski  and  Millikan)  is  very  large,  chiefly  owing  to  the  value 
which  the  phenomenon  is  shewn  to  have  in  determining  the  size  of  the  atom  or 
the  charge  on  an  electron,  and  of  giving,  as  Ostwald  said,  experimental  proof  of 
the  atomic  theory. 


II]  OF  THE  BROWNIAN  MOVEMENT  75 

progression  as  we  ascend  to  higher  and  higher  layers,  so  is  it  with 
our  particles  within  the  narrow  Umits  of  the  little  portion  of  fluid 
under  our  microscope. 

It  is  only  in  regard  to  particles  of  the  simplest  form  that  these 
phenomena  have  been  theoretically  investigated*,  and  we  may  take 
it  as  certain  that  more  complex  particles,  such  as  the  twisted  body 
of  a  Spirillum,  would  shew  other  and  still  more  comphcated  mani- 
festations. It  is  at  least  clear  that,  just  as  the  early  microscopists 
in  the  days  before  Robert  Brown  never  doubted  but  that  these 
phenomena  were  purely  vital,  so  we  also  may  still  be  apt  to  confuse, 
in  certain  cases,  the  one  phenomenon  with  the  other.  We  cannot, 
indeed,  without  the  most  careful  scrutiny,  decide  whether  the 
movements  of  our  minutest  organisms  are  intrinsically  "vital"  (in 
the  sense  of  being  beyond  a  physical  mechanism,  or  working  model) 
or  not.  For  example,  Schaudinn  has  suggested  that  the  undulating 
movements  of  Spirochaete  pallida  must  be  due  to  the  presence  of  a 
minute,  unseen,  "undulating  membrane";  and  Doflein  says  of  the 
same  species  that  "sie  verharrt  oft  mit  eigenthiimlich  zitternden 
Bewegungen  zu  einem  Orte."  Both  movements,  the  trembling  or 
quivering  movement  described  by  Doflein,  and  the  undulating  or 
rotating  movement  described  by  Schaudinn,  are  just  such  as  may 
be  easily  and  naturally  interpreted  as  part  and  parcel  of  the  Brownian 
phenomenon. 

While  the  Brownian  movement  may  thus  simulate  in  a  deceptive 
way  the  active  movements  of  an  organism,  the  reverse  statement 
also  to  a  certain  extent  holds  good.  One  sometimes  Hes  awake  of 
a  summer's  morning  watching  the  flies  as  they  dance  under  the 
ceiling.  It  is  a  very  remarkable  dance.  The  dancers  do  not  whirl  or 
gyrate,  either  in  company  or  alone;  but  they  advance  and  retire; 
they  seem  to  jostle  and  rebound;  between  the  rebounds  they  dart 
hither  or  thither  in  short  straight  snatches  of  hurried  flight,  and 
turn  again  sharply  in  a  new  rebound  at  the  end  of  each  little  rushf. 

*  Cf.  R.  Gans,  Wie  fallen  Stabe  und  Scheiben  in  einer  reibenden  Fliissigkeit? 
Miinchener  Bericht,  1911,  p.  191;  K.  Przibram,  Ueber  die  Brown'sche  Bewegung 
nicht  kugelformiger  Teilchen,  Wiener  Bericht,  1912,  p.  2339;   1913,  pp.  1895-1912. 

t  As  Clerk  Maxwell  put  it  to  the  British  Association  at  Bradford  in  1873,  "We 
cannot  do  better  than  observe  a  swarm  of  bees,  where  every  individual  bee  is 
flying  furiously,  first  in  one  direction  and  then  in  another,  while  the  swarm  as 
a  whole  is  either  at  rest  or  sails  slowly  through  the  air." 


76  ON  MAGNITUDE  [ch. 

Their  motions  are  erratic,  independent  of  one  another,  and 
devoid  of  common  purpose*.  This  is  nothing  else  than  a 
vastly  magnified  picture,  or  simulacrum,  of  the  Brownian  move- 
ment; the  parallel  between  the  two  cases  lies  in  their  complete 
irregularity,  but  this  in  itself  implies  a  close  resemblance.  One 
might  see  the  same  thing  in  a  crowded  market-place,  always  provided 
that  the  busthng  crowd  had  no  business  whatsoever.  In  like 
manner  Lucretius,  and  Epicurus  before  him,  watched  the  dust-motes 
quivering  in  the  beam,  and  saw  in  them  a  mimic  representation, 
rei  simulacrum  et  imago,  of  the  eternal  motions  of  the  atoms.  Again 
the  same  phenomenon  may  be  witnessed  under  the  microscope,  in 
a  drop  of  water  swarming  with  Paramoecia  or  such-Hke  Infusoria; 
and  here  the  analogy  has  been  put  to  a  numerical  test.  Following 
with  a  pencil  the  track  of  each  little  swimmer,  and  dotting  its  place 
every  few  seconds  (to  the  beat  of  a  metronome),  Karl  Przibram 
found  that  the  mean  successive  distances  from  a  common  base-hne 
obeyed  with  great  exactitude  the  "Einstein  formula,"  that  is  to 
say  the  particular  form  of  the  "law  of  chance"  which  is  apphcable 
to  the  case  of  the  Brownian  movement  f.  The  phenomenon  is  (of 
course)  merely  analogous,  and  by  no  means  identical  with  the 
Brownian  movement;  for  the  range  of  motion  of  the  little  active 
organisms,  whether  they  be  gnats  or  infusoria,  is  vastly  greater  than 
that  of  the  minute  particles  which  are  passive  under  bombardment ; 
nevertheless  Przibram  is  inclined  to  think  that  even  his  compara- 
tively large  infusoria  are  small  enough  for  the  molecular  bombard- 
ment to  be  a  stimulus,  even  though  not  the  actual  cause,  of  their 
irregular  and  interrupted  movements  {. 

*  Nevertheless  there  may  be  a  certain  amount  of  bias  or  direction  in  these 
seemingly  random  divagations:  cf.  J.  Brownlee,  Proc.  R.S.E.  xxxi,  p.  262, 
1910-11;  F.  H.  Edgeworth,  Metron,  i,  p.  75,  1920;  Lotka,  Elem.  of  Physical 
Biology,  1925,  p.  344. 

t  That  is  to  say,  the  mean  square  of  the  displacements  of  a  particle,  in  any 
direction,  is  proportional  to  the  interval  of  time.  Cf.  K.  Przibram,  Ueber  die 
ungeordnete  Bewegung  niederer  Tiere,  Pfliigefs  Archiv,  cliii,  pp.  401-405,  1913; 
Arch.  f.  Entw.  Mech.  xliii,  pp.  20-27,  1917. 

X  All  that  is  actually  proven  is  that  "pure  chance"  has  governed  the  movements 
of  the  little  organism.  Przibram  has  made  the  analogous  observation  that 
infusoria,  when  not  too  crowded  together,  spread  or  diffuse  through  an  aperture 
from  one  vessel  to  another  at  a  rate  very  closely  comparable  to  the  ordinary  laws 
of  molecular  diffusion. 


II]  OF  THE  EFFECTS  OF  SCALE  77 

George  Johnstone  Stoney,  the  remarkable  man  to  whom  we  owe 
the  name  and  concept  of  the  electron,  went  further  than  this;  for 
he  supposed  that  molecular  bombardment  might  be  the  source  of 
the  life-energy  of  the  bacteria.  He  conceived  the  swifter  moving 
molecules  to  dive  deep  into  the  minute  body  of  the  organism,  and 
this  in  turn  to  be  able  to  make  use  of  these  importations  of  energy*. 

We  draw  near  the  end  of  this  discussion.  We  found,  to  begin 
with,  that  "scale"  had  a  marked  eifect  on  physical  phenomena,  and 
that  increase  or  diminution  of  magnitude  migl^t  mean  a  complete 
change  of  statical  or  dynamical  equiUbrium.  In  the  end  we  begin 
to  see  that  there  are  discontinuities  in  the  scale,  defining  phases  in 
which  different  forces  predominate  and  different  conditions  prevail. 
Life  has  a  range  of  magnitude  narrow  indeed  compared  to  that  with 
which  physical  science  deals ;  but  it  is  wide  enough  to  include  three 
such  discrepant  conditions  as  those  in  which  a  man,  an  insect  and 
a  bacillus  have  their  being  and  play  their  several  roles.  Man  is 
ruled  by  gravitation,  and  rests  on  mother  earth.  A  water-beetle 
finds  the  surface  of  a  pool  a  matter  of  Ufe  and  death,  a  perilous 
entanglement  or  an  indispensable  support.  In  a  third  world, 
where  the  bacillus  Hves,  gravitation  is  forgotten,  and  the  viscosity 
of  the  Hquid,  the  resistance  defined  by  Stokes's  law,  the  molecular 
shocks  of  the  Brownian  movement,  doubtless  also  the  electric 
charges  of  the  ionised  medium,  make  up  the  physical  environment 
and  have  their  potent  and  immediate  influence  on  the  organism. 
The  predominant  factors  are  no  longer  those  of  our  scale ;  we  have 
come  to  the  edge  of  a  world  of  which  we  have  no  experience,  and 
where  all  our  preconceptions  must  be  recast. 

*  Phil.  Mag.  April  1890. 


CHAPTER  III, 

THE  RATE  OF  GROWTH 

When  we  study  magnitude  by  itself,  apart  from  the  gradual 
changes  to  which  it  may  be  subject,  we  are  deahng  with  a  something 
which  may  be  adequately  represented  by  a  number,  or  by  means 
of  a  Hne  of  definite  length ;  it  is  what  mathematicians  call  a  scalar 
phenomenon.  When  we  introduce  the  conception  of  change  of 
magnitude,  of  magnitude  which  varies  as  we  pass  from  one  point 
to  another  in  space,  or  from  one  instant  to  another  in  time,  our 
phenomenon  becomes  capable  of  representation  by  means  of  a  line 
of  which  we  define  both  the  length  and  the  direction;  it  is  (in  this 
particular  aspect)  what  is  called  a  vector  phenomenon. 

When  we  deal  with  magnitude  in  relation  to  the  dimensions  of  space, 
our  diagram  plots  magnitude  in  one  direction  against  magnitude  in 
another — length  against  height,  for  instance,  or  against  breadth ;  and 
the  result  is  what  we  call  a  picture  or  outhne,  or  (more  correctly) 
a  "plane  projection"  of  the  object.  In  other  words,  what  we  call 
Form  is  a  ratio  of  magnitudes*  referred  to  direction  in  space. 

When,  in  deahng  with  magnitude,  we  refer  its  variations  to 
successive  intervals  of  time  (or  when,  as  it  is  said,  we  equate  it  with 
time),  we  are  then  dealing  with  the  phenomenon  of  growth;  and 
it  is  evident  that  this  term  growth  has  wide  meanings.  For  growth 
may  be  positive  or  negative,  a  thing  may  grow  larger  or  smaller, 
greater  or  less;  and  by  extension  of  the  concrete  signification  of 
the  word  we  easily  arid  legitimately  apply  it  to  non-material  things, 
such  as  temperature,  and  say,  for  instance,  that  a  body  "grows" 
hot  or  cold.  When  in  a  two-dimensional  diagram  we  represent  a 
magnitude  (for  instance  length)  in  relation  to  time  (or  "plot"  length 
against  time,  as  the  phrase  is),  we  get  that  kind  of  vector  diagram 
which  is  known  as  a  "curve  of  growth."  We  see  that  the  pheno- 
menon which  we  are  studying  is  a  velocity  (whose  "dimensions"  are 
space/time,  or  L/T),  and  this  phenomenon  we  shall  speak  of,  simply, 
as  a  rate  of  growth. 

In  various  conventional  wa-ys  we  convert  a  two-dimensional  into 

*  In  Aristotelian  logic.  Form  is  a  quality.  None  the  less,  it  is  related  to  qiuirUity't 
and  we  jfind  the  Schoolmen  speaking  of  it  as  qualitas  circa  quantitatem. 


CH.  Ill]  OF  CHANGE  OF  MAGNITUDE  79 

a  three-dimensional  diagram.  We  do  so,  for  example,  when,  by 
means  of  the  geometrical  method  of  "perspective,"  we  represent 
upon  a  sheet  of  paper  the  length,  breadth  and  depth  of  an  object 
in  three-dimensional  space,  but  we  do  it  better  by  means  of  contour- 
Hnes  or  "isopleths."  By  contour-lines  superposed  upon  a  map  of 
a  country,  we  shew  its  hills  and  valleys;  and  by  contour-Unes  we 
may  shew  temperature,  rainfall,  population,  language,  or  any  other 
*' third  dimension"  related  to  the  two  dimensions  of  the  map.  Time 
is  always  impHcit,  in  so  far  as  each  map  refers  to  its  own  date  or 
epoch;  but  Time  as  a  dimension  can  only  be  substituted  for  one  of 
the  three  dimensions  already  there.  Thus  we  may  superpose  upon 
our  map  the  successive  outlines  of  the  coast  from  remote  antiquity, 
or  of  any  single  isotherm  or  isobar  from  day  to  day.  And  if  in  hke 
manner  we  superpose  on  one  another,  or  even  set  side  by  side,  the 
outhnes  of  a  growing  organism — for  instance  of  a  young  leaf  and 
an  old,  we  have  a  three-dimensional  diagram  which  is  a  partial 
representation  (limited  to  two  dimensions  of  space)  of  the  organism's 
gradual  change  of  form,  or  course  of  development;  in  such  a  case 
our  contours  may,  for  the  purposes  of  the  embryologist,  be  separated 
by  time-intervals  of  a  few  hours  or  days,  or,  for  the  palaeontologist, 
by  interspaces  of  unnumbered  and  innumerable  years*. 

Such  a  diagram  represents  in  two  of  its  three  dimensions  form, 
and  in  two  (or  three)  of  its  dimensions  growth,  and  we  see  how 
intimately  the  two  concepts  are  correlated  or  interrelated  to  one 
another.  In  short  it  is  obvious  that  the  form  of  an  organism  is 
determined  by  its  rate  of  growth  in  various  directions ;  hence  rate 
of  growth  deserves  to  be  studied  as  a  necessary  preliminary  to  the 
theoretical  study  of  form,  and  organic  form  itself  is  found, 
mathematically  speaking,  to  be  di  function  of  time'\. 

*  Sometimes  we  find  one  and  the  same  diagram  suffice,  whether  the  time-intervals 
be  great  or  small;  and  we  then  invoke  " Wolff's  law"  (or  Kielmeyer's),  and  assert 
that  the  life-history  of  the  individual  repeats,  or  recapitulates,  the  history  of  the 
race.  This  "recapitulation  theory"  was  alt-important  in  nineteenth -century 
embryology,  but  was  criticised  by  Adam  Sedgwick  {Q.J. M.S.  xxxvi,  p.  38,  1894) 
and  many  later  authors;  cf.  J.  Needham,  Chemical  Embryology,  1931,  pp.  1629-1647. 

t  Our  subject  is  one  of  Bacon's  "Instances  of  the  Course"  or  studies  wherein 
we  "measure  Nature  by  periods  of  Time."  In  Bacpji's  Catalogue  of  Particular 
Histories,  one  of  the  odd  hundred  histories  or  investigations  which,  he  foreshadows 
is  precisely  that  which  we  are  engaged  on,  viz.  a  "History  of  the  Growth  and 
Increase  of  the  Body,  in  the  whole  and  in  its  parts." 


80  THE  RATE  OF  GROWTH  [ch. 

At  the  same  time,  we  need  only  consider  this  large  part  of  our 
subject  somewhat  briefly.  Though  it  has  an  essential  bearing  on 
the  problems  of  morphology,  it  is  in  greater  degree  involved  with 
physiological  problems;  also,  the  statistical  or  numerical  aspect  of 
the  question  is  peculiarly  adapted  to  the  mathematical  study  of 
variation  and  correlation.  These  important  subjects  we  must  not 
neglect;  but  our  main  purpose  will  be  served  if  we  consider  the 
characteristics  of  a  rate  of  growth  in  a  few  illustrative  cases,  and 
recognise  that  this  rate  of  growth  is  a  very  important  specific 
property,  with  its  own  characteristic  value  in  this  organism  or  that, 
in  this  or  that  part  of  each  organism,  and  in  this  or  that  phase  of 
its  existence. 

The  statement  which  we  have  just  made  that  "the  form  of  an 
organism  is  determined  by  its  rate  of  growth  in  various  directions," 
is  one  which  calls  for  further  explanation  and  for  some  measure  of 
quahfication. 

Among  organic  forms  we  shall  have  many  an  occasion  to  see  that 
form  may  be  due  in  simple  cases  to  the  direct  action  of  certain 
molecular  forces,  among  which  surface-tension  plays  a  leading  part. 
Now  when  surface-tension  causes  (for  instance)  a  minute  semifluid 
organism  to  assume  a  spherical  form,  or  gives  to  a  film  of  protoplasm 
the  form  of  a  catenary  or  of  an  elastic  curve,  or  when  it  acts  in 
various  other  ways  productive  of  definite  contours — just  as  it  does 
in  the  making  of  a  drop,  a  splash  or  a  jet — this  is  a  process  of  con- 
formation very  diiferent  from  that  by  which  an  ordinary  plant  or 
animal  grows  into  its  specific  form.  In  both  cases  change  of  form 
is  brought  about  by  the  movement  of  portions  of  matter,  and  in 
both  cases  it  is  ultimately  due  to  the  action  of  molecular  forces; 
but  in  the  one  case  the  movements  of  the  particles  of  matter  lie  for 
the  most  part  within  molecular  range,  while  in  the  other  we  have 
to  deal  with  the  transference  of  portions  of  matter  into  the  system 
from  without,  and  from  one  widely  distant  part  of  the  organism  to 
another.  It  is  to  this  latter  class  of  phenomena  that  we  usually 
restrict  the  term  growth;  it  is  in  regard  to  them  that  we  are  in  a 
position  to  study  the  rate  of  action  in  different  directions  and  at 
different  times,  and  to  realise  that  it  is  on  such  differences  of  rate 
that  form  and  its  modifications  essentially  and  ultimately  depend. 


Ill]  OF  RATE  OF  ACTION  81 

The  difference  between  the  two  classes  of  phenomena  is  akin  to 
the  difference  between  the  forces  which  determine  the  form  of  a 
raindrop  and  those  which,  by  the  flowing  of  the  waters  and  the 
sculpturing  of  the  solid  earth,  have  brought  about  the  configuration 
of  a  river  or  a  hill ;  molecular  forces  are  paramount  in  the  one,  and 
wolar  forces  are  dominant  in  the  other. 

At  the  same  time,  it  is  true  that  all  changes  of  form,  inasmuch 
as  they  necessarily  involve  changes  of  actual  and  relative  magnitude, 
may  in  a  sense  be  looked  upon  as  phenomena  of  growth ;  and  it  is 
also  true,  since  the  movement  of  matter  must  always  involve  an 
element  of  time*,  that  in  all  cases  the  rate  of  growth  is  a  phenomenon 
to  be  considered.  Even  though  the  molecular  forces  which  play 
their  part  in  modifying  the  form  of  an  organism  exert  an  action 
which  is,  theoretically,  all  but  instantaneous,  that  action  is  apt  to 
be  dragged  out  to  an  appreciable  interval  of  time  by  reason  of 
viscosity  or  some  other  form  of  resistance  in  the  material.  From 
the  physical  or  physiological  point  of  view  the  rate  of  action  may  be 
well  worth  studying  even  in  such  cases  as  these;  for  example,  a 
study  of  the  rate  of  cell-division  in  a  segmenting  egg  may  teach  us 
something  about  the  w^ork  done,  and  the  various  energies  concerned. 
But  in  such  cases  the  action  is,  as  a  rule,  so  homogeneous,  and  the 
form  finally  attained  is  so  definite  and  so  little  dependent  on  the 
time  taken  to  effect  it,  that  the  specific  rate  of  change,  or  rate  of 
growth,  does  not  enter  into  the  morphological  problem. 

We  are  deahng  with  Form  in  a  very  concrete  way.  To  Aristotle 
it  was  a  metaphysical  concept;  to  us  it  is  a  quasi-mechanical  effect 
on    Matter    of   the    operation    of   chemico-physical    forces  f.     To 


*  Cf.  Aristotle,  Phys.  VI,  5,  235a,  11,  eirel  yap  awaaa  Kiurjffis  ev  XP^'^V*  kt\.;  he  had 
already  told  us  that  natural  science  deals  with  magnitude,  with  motion  and  with 
time:  ^<ttiu  ij  irepl  (pvaeus  eiriar-qtnj  irepl  fxiyedos  Kai  klvt^ctlv  koL  xP^^^^-  Hence 
omnis  velocitas  tempore  durat  became  a  scholastic  aphorism.  Bacon  emphasised,  in 
like  manner,  the  fact  that  "all  motion  or  natural  action  is  performed  in  time: 
some  more  quickly,  some  more  slowly,  but  all  in  periods  determined  and  fixed  in 
the  nature  of  things.  Even  those  actions  which  seem  to  be  performed  suddenly, 
and  (as  we  say)  in  the  twinkling  of  an  eye,  are  found  to  admit  of  degree  in 
respect  of  duration"  {Nov.  Organon,  xlvi).  That  infinitely  small  motions  take 
place  in  infinitely  small  intervals  of  time  is  the  concept  which  lies  at  the  root  of  the 
calculus.     But  there  is  another  side  to  the  story. 

t  Cf.  N.  K.  KoltzotF,  Physikalisch-chemische  Grundlage  der  Morphologie, 
Biol.  Centralbl.  1928,  pp.  345-369. 


82  THE  RATE  OF  GROWTH  [ch. 

Aristotle  its  Form  was  the  essence,  the  archetype,  the  very  "nature" 
of  a  thing,  and  Matter  and  Form  were  an  inseparable  duahty. 
Even  now,  when  we  divide  our  science  into  Physiology  and  Mor- 
phology, we  are  harking  back  to  the  old  Aristotehan  antithesis. 

To  sum  up,  we  may  lay  down  the  following  general  statements. 
The  form  of  organisms  is  a  phenomenon  to  be  referred  in  part  to 
the  direct  action  of  molecular  forces,  in  larger  part  to  a  more  complex 
and  slower  process,  indirectly  resulting  from  chemical,  osmotic  and 
other  forces,  by  which  material  is  introduced  into  the  organism 
and  transferred  from  one  part  of  it  to  another.  It  is  this  latter 
complex  phenomenon  which  we  usually  speak  of  as  "growth." 

Every  growing  organism;  and  every  part  of  such  a  growing 
organism,  has  its  own  specific  rate  of  growth,  referred  to  this  or 
that  particular  direction ;  and  it  is  by  the  ratio  between  these  rates 
in  different  directions  that  we  must  account  for  the  external  forms 
oiall  save  certain  very  minute  organisms.  This  ratio  may  sometimes 
be  of  a  simple  kind,  as  when  it  results  in  the  mathematically 
definable  outhne  of  a  shell,  or  the  smooth  curve  of  the  margin  of  a 
leaf.  It  may  sometimes  be  a  very  constant  ratio,  in  which  case  the 
organism  while  growing  in  bulk  suffers  httle  or  no  perceptible  change 
in  form;  but  such  constancy  seldom  endures  beyond  a  season,  and 
when  the  ratios  tend  to  alter,  then  we  have  the  phenomenon  of 
morphological  ''development,''  or  steady  and  persistent  alteration  of 
form. 

This  elementary  concept  of  Form,  as  determined  by  varying  rates 
of  Growth,  was  clearly  apprehended  by  the  mathematical  mind  of 
Haller — who  had  learned  his  mathematics  of  the  great  John 
Bemoulh,  as  the  latter  in  turn  had  learned  his  physiology  from  the 
writings  of  Borelh*  It  was  this  very  point,  the  apparently  un- 
limited extent  to  which,  in  the  development  of  the  chick,  inequalities 
of  growth  could  and  did  produce  changes  of  form  and  changes  of 
anatomical  structure,  that  led  Haller  to  surmise  that  the  process 
was  actually  without  Hmits,  and  that  all  development  was  but  an 
unfolding  or  "evolutio,"  in  which  no  part  came  into  being  which 

*  "Qua  in  re  Incomparabilis  Viri  Joh.  Alph.  Borelli  vestigiis  insistemus." 
Joh.  Bernoulli,  De  motu  musculorum,  1694. 


Ill]  THE  DOCTRINE  OF   PREFORMATION  83 

had  not  essentially  existed  before*.  In  short  the  celebrated  doctrine 
of  "preformation"  implied  on  the  one  hand  a  clear  recognition  of 
what  growth  can  do  throughout  the  several  stages  of  development, 
by  hastening  the  increase  in  size  of  one  part,  hindering  that  of 
another,  changing  their  relative  magnitudes  and  positions,  and  so 
altering  their  forms;  while  on  the  other  hand  it  betrayed  a  failure 
(inevitable  in  those  days)  to  recognise  the  essential  difference 
between  these  movements  of  masses  and  the  molecular  processes 
which  precede  and  accompany  them,  and  which  are  characteristic 
of  another  order  of  magnitude. 

The  general  connection,  between  growth  and  form  has  been 
recognised  by  other  writers  besides  Haller.  Such  a  connection  is 
imphcit  in  the  "proportional  diagrams"  by  which  Diirer  and  his 
brother-artists  illustrated  the  changes  in  form,  or  of  relative 
dimensions,  which  mark  the  child's  growth  to  boyhood  and  to 
manhood.  The  same  connection  was  recognised  by  the  early 
embryologists,  and  appears,  as  a  survival  of  the  doctrine  of  pre- 
formation, in  Pander's  I  study  of  the  development  of  the  chick. 
And  long  afterwards,  the  embryological  aspect  of  the  case  was 
emphasised  by  His  J,  who  pointed  out  that  the  foldings  of  the 
blastoderm,  by  which  the  neural  and  amniotic  folds  are  brought 
into  being,  were  the  resultant  of  unequal  rates  of  growth  in  what 
to  begin  with  was  a  uniform  layer  of  embryonic  tissue.  If  a  sheet 
of  paper  be  made  to  expand  here  and  contract  there,  as  by  moisture 
or  evaporation,  the  plane  surface  becomes  dimpled,  or  folded,  or 
buckled,  by  the  said  expansions  and  contractions;  and  the  dis- 
tortions to  which  the  surface  of  the  "germinal  disc"  is  subject  are, 
as  His  shewed  once  and  for  all,  precisely  analogous.     There  are 

*  Cf.  (e.g.)  Elem.  Physiologiae,  ed.  1766,  vm,  p.  114,  "Ducimur  autem  ad 
evolutionem  potissimum,  quando  a  perfecto  animale  retrorsum  progredimuis  et 
incrementonim  atque  mutationum  seriem  relegimus.  Ita  inveniemus  perfectum 
illud  animal  fuisse  imperfectius,  alterius  figurae  et  fabricae,  et  denique  rude  et 
informe :  et  tamen  idem  semper  animal  sub  iis  diversis  phasibus  fuisse,  quae  absque 
ullo  saltu  perpetuos  parvosque  per  gradus  cohaereant." 

t  Beitrdge  zur  Entwickelungsgeschichte  des  Huhnchens  im  Ei,  1817,  p.  40.  Roux 
ascribes  the  same  views  also  to  Von  Baer  and  to  R.  H.  Lotze  {Allgem.  Physiologie, 
1851,  p.  353). 

I  W.  His,  Unsere  Korperform,  und  das  physiologische  Problem  ihrer  Entstehung, 
1874.  See  also  Archiv  f.  Anatomie,  1894;  and  cf.  C.  B.  Davenport,  Processes  con- 
cerned in  Ontogeny,  Bull.  Mus.  Comp.  Anat,  xxvn,  1895;  also  G.  Dehnel  and 
Jan  Tur,  De  Embryonum  evolutionis  progress u  ineqvuli:  Kosmos  (Lwow),  lhi,  1928. 


84  THE  RATE  OF  GROWTH  [ch. 

certain  Nostoc-algae  in  which  unequal  growth,  ceasing  towards  the 
periphery  of  a  disc  and  increasing  here  and  there  within,  gives  rise 
to  folds  and  bucklings  curiously  Hke  those  of  our  own  ears:  which 
indeed  owe  their  shape  and  characteristic  folding  to  an  identical  or 
analogous  cause. 

An  experimental  demonstration  comparable  to  the  actual  case  is 
obtained  by  making  an  "artificial  blastoderm"  of  Uttle  pills  or 
pellets  of  dough,  which  are  caused  to  grow  at  varying  rates  by  the 
addition  of  varying  quantities  of  yeast.  Here,  as  Roux  is  careful 
to  point  out,*  it  is  not  only  the  growth  of  the  individual  cells,  but 
the  traction  exercised  on  one  another  through  their  mutual  inter- 
connections, which  brings  about  foldings,  wrinklings  and  other 
distortions  of  the  structure.  But  this  again,  or  such  as  this,  had  been 
in  Haller's  mind,  and  formed  an  essential  part  of  his  embry illogical 
doctrine.  For  he  has  no  sooner  treated  of  incrementum,  or  celeritas 
incrementi,  than  he  proceeds  to  deal  with  the  contributory  and 
complementary  phenomena  of  expansion,  traction  (adtractio)^  and 
pressure,  and  the  more  subtle  influences  which  he  denominates  vis 
derivationis  et  revulsionis%'.  these  latter  being  the  secondary  and 
correlated  effects  on  growth  in  one  part,  brought  about  by  such 
changes  as  are  produced,  for  instance  in  the  circulation,  by  the 
growth  of  another. 

We  have  to  do  with  growth,  with  exquisitely  graded  or  balanced 
growth,  and  with  forces  subtly  exerted  by  one  growing  part  upon 
another,  in  so  wonderful  a  piece  of  work  as  the  development  of  the 
eye :  as  its  primary  vesicle  expands  and  then  dimples  in,  as  the  lens 
appears  and  fits  into  place,  as  the  secondary  vesicle  closes  over  to 
form  iris  and  pupil,  and  in  all  the  rest  of  the  story. 

Let  us  admit  that,  on  the  physiological  side,  Haller's  or  His's 
methods  of  explanation  carry  us  but  a  little  way;  yet  even  this 
little  way  is  something  gained.     Nevertheless,  I  can  well  remember 

*  Roux,  Die  Entwickelungsmechanik,  1905,  p.  99. 

t  Op.  cit.  p.  302,  "Magnum  hoc  naturae  instrumentum,  etiam  in  corpore  animato 
evolvendo  potenter  operatur,  etc."  The  recurrent  laryngeal  nerve,  drawn  down 
as  its  arch  of  the  aorta  descends,  is  a  simple  instance  of  anatomical  traction.  The 
vitelline  and  omphalomesenteric  arteries  lead,  by  more  complicated  constraints 
and  tractions,  to  the  characteristic  loops  of  the  intestinal  blood  vessels,  and  of  the 
intestine  itself.     Cf.  G.  Enbom,  Lunds  Univ.  Arsskrift,  1939. 

X  Ibid.  p.  306,  "Subtiliora  ista,  et  aliquantum  hypothesi  mista,  tamen  magnam 
mihi  videntur  speciem  veri  habere." 


Ill]  OF  PHYSICS  AND  EMBRYOLOGY  85 

the  harsh  criticism  and  even  contempt  which  His's  doctrine  met 
with,  not  merely  on  the  ground  that  it  was  inadequate,  but  because 
such  an  explanation  was  deemed  wholly  inappropriate,  and  was 
utterly  disavowed*.  Oscar  Hertwig,  for  instance,  asserted  that,  in 
embryology,  when  we  find  one  embryonic  stage  preceding  another, 
the  existence  of  the  former  is,  for  the  embryologist,  an  all-sufficient 
"causal  explanation"  of  the  latter.  "We  consider  (he  says)  that 
we  are  studying  and  explaining  a  causal  relation  when  we  have 
demonstrated  that  the  gastrula  arises  by  invagination  of  a  blasto- 
sphere,  or  the  neural  canal  by  the  infolding  of  a  cell-plate  so  as  to 
constitute  a  tubef."  For  Hertwig,  then,  as  Roux  remarks,  the  task 
of  investigating  a  physical  mechanism  in  embryology — "der  Ziel  das 
Wirken  zu  erforschen" — has  no  existence  at  all.  For  Balfour  also, 
as  for  Hertwig,  the  mechanical  or  physical  aspect  of  organic  develop- 
ment had  little  or  no  attraction.  In  one  notable  instance,  Balfour 
himself  adduced  a  physical,  or  quasi-physical,  explanation  of  an 
organic  process,  when  he  referred  the  various  modes  of  segmentation 
of  an  ovum,  complete  or  partial,  equal  or  unequal  and  so  forth,  to 
the  varying  amount  or  varying  distribution  of  food-yolk  associated 
with  the  germinal  protoplasm  of  the  egg.  But  in  the  main,  like  all 
the  other  embryologists  of  his  day,  Balfour  was  engrossed  in  the 

*  Cf.  His,  On  the  Principles  of  Animal  Morphology,  Proc.  R.S.E.  xv, 
p.  294,  1888:  "My  own  attempts  to  introduce  some  elementary  mechanical  or 
physiolbgical  conceptions  into  embryology  have  not  generally  been  agreed  to  by 
morphologists.  To  one  it  seemed  ridiculous  to  speak  of  the  elasticity  of  the  germinal 
layers;  another  thought  that,  by  such  considerations,  we  'put  the  cart  before 
the  horse';  and  one  more  recent  author  states,  that  we  have  better  things  to  do 
in  embryology  than  to  discuss  tensions  of  germinal  layers  and  similar  questions, 
since  all  explanations  must  of  necessity  be  of  a  phylogenetic  nature.  This  opposition 
to  the  application  of  the  fundamental  principles  of  science  to  embryological  questions 
would  scarcely  be  intelligible  had  it  not  a  dogmatic  background.  No  other  explana- 
tion of  living  forms  is  allowed  than  heredity,  and  any  which  is  founded  on  another 
basis  must  be  rejected.. .  .To  think  that  heredity  will  build  organic  beings  without 
mechanical  means  is  a  piece  of  unscientific  mysticism."  Even  the  school  of 
Entwickelungsmechanik  showed  a  certain  reluctance,  or  extreme  caution,  in  speaking 
of  the  physical  forces  in  relation  to  embryology  or  physiology.  This  reluctant 
caution  is  well  exemplified  by  Martin  Heidenhain,  writing  on  "Formen  und  Krafte 
in  der  lebendigen  Natur"  in  Roux's  Vortrdge,  xxxir,  1923.  Speaking  of  "die 
Krafte  welche  die  Entwickelung  und  den  fertigen  Zustand  der  Formen  bedingen", 
he  says:  "letztere  kann  man  aber  nicht  auf  dem  Felde  der  Physik  suchen,  sondern 
nur  im  Umkreis  der  Lebendigen,  obwohl  anzunehmen  ist,  dass  diese  Krafte  spater 
einmal  '  analogienhaft '  nach  dem  Vorbilde  der  Physik  beschreibbar  sein  werden" 

t  0.  Hertwig,  Zeit-  und  Streitfragen  der  Biologie,  ii,  1897. 


86  THE  RATE  OF  GROWTH  [ch. 

problems  of  phylogeny,  and  he  expressly  defined  the  aims  of  com- 
parative embryology  (as  exemphfied  in  his  own  textbook)  as  being 
*' twofold:  (1)  to  form  a  basis  for  Phylogeny,  and  (2)  to  form  a  basis 
for  Organogeny,  or  the  origin  and  evolution  of  organs*." 

It  has  been  the  great  service  of  Roux  and  his  fellow-workers  of 
the  school  of  "Entwickelungsmechanik,"  and  of  many  other  students 
to  whose  work  we  shall  refer,  to  try,  as  His  tried,  to  import  into 
embryology,  wherever  possible,  the  simpler  concepts  of  physics,  to 
introduce  along  with  them  the  method  of  experiment,  and  to  refuse 
to  be  bound  by  the  narrow  hmitations  which  such  teaching  as  that 
of  Hertwig  would  of  necessity  impose  on  the  work  and  the  thought 
and  the  whole  philosophy  of  the  biologist. 

Before  we  pass  from  this  general  discussion  to  study  some  of  the 
particular  phenomena  of  growth,  let  m.e  give  an  illustration,  from 
Darwin,  of  a  point  of  view  which  is  in  marked  contrast  to  Haller's 
simple  but  essentially  mathematical  conception  of  Form. 

There  is  a  curious  passage  in  the  Origin  of  Species  "f,  where  Darwin 
is  discussing  the  leading  facts  of  enibryology,  and  in  particular 
Von  Baer's  "law  of  embryonic  resemblance."  Here  Darwin  says: 
"We  are  so  much  accustomed  to  see  a  difference  in  structure  between 
the  embryo  and  the  adult  that  we  are  tempted  to  look  at  this 
difference  as  in  some  necessary  manner  contingent  on  growth.  But 
there  is  no  reason  why,  for  instance,  the  wing  of  a  bat,  or  the  fin 
of  a  porpoise,  should  not  have  been  sketched  out  with  all  their  parts 
in  proper  proportion,  as  soon  as  any  part  became  visible."  After 
pointing  out  various  exceptions,  with  his  habitual  care,  Darwin 
proceeds  to  lay  down  two  general  principles,  viz.  "that  shght 
variations  generally  appear  at  a  not  very  early  period  of  Hfe,"  and 
secondly,  that  "at  whatever  age  a  variation  first  appears  in  the 
parent,  it  tends  to  reappear  at  a  corresponding  age  in  the  offspring." 
He  then  argues  that  it  is  with  nature  as  with  the  fancier,  who  does 
not  care  what  his  pigeons  look  Uke  in  the  embryo  so  long  as  the 
full-grown  bird  possesses  the  desired  quahties :  and  that  the  process 
of  selection  takes  place  when  the  birds  or  other  animals  are  nearly 

*  Treatise  on  Comparative  Embryology,  i,  p.  4,  1881. 

t  Ist  ed.  p.  444;  6th  ed.  p.  390.  The  student  should  not  fail  to  consult  the 
passage  in  question ;  for  there  is  always  a  risk  of  misunderstanding  or  misinterpreta- 
tion when  one  attempts  to  epitomise  Darwin's  carefully  condensed  arguments. 


Ill]  A  PASSAGE  IN  DARWIN  87 

grown  up — at  least  on  the  part  of  the  breeder,  and  presumably  in 
nature  as  a  general  rule.  The  illustration  of  these  principles  is  set 
forth  as  follows:  "Let  us  take  a  group  of  birds,  descended  from 
some  ancient  form  and  modified  through  natural  selection  for 
different  habits.  Then,  from  the  many  successive  variations  having 
supervened  iA  the  several  species  at  a  not  very  early  age,  and  having 
been  inherited  at  a  corresponding  age,  the  young  will  still  resemble 
each  other  much  more  closely  than  do  tl^e  adults — just  as  we  have 
seen  with  the  breeds  of  the  pigeon. . . .  Whatever  influence  long- 
continued  use  or  disuse  may  have  had  in  modifying  the  limbs  or 
other  parts  of  any  species,  this  will  chiefly  or  solely  have  affected 
it  when  nearly  mature,  when  it  was  compelled  to  use  its  full  powers 
to  gain  its  own  living ;  and  the  effects  thus  produced  will  have  been 
transmitted  to  the  offspring  at  a  corresponding  nearly  mature  age. 
Thus  the  young  will  not  be  modified,  or  will  be  modified  only  in  a 
shght  degree,  through  the  effects  of  the  increased  use  or  disuse*  of 
parts."  This  whole  argument  is  remarkable,  in  more  ways  than 
we  need  try  to  deal  with  here;  but  it  is  especially  remarkable  that 
Darwin  should  begin  by  casting  doubt  upon  the  broad  fact  that  a 
"difference  in  structure  between  the  embryo  and  the  adult"  is 
"in  some  necessary  matter  contingent  on  growth";  and  that  he 
should  see  no  reason  why  compUcated  structures  of  the  adult 
"should  not  have  been  sketched  out  with  all  their  parts  in  proper 
proportion,  as  soon  as  any  part  became  visible."  It  would  seem  to 
me  that  even  the  most  elementary  attention  to  form  in  its  relation 
to  growth  would  have  removed  most  of  Darwin's  difficulties  in  regard 
to  the  particular  phenomena  which  he  is  considering  here.  For 
these  phenomena  are  phenomena  of  form,  and  therefore  of  relative 
magnitude ;  and  the  magnitudes  in  question  are  attained  by  growth, 
proceeding  with  certain  specific  velocities,  and  lasting  for  certain 
long  periods  of  time.  And  it  seems  obvious  accordingly  that  in  any 
two  related  individuals  (whether  specifically  identical  or  not)  the 
differences  between  them  must  manifest  themselves  gradually,  and 
be  but  Httle  apparent  in  the  young.  It  is  for  the  same  simple 
reason  that  animals  which  are  of  very  different  sizes  when  adult 
differ  less  and  less  in  size  (as  well  as  form)  as  we  trace  them  back- 
wards to  their  early  stages. 

Though  we  study  the  visible  effects  of  varying  rates  of  growth 


88  THE  RATE  OF  GROWTH  [ch. 

throughout  wellnigh  all  the  problems  of  morphology,  it  is  not  very 
often  that  we  can  directly  measure  the  velocities  concerned.  But 
owing  to  the  obvious  importance  of  the  phenomenon  to  the  morpho- 
logist  we  must  make  shift  to  study  it  where  we  can,  even  though 
our  illustrative  cases  may  seem  sometimes  to  have  little  bearing 
on  the  morphological  problem*. 

In  a  simple  spherical  organism,  such  as  the  single  spherical  cell  of 
Protococcus  or  of  Orbulina,  growth  is  reduced  to  its  simplest  terms, 
and  indeed  becomes  so  simple  in  its  outward  manifestations  that  it 
loses  interest  to  the  morphologist.  The  rate  of  growth  is  measured 
by  the  rate  of  change  in  length  of  a  radius,  i.e.  F  =  {R'  —  R)IT,  and 
from  this  we  may  calculate,  as  already  indicated,  the  rate  in  terms 
of  surface  and  of  volume.  The  growing  body  remains  of  constant 
form,  by  the  symmetry  of  the  system;  because,  that  is  to  say,  on 
the  one  hand  the  pressure  exerted  by  the  growing  protoplasm  is 
exerted  equally  in  all  directions,  after  the  manner  of  a  hydrostatic 
pressure,  which  indeed  it  actually  is ;  while  on  the  other  hand  the 
"skin"  or  surface  layer  of  the  cell  is  sufficiently  homogeneous  to  exert 
an  approximately  uniform  resistance.  Under  these  simple  conditions, 
then,  the  rate  of  growth  is  uniform  in  all  directions,  and  does  not 
affect  the  form  of  the  organism." 

But  in  a  larger  or  a  more  complex  organism  the  study  of  growth, 
and  of  the  rate  of  growth,  presents  us  with  a  variety  of  problems, 
and  the  whole  phenomenon  (apart  from  its  physiological  interest) 
becomes  a  factor  of  great  morphological  importance.  We  no  longer 
find  that  growth  tends  to  be  uniform  in  all  directions,  nor  have  we 
any  reason  to  expect  it  should.  The  resistances  which  it  meets  with 
are  no  longer  uniform.  In  one  direction  but  not  in  others  it  will 
be  opposed  by  the  important  resistance  of  gravity;  within  the 
growing  system  itself  all  manner  of  structural  differences  come  into 
play,  and  set  up  unequal  resistances  to  growth  in  one  direction  or 
another.  At  the  same  time  the  actual  sources  of  growth,  the 
chemical  and  osmotic  forces  which  lead  to  the  intussusception  of 
new  matter,  are  not  uniformly  distributed;  one  tissue  or  one  organ 
may  well  increase  while  another  does  not;  a  set  of  bones,  their 
intervening  cartilages  and  their  surrounding  muscles,  may  all  be 

*  "In  omni  rerum  naturalium  historia  utile  est  mensuras  definiri  et  numeros,^^ 
Haller,  Elem.  Physiol,  ii,  p.  258,  1760.     Cf.  Hales,  Vegetable  Staticks,  Introduction. 


Ill]  ADOLPHE  QUETELET  89 

capable  of  very  different  rates  of  increment.  The  changes  of  form 
which  result  from  these  differences  in  rate  are  especially  manifested 
during  that  phase  of  life  when  growth  itself  is  rapid:  when  the 
organism^  as  we  say,  is  undergoing  its  development. 

When  growth  in  general  has  slowed  down,  the  differences  in  rate 
between  different  parts  of  the  organism  may  still  exist,  and  may  be 
made  manifest  by  careful  observation  and  measurement,  but  the 
resultant  change  of  form  is  less  apt  to  strike  the  eye.  Great  as  are 
the  differences  between  the  rates  of  growth  in  different  parts  of  a 
complex  organism,  the  marvel  is  that  the  ratios  between  them  are 
so  nicely  balanced  as  they  are,  and  so  capable  of  keeping  the  form 
of  the  growing  organism  all  but  unchanged  for  long  periods  of  time, 
or  of  slowly  changing  it  in  its  own  harmonious  way.  There  is  the 
nicest  possible  balance  of  forces  and  resistances  in  every  part  of 
the  complex  body;  and  when  this  normal  equilibrium  is  disturbed, 
then  we  get  abnormal  growth,  in  the  shape  of  tumours  and  exostoses, 
and  other  malformations  and  deformities  of  every  kind. 

The  rate  of  growth  in  man 
Man  will  serve  us  as  well  as  another  organism  for  our  first  illus- 
trations of  rate  of  growth,  nor  can  we  easily  find  another  which  we 
can  better  study  from  birth  to  the  utmost  Hmits  ofold  age.  Nor 
can  we  do  better  than  go  for  our  first  data  concerning  him  to 
Quetelet's  Essai  de  Physique  Sociale,  an  epoch-making  book  for  the 
biologist.  For  it  is  packed  with  information,  some  of  it  unsurpassed, 
in  regard  to  human  growth  and  form;  and  it  stands  out  as  the 
first  great  essay  in  which  social  statistics  and  organic  variation  are 
dealt  with  from  the  point  of  view  of  the  mathematical  theory  of 
probabilities.  How  on  the  one  hand  Quetelet  followed  Da  Vinci, 
Luca  Pacioli  and  Dlirer  in  studying  the  growth  and  proportions  of 
man :  and  how  on  the  other  he  simplified  and  extended  the  ideas  of 
James  Bernoulli,  of  d'Alembert,  Laplace,  Poisson  and  the  rest,  is 
another  and  a  vastly  interesting  story*. 

*  Quetelet,  Sur  V Homme,  ...,  ou  Essai  de  Physique  Sociale,  Bruxelles,  1835: 
trans.  Edinburgh,  1842;  a,\so  Instructions populaires sur  le  calcul des probabilites,  1828; 
Lettres.  .  .sur  la  theorie  des  probabilites  appliquee  aux  sciences  morales  et  politiques, 
1846 ;  and  Anthropometrie,  1871 .  For  an  account  of  his  life  and  writings,  see  Lottin's 
Quetelet,  statisticien  et  sociologue,  Louvain,  1912;  also  J.  M.  Keynes.  Treatise  on 
Probability,  1921. 


90 


THE  RATE  OF  GROWTH 


[CH. 

The  meaning  of  the  word  "statistics"  is  curiously  changed.  For 
Shakespeare  or  for  Milton  a  statist  meant  (so  Dr  Johnson  says) 
"a  pohtician,  a  statesman;  one  skilled  in  government."  The 
eighteenth-century  Statistical  Account  of  Scotland  was  a  description 
of  the  State  and  of  its  people,  its  wealth,  its.  agriculture  and  its  trade. 


Stature  and  weight*  of  man  (from  QuMeleVs  Belgian  data, 
Essai,  II,  pp.  23-43;  Anthropometric,  p.  346)  f 


Stature  in  metres 

A 

Weight  in  kgm. 

A. 

WjU 

xlOO 

A 

Age 

Male 

Female 

%  F/M 

Male 

Female 

%  F/M 

Male 

Female 

0 

0-50 

0-48 

96-0 

3-20 

2-91 

90-9 

2-56 

2-64 

1 

0-70 

0-69 

98-6 

1000 

9-30 

93-0 

2-92 

283 

2 

0-80 

0-78 

97-5 

1200 

11-40 

95-0 

2-35 

2-40 

3 

0-86 

0-85 

98-8 

13-21 

12-45 

94-2 

2-09 

203 

4 

0-93 

0-91 

97-6 

1507 

1418 

94-1 

1-84 

1-88 

6 

0-99 

0-97 

98-4 

16-70 

15-50 

92-8 

1-89 

1-69 

6 

105 

103 

98-6 

18-04 

16-74 

92-8 

1-56 

1-53 

1 

Ml 

MO 

98-6 

20-16 

18-45 

91-5 

1-48 

1-39 

8 

M7 

114 

97-3 

22-26 

19-82 

89-0 

1-39 

1-34 

9 

1-23 

1-20 

97-8 

24-09 

22-44 

93-2 

1-29 

1-30 

10 

1-28 

1-25 

97-3 

2612 

24-24 

92-8 

1-25 

1-24 

11 

1-33 

1-28 

96-1 

27-85 

26-25 

94-3 

1-18 

1-25 

12 

1-36 

1-33 

97-6 

31-00 

30-54 

98-5 

1-23 

1-38 

13 

1-40 

1-39 

98-8 

35-32 

34-65 

98-1 

1-29 

1-29 

14 

1-49 

1-45 

97-3 

40-50 

38-10 

94-1 

1-21 

1-25 

15 

1-56 

1-47 

94-6 

46-41 

41-30 

89-0 

1-22 

1-30 

16 

1-61 

1-52 

93-2 

53-39 

44-44 

83-2 

1-20 

1-32 

17 

1-67 

1-54 

92-5 

57-40 

49-08 

85-5 

1-23 

i-a4 

18 

1-70 

1-56 

91-9 

61-26 

5310 

86-.7 

1-24 

1-40 

19 

1-71 





63-32 





1-20 

. 

20 

1-71 

1-57 

91-8 

65-00 

54-46 

83-8 

1-30 

1-41 

25 

1-72 

1-58 

91-6 

68-29 

5508 

80-7 

1-39 

1-39 

30 

172 

1-58 

91-7 

68-90 

5514 

80-0 

1-35 

1-39 

40 

1-71 

1-56 

90-8 

68-81 

56-65 

82-3 

1-38 

1-49 

50 

1-67 

1-54 

91-8 

67-45 

58-45 

86-7 

1-45 

1-59 

60 

1-64 

1-52 

92-5 

65-50 

56:73 

86-6 

1-48 

1-61 

70 

1-62 

1-51 

93-3 

6303 

53-72 

85-2 

1-48 

1-58 

80 

1-61 

1-51 

93-4 

61-22 

51-52 

84-1 

1-46 

1-50 

This  is  what  Sir  Wilham  Petty  had  meant  in  the  seventeenth  century 
by  his  Political  Arithmetic,  and  what  Quetelet  meant  in  the  nine.teenth 
by  his  Physique  Sociale.  But  "statistics"  nowadays  are  counts  and 
measures  of  all  sorts  of  things;    and  statistical  science  arranges, 

*  The  figures  for  height  and  weight  given  in  my  first  edition  were  Quetelet's 
smoothed  or  adjusted  values.     I  have  gone  back  to  his  original  data. 

t  This  "almost  steady  growth,"  from  about  seven  years  old  to  eleven,  means 
that  the  curve  of  growth  is  a  nearly  straight  line  during  this  period:  a  result 
already  found  by  Elderton  for  Glasgow  children  {Biometrika,  x,  p.  293,  1914-15), 
by  Fessard  and  Laufer  in  Paris  {Nouvelles  Tables  de  Croissance,  1935,  p.  13),  etc. 


Ill] 


OF  STATISTICAL  METHODS 


91 


explains,  and  draws  deductions  from,  the  resulting  series  and  arrays 
of  numbers.  It  deals  with  simple  and  measurable  effects,  due  to 
complex  and  often  unknown  causes ;  and  when  experiment  is  not  at 
hand  to  disentangle  these  causes,  statistical  methods  may  still  do 
something  to  elucidate  them. 

Now  as  to  the  growth  of  man,  if  the  child  be  some  20  inches,  or 
say  50  cm.,  tall  at  birth,  and  the  man  some  six  feet,  or  180  cm., 
high  at  twenty,  we  may  say  that  his  average  rate  of  growth  had 


10  13 

Time  in  years 


20 


Fig.  4.     Curve  of  growth  in  man.   From  Quet^let's  Belgian  data. 
The  curve  H*  is  proportional  to  the  height  cubed. 

been  (180  -  50)/20  cm.,  or  6-5  cm.  per  annum.  But  we  well  know 
that  this  is  but  a  rough  preliminary  statement,  and  that  growth 
was  surely  quick  during  some  and  slow  during  other  of  those  twenty 
years;  we  must  learn  not  only  the  result  of  growth  but  the  course 
of  growth ;  we  must  study  it  in  its  continuity.  This  we  do,  in  the 
first  instance,  by  the  method  of  coordinates,  plotting  magnitude 
against  time.  We  measure  time  along  a  certain  axis  (x),  and  the 
magnitude  in  question  along  a  coordinate  axis  (y);  a  succession  of 
points  defines  the  magnitudes  reached  at  corresponding  epochs,  and 
these  points  constitute  a  ''curve  of  growth''  when  we  join  them 
together. 


92  THE  RATE  OF  GROWTH  [ch. 

Our  curve  of  growth,  whether  for  weight  or  stature,  has  a  definite 
form  or  characteristic  curvature:  this  being  a  sign  that  the  rate  of 
growth  is  not  always  the  same  but  changes  as  time  goes  on.  Such 
as  it  is,  the  curvature  alters  in  an  orderly  way;  so  that,  apart  from 
minor  and  "fortuitous"  irregularities,  our  curves  of  growth  tend  to 
be  smooth  curves.  And  the  fact  that  they  are  so  is  an  instance  of 
that  "principle  of  continuity"  which  is  the  foundation  of  all  physical 
and  natural  science. 

The  curve  of  growth  (Fig.  4)  for  length  or  stature  in  man  indicates 
a  rapid  increase  at  the  outset,  during  the  quick  growth  of  babyhood ; 
a  long  period  of  slower  but  almost  steady  growth  in  boyhood;  as 
a  rule  a  marked  quickening  in  his  early  teens,  when  the  boy  comes 
to  the  "growing  age";  and  a  gradual  arrest  of  growth  as  he  "comes 
to  his  full  height"  and  reaches  manhood.  If  we  carried  the  curve 
farther,  we  should  see  a  very  curious  thing.  We  should  see  that  a 
man's  full  stature  endures  but  for  a  spell;  long  before  fifty*  it  has 
begun  to  abate,  by  sixty  it  is  notably  lessened,  in  extreme  old  age 
the  old  man's  frame  is  shrunken  and  it  is  but  a  memory  that  "he 
once  was  tall";  the  dechne  sets  in  sooner  in  women  than  in  men, 
and  "a  httle  old  woman"  is  a  household  word.  We  have  seen, 
and  we  see  again,  that  growth  may  have  a  negative  value,  pointing 
towards  an  inevitable  end.  The  phenomenon  of  negative  growth 
extends  to  weight  also;  it  is  largely  chemical  in  origin;  the  meta- 
bolism of  the  body  is  impaired,  and  the  tissues  keep  pace  no  longer 
with  senile  wastage  and  decay. 

We  must  be  very  careful,  however,  how  we  interpret  such  a  Table 
as  this;  for  it  records  the  character  of  a  population,  and  we  are  apt 
to  read  in  it  the  life-history  of  the  individual.  The  two  things  are 
not  necessarily  the  same.  That  a  man  grows  less  as  he  grows  older 
all  old  men  know;  but  it  may  also  be  the  case,  and  our  Table  may 
indicate  it,  that  the  short  men  live  longer  than  the  tall. 

Our  curve  of  growth  is,  by  implication,  a  "time-energy"  diagram  f 
or  diagram  of  activity.  As  man  grows  he  is  absorbing  energy 
beyond  his  daily  needs,  and  accumulating  it  at  a  rate  depicted  in 

*  Dr  Johnson  was  not  far  wrong  in  saying  that  "life  declines  from  thirty-five"; 
though  the  Autocrat  of  the  Breakfast-table  declares,  like  Cicero,  that  "the  furnace 
is  in  full  blast  for  ten  years  longer". 

t  J.  Joly,  The  Abundance  of  Life,  1915  (1890),  p.  86. 


Ill]  OF  MAN'S  GROWTH  AND  STATURE  93 

our  curve ;  till  the  time  comes  when  he  accumulates  no  longer,  and 
is  constrained  to  draw  upon  his  dwindhng  store.  But  in  part,  tte 
slow  decline  in  stature  is  a  sign  of  the  unequal  contest  between  our 
bodily  powers  and  the  unchanging  force  of  gravity,  which  draws  us 
down  when  we  would  fain  rise  up*;  we  strive  against  it  all  our 
days,  in  every  movement  of  our  limbs,  in  every  beat  of  our  hearts. 
Gravity  makes  a  difference  to  a  man's  height,  and  no  slight  one, 
between  the  morning  and  the  evening ;  it  leaves  its  mark  in  sagging 
wrinkles,  drooping  mouth  and  hanging  breasts ;  it  is  the  indomitable 
force  which  defeats  us  in  the  end,  which  lays  us  on  our  death-bed 
and  lowers  us  to  the  grave  f.  But  the  grip  in  which  it  holds  us  is 
the  title  by  which  we  live;  were  it  not  for  gravity  one  man  might 
hurl  another  by  a  puff  of  his  breath  into  the  depths  of  space,  beyond 
recall  for  all  eternity  {. 

Side  by  side  with  the  curve  which  represents  growth  in  length, 
or  height  or  stature,  our  diagram  shews  the  corresponding  curve  of 
weight.  That  this  curve  is  of  a  different  shape  from  the  former  one 
is  accounted  for  in  the  main  (though  not  wholly)  by  the  fact — 
which  we  have  already  dealt  with — that  in  similar  bodies  volume, 
and  therefore  weight,  varies  as  the  cubes  of  the  linear  dimensions; 
and  drawing  a  third  curve  to  represent  the  cubes  of  the  corresponding 
heights,  it  now  resembles  the  curve  of  weight  pretty  closely,  but 
still  they  are  not  quite  the  same.  There  is  a  change  of  direction, 
or  "point  of  inflection,"  in  the  curve  of  weight  at  one  or  two  years 
old,  and  there  are  certain  other  features  in  our  curves  which  the 
scale  of  the  diagram  does  not  make  clear;  and  all  these  differences 
are  due  to  the  fact  that  the  child  is  changing  shape  as  he  grows, 
that    other   linear    dimensions    grow    somewhat   differently   from 

*  "Lou  pes,  mestre  de  tout  (Le  poids,  maitre  de  tout),  m^stre  senso  vergougno, 
Que  te  tirasso  en  bas  de  sa  brutalo  pougno."  J.  H.  Fabre,  Oubreto  prouven^alo, 
p.  61. 

t  The  continuity  of  the  phenomenon  of  growth,  and  the  natural  passage  from 
the  phase  of  increase  to  that  of  decrease  or  decay,  are  admirably  discussed  by 
Enriques,  in  La  Morte,  Rivista  di  Scienza,  1907,  and  in  Wachstum  und  seine 
analytische  Darstellung,  Biol.  Centralbl.  June,  1909.  Haller  [Elementa,  vii, 
p.  68)  recognised  decrementum  as  a  phase  of  growth,  not  less  important  (theoretically) 
than  incrementum ;  ''Hristis,  sed  copiosa,  haec  est  materies." 

X  Boscovich,  Theoria,  para.  552,  "Homo  hominem  arreptum  a  Tellure,  et 
utcumque  exigua  impulsum  vi  vel  uno  etiam  oris  flatu  impetitum,  ab  hominum 
omnium  commercio  in  infinitum  expelleret,  nunquam  per  totam  aeternitatem 
rediturum." 


94 


THE  RATE  OF  GROWTH 


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Ill]  OF  CURVES  OF  GROWTH  95 

length  or  stature,  and  in  short  that  infant,  boy  and  man  are  not 
similar  figures*.  The  change  of  form  seems  sHght  and  gradual,  but 
behind  it  he  other  and  more  complex  things.  Th6  changing  ratio 
between  height  and  weight  imphes  changes  in  the  child's  metabolism^ 
in  the  income  and  expenditure  of  the  body.  The  infant  stores  up 
fat,  and  the  active  child  "runs  it  off"  again;  at  four  years  old  or 
five,  bodily  metaboUsm  and  increase  of  weight  are  at  a  minimum; 
but  a  fresh  start  is  made,  a  new  "nutritional  period"  sets  in,  and  the 
small  schoolboy  grows  stout  and  sJ3rong|. 

Our  curve  of  growth  shews  at  successive  epochs  of  time  the  height 
or  weight  which  has  been  reached  by  then;  it  plots  changing 
magnitude  (y)  against  advancing  time  (x).  It  is  essentially  a 
cumulative  or  summation  curve;  it  sums  up  or  "integrates"  all  the 
successive  magnitudes  which  have  been  added  in  all  the  foregoing 
intervals  of  time.  Where  the  curve  is  steep  it  means  that  growth 
was  rapid,  and  when  growth  ceases  the  curve  becomes  a  horizontal 
line.  It  follows  that,  by  measuring  the  slope  or  steepness  of  our 
curve  of  growth  at  successive  epochs,  we  shall  obtain  a  picture  of 
the  successive  velocities  or  growth-rates. 

The  steepness  of  a  curve  is  measured  by  its  "gradient  J,"  or  we 
may  roughly  estimate  it  by  taking  for  equal  intervals  of  time 
(strictly  speaking,  for  each  infinitesimal  interval  of  time)  the  incre- 
ment added  during  that  interval;  and  this  amounts  in  practice  to 
taking  the  differences  between  the  values  given  for 'the  successive 
epochs,  or  ages,  which  we  have  begun  by  studying.  Plotting  these 
successive  differences  against  time,  we  obtain  a  curve  each  point  on 
which  represents  a  certain  rate  at  a  certain  time;  and  while  the 
former  curve  shewed  a  continuous  succession  of  varying  magnitudes, 
this  shews  a  succession  of  varying  velocities.  The  mathematician 
calls  it  a  curve  of  first  differences ;  we  may  call  it  a  curve  of  annual 
(or  other)  increments ;  but  we  shall  not  go  wrong  if  we  call  it  a  curve 
of  the  rate  (or  rates)  of  growth,  or  still  more  simply,  a  velocity-curve. 

*  According  to  Quetelet's  data,  man's  stature  is  multiplied  by  3-4  and  his  weight 
by  20-3,  between  birth  and  the  age  of  twenty-one.  But  the  cube  of  3-4  is  nearly 
40;  so  the  weight  at  birth  should  be  multiplied  forty  times  by  the  age  of 
twenty-one,  if  infant,  boy  and  man  were  similar  figures. 

t  Cf.  T.  W.  Adams  and  E.  P.  Poulton,  Heat  production  in  man,  Ouy's  Hospital 
Reports  (4),  xvn,  1937,  and  works  quoted  therein. 

I  That  is,  by  its  trigonometrical  tangent,  referred  to  the  base-line. 


96 


THE  RATE  OF  GROWTH 


[CH. 


We  have  now  obtained  two  different  but  closely  related  curves, 
based  on  the  selfsame  facts  or  observations,  and  illustrating  them 
in  different  ways.  One  is  the  inverse  of  the  other ;  one  is  the  integral 
and  one  the  differential  of  the  other;  and  each  makes  clear  to  the 
eye  phenomena  which  are  imphcit,  but  are  less  conspicuous,  in  the 
other.  We  are  using  mathematical  terms  to  describe  or  designate 
them;  but  these  "curves  of  growth"  are  more  comphcated  than 
the  curves  with  which  mathematicians  are  wont  to  deal.  In  our 
study  of  growth  we  may  well  hope  to  find  curves  simpler  than  these; 


20» 


3       5        7       9       II      13      15      17      19     21 
Age  in  years 
Fig.  5.     Annual  increments  of  growth  in  man.     From  Quetelet's  Belgian  data. 

but  in  the  successive  annual  increments  of  a  boy's  growth  (as  Fig.  5 
exhibits  them)  we  are  deahng  with  no  one  continuous  operation 
(such  as  a  mathematical  formula  might  define),  but  with  a  succession 
of  events,  changing  as  times  and  circumstances  change. 

Our  curve  of  increments,  or  of  first  differences,  for  man's  stature 
(Fig.  5)  is  based,  perforce,  on  annual  measurements,  and  growth 
alters  quickly  enough  at  certain  ages  to  make  annual  intervals  unduly 
long;  nevertheless  our  curve  shews  several  important  things.  It 
suffices  to  shew,  for  length  or  stature,  that  the  growth-rate  in  early 
infancy  is  such  as  is  never  afterwards  re-attained.  From  this  high 
early  velocity  the  rate  on  the  whole  falls  away,  until  growth  itself 


Ill] 


OF  HUMAN  STATURE 


97 


comes  to  an  end  * ;  but  it  does  so  subject  to  certain  important  changes 
and  interruptions,  which  are  much  the  same  whether  we  draw  them 
from  Quetelet's  Belgian  data,  or  from  the  British,  American  and 
other  statistics  of  later  writers.  The  curve  falls  fast  and  steadily 
during  the  first  couple  of  years  of  the  child's  hfe  (a).  It  runs  nearly 
level  during  early  boyhood,  from  four  or  five  years  old  to  nine  or 
ten  (6).  Then,  after  a  brief  but  unmistakable  period  of  depression! 
during  which  growth  slows  down  still  more  (c),  the  boy  enters  on 

Annual  increments  of  stature  and  of  weight  in  man 
{After  Quetelet;  see  Table,  p.  90) 


Stature 

A 

(cm.) 

Weight 
Male 

(kgm.) 

Age 

Male 

Female 

Fema 

0-  1 

20 

21 

6-8 

6-4 

1-  2 

10 

9 

2-0 

1-9 

2-  3 

6 

7 

1-2 

11 

3-  4 

7 

6 

1-9 

1-7 

4-  5 

6 

6 

1-6 

1-3 

5-  6 

6 

6 

1-3 

1-2 

6-  7 

6 

7 

2-1 

1-7 

7-  8 

6 

4 

21 

1-4 

8-  9 

6 

6 

1-8 

2-6 

9-10 

5 

5 

2-0 

1-8 

10-11 

5 

3 

1-7 

2-0 

11-12 

3 

5 

3-2 

4-3 

12-13 

4 

6 

4-3 

4-1 

13-14 

9 

6 

5-2 

3-5 

14-15 

7 

2 

5-9 

3-2 

15-16 

5 

3 

7-0 

2-1 

16-17 

6 

4 

40 

4-6 

17-18 

3 

2 

3-9 

40 

18-19 

1 

1 

21 

1-4 

19-20 

0 

0 

1-7 

— 

his  teens  and  begins  to  "grow  out  of  his  clothes";  it  is  his  "growing 
age",  and  comes  to  its  height  when  he  is  about  thirteen  or  fourteen 
years  old  (d).  The  lad  goes  on  growing  in  stature  for  some  years  more, 
but  the  rate  begins  to  fall  off  (e),  and  soon  does  so  with  great  rapidity. 
The  corresponding  curve  of  increments  in  weight  is  not  very 
different  from  that  for  stature,  but  such  differences  as  there  are 

*  As  Haller  observed  it  to  do  in  the  chick f  "Hoc  iterum  incrementum  miro 
ordine  distribuitur,  ut  in  principio  incubationis  maximum  est;  inde  perpetuo 
minuatur"  {Elementa  Pkysiologiae,  viii,  p.  294).  Or  as  Bichat  says,  "II  y  a 
surabondance  de  vie  dans  I'enfant"  {Sur  la  Vie  et  la  Mort,  p.  1). 

t  This  depression,  or  slowing  down  before  puberty,  seems  to  be  a  universal 
phenomenon,  common  to  all  races  of  men.  It  is  a  curious  thing  that  Quetelet's 
"adjusted  figures"  (which  I  used  in  my  first  edition)  all  but  smooth  out  of 
recognition  this  characteristic  feature  of  his  own  observations. 


98  THE  RATE  OF  GROWTH  [ch. 

between  them  are  significant  enough.  There  is  some  tendency  for 
growth  in  weight  to  fall  off  or  fluctuate  at  four  or  five  years  old, 
before  the  small  boy  goes  to  school ;  but  there  is,  or  should  be,  httle 
retardation  of  weight  when  growth  in  height  slows  down  before 
he  enters  on  his  teens*.  The  healthy  lad  puts  on  weight  again 
more  and  more  rapidly,  for  some  httle  while  after  gjowth  in  stature 
has  slowed  down;  and  normal  increase  of  weight  goes  on,  more 
slowly,  while  the  man  is  "fiUing  out,"  long  after  growth  in  stature 
has  come  to  an  end.  But  somewhere  about  thirty  he  begins  losing 
weight  a  httle ;  and  such  subsequent  slow  changes  as  men  commonly 
undergo  we  need  not  stop  to  deal  with. 

The  differences  in  stature  and  build  between  one  race  and  another 
are  in  hke  manner  a  question  of  growth-rate  in  the  main.  Let  us 
take  a  single  instance,  and  compare  the  annual  increments  of 
growth  in  Chinese  and  Enghsh  boys.  The  curves  are  much  the 
same  in  form,  but  differ  in  amplitude  and  phase.  The  Enghsh  boy 
is  growing  faster  all  the  while;  but  the  minimal  rate  and  the 
maximal  rate  come  later  by  a  year  or  more  than  in  the  Chinese 
curve  t  (Fig.  6). 

Quetelet  was  not  the  first  to  study  man's  growth  and  stature, 
nor  was  he  the  first  student  of  social  statistics  and  "demography." 
The  foundations  of  modern  vital  statistics  had  been  laid  by  Graunt 
and  Petty  in  the  seventeenth  century  J;  the  economists  developed 
the  subject  during  the  eighteenth §,  and  parts  of  it  were  studied 

♦  That  the  annual  increments  of  weight  in  boys  are  nearly  constant,  and  the 
curve  of  growth  nearly  a  straight  line  at  this  age,  especially  from  about  8  to  11, 
has  been  repeatedly  noticed.  Cf.  Elderton,  Glasgow  School-children,  Biometrika, 
X,  p.  283,  1914-15;  Fessard  and  others,  Croissance  des  Ecoliers  Pariaiens,  1934, 
p.  13.  But  careful  measurements  of  American  children,  by  Katherine  Simmons 
and  T.  Wingate  Todd,  shew  steadily  increasing  increments  from  four  years  old 
tiU  puberty  {Growth,  n,  pp.  93-133,  1938). 

t  For  copious  bibliography,  see  J.  Needham,  op.  cit.,  also  Gaston  Backmah,  Das 
Wachstum der  Korperlange  des Menschen,  K.  Sv.Vetensk.  Akad.  Hdlgr.  (3),  xiv,  1934. 

X  Cf.  John  Graunt's  Natural  and  Political  Observations. .  .upon  the  Bills  of 
Mortality,  London,  1662;  The  Economic  Writings  of  Sir  William  Petty,  ed.  by 
C.  H.  Hull,  2  vols,,  Cambridge,  Mass.,  1927.  Concerning  Graimt  and  Petty — two 
of  the  original  Fellows  of  the  Royal  Society — see  {int.  al.)  H.  Westergaard,  History 
of  Statistics,  1932,  and  L.  Hogben  (and  others).  Political  Arithmetic,  1938. 

§  Besides  the  many  works  of  the  economists,  cf.  J.  G.  Roederer,  Sermo  de 
pondere  et  longitudine  recens-natorum.  Comment.  Soe.  Beg.  Sci.  Oottingae,  m, 
1753;  J.  F.  G.  Dietz,  De  temporum  in  graviditate  et  partu  aestimatione,  Diss., 
Gottingen,  1757. 


Ill] 


OF  HUMAN  STATURE 


99 


7 

1         1         1         1         1 

1          1         1 

1 

6 

*x   N^ 

- 

5 

\        y^ 

\      \ 
\      \ 

a  4 

^^        '>            / 

\      \ 

a 

v         r 

6  ^ 

Chinese 

V 

2 

\ 

1 
0 

1        f,       1        1        1 

1      1      1 

1 

7 

k               9/,o                        12/,  3 

•6/l7 

I7/|8 

Age 

ig.  6.     Annual  increments  of  stature. 

From  Roberts 

(English) 

and  Appleton's  (Chinese)  data. 

•Date 
1760*61  ^62'63'64'65  W67^68*6970'7r72*73'74V5*76  1777 

|6'0 


0    1     2    3    4    5    6    7    8    9   10  n  12  13  14  15  16  17  18 

Age  in  years 

Fig.  7.     Curve  of  growth  of  a  French  boy  of  the  eighteenth  century. 

From  Scammon,  after  BufFon. 


100 


THE  RATE  OF  GROWTH 


CH. 


eagerly  in  the  early  nineteenth,  when  the  exhaustion  of  the  armies 
of  France  and  the  evils  of  factory  labour  in  England  drew  attention 
to  the  stature  and  physique  of  man  and  to  the  difference  between 
the  healthy  and  the  stunted  child*. 

A  friend  of  BufPon's,  the  Count  Philibert  Gueneau  de  Montbeillard, 
kept  careful  measurements  of  his  own  son;  and  Buifon  pubhshed 
these  in  1777,  in  a  supplementary  volume  of  the  Histoire  Naturelle'f. 
The  child  was  born  in  April  1759;  it  was  measured  every  six  months 


1 
-  \ 

1  1  I 

1  1  1 

1      '      ' 

.      1      .      . 

20 

A 

— 

_  \ 

- 

- 

d 

- 

- 

\" 

- 

- 

A 

- 

10 

1 

1  1  1 

b        . 

1      1 

A 

1     1     1     1 

1     1 

10 


20 


Fig.  8.     Annual  increments  of  stature  of  the  said  French  boy. 

for  seventeen  years,  and  the  record  gives  a  curve  of  great  interest 
and  beauty  (Fig.  7).  There  are  two  ways  of  studying  such  a 
phenomenon — the  statistical  method  based  on  large  numbers,  and 
the  careful  study  of  the  individual  case;  the  curve  of  growth  of  this 
one  French  child  is  to  all  intents  and  purposes  identical,  save  that 
the  boy  was  throughout  a  trifle  taller,  with  the  mean  curve  yielded 
by  a  recent  study  of  forty-four  thousand  Uttle  Parisians  {. 

In  young  Montbeillard's  case  the  "curve  of  first  differences,"  or 
of  the  successive  annual  increments  of  stature  (Fig.  8),  is  clear  and 
beautiful.    It  shews  (a)  the  rapid,  but  rapidly  diminishing,  rate  of 

*  Cf.  M.  Hargenvilliers,  Recherches . . . sur . .  .le  recrutement  de  Varmie  en  France, 
1817;  J.  W.  Cowell,  Measurements  of  children  in  Manchester  and  Stockport, 
Factory  Reports,  i;  and  works  referred  to  by  Quetelet. 

t  See  Richard  E.  Scammon,  The  first  seriatim  study  of  human  growth,  Amer. 
Journ.  of  Physical  Anthropology,  x,  pp.  329-336,  1927. 

J  MM.  Variot  et  Chaumet,- Tables  de  croissance,  dressees. .  .d'apr^s  les  mensura- 
tions de  44,000  enfants  parisiens,  Bull,  et  M4m.  Soc.  d'Anthropologie,  iii,  p.  55, 
1906. 


Ill]  OF  GROWTH  IN  MAN  AND  WOMAN  101 

growth  in  infancy ;  (6)  the  steady  growth  in  early  boyhood ;  (c)  the 
period  of  retardation  which  precedes,  and  (d)  the  rapid  growth  which 
accompanies,  puberty. 

Buifon,  with  his  usual  wisdom,  adds  some  remarks  of  his  own, 
which  include  two  notable  discoveries.  He  had  observed  that  a 
man's  stature  is  measurably  diminished  by  fatigue,  and  the  loss  soon 
made  up  for  in  repose;  long  afterwards  Quetelet  said,  to  the  same 
effect,  "le  lit  est  favorable  a  la  croissance,  et  le  matin  un  homme  est 
un  peu  plus  grand  que  le  soir."  Buff  on  asked  whether  growth 
varied  with  the  seasons,  and  Montbeillard's  data  gave  him  his  reply. 
Growth  was  quicker  from  April  to  October  than  during  the  rest  of 
the  year:  shewing  that  "la  chaleur,  qui  agit  generalement  sur  le 
developpement  de  tous  les  etres  organisees,  influe  considerablement 
sur  I'accroissement  du  corps  humain."  Between  five  years  old  and 
ten,  the  child  grew  seven  inches  during  the  five  summers,  but  during 
the  five  winters  only  four;  there  was  a  Hke  difference  again,  though 
not  so  great,  while  the  boy  was  growing  quickly  in  his  teens;  but 
there  were  no  seasonal  differences  at  all  from  birth  to  five  years  old, 
when  the  child  was  doubtless  sheltered  from  both  heat  and  cold*. 

On  rate  of  growth  in  man  and  ivoman 

That  growth  follows  a  different  course  in  boyhood  and  in  girlhood 
is  a  matter  of  common  knowledge;  but  differences  in  the  curves  "of 
growth  are  not  very  apparent  on  the  scale  of  our  diagrams.  They 
are  better  seen  in  the  annual  increments,  or  first  differences;  and 
we  may  further  simplify  the  comparison  by  representing  the  girl's 
weight  or  stature  as  a  percentage  of  the  boy's. 

Taking  weight  to  begin  with  (Fig.  9),  the  girl's  growth-rate  is 
steady  in  childhood,  from  two  or  three  to  six  or  seven  years  old, 

*  Growth-rates  based  on  the  continuqus  study  of  a  single  individual  are  rare; 
we  depend  mostly  on  average  measurements  of  many  individuals  grouped  according 
to  their  average  age.  That  this  is  a  sound  method  we  take  for  granted,  but  we  may 
lose  by  it  as  well  as  gain.  (See  above,  p.  92.)  The  chief  epochs  of  growth,  the  chief 
singularities  of  the  curve,  will  come  out  much  the  same  in  the  individual  and  in  the 
average  curve.  But  if  the  individual  curves  be  skew,  averaging  them  will  tend  to 
smooth  the  skewnesa  away;  and,  more  curiously,  if  they  be  all  more  or  less  diverse, 
though  all  symmetrical,  a  certain  skewness  will  tend  to  develop  in  the  composite  or 
average  curve.  Cf.  Margaret  Merrill,  The  relationship  of  individual  to  average 
growth,  Human  Biology,  iii.  pp.  37-70,  1931. 


102 


THE  RATE  OF  GROWTH 


[CH. 


J \ I L 


J \ L 


0         2        4         6         8        10       12       14        16       18  19 
Age 
Fig.  9.    Annual  increase  in  weight  of  Belgian  boys  and  girls. 
From  Quetelet's  data.    (Smoothed  curves.) 


10  15 

Age 

Fig.  10.    Percentage  ratio  of  female  weight  and  stature  to  male. 
From  Quetelet's  Belgian  data. 


23 


Ill] 


OF  GROWTH  IN  MAN  AND  WOMAN 


103 


just  as  is  the  boy's;  but  her  curve  stands  on  a  lower  level,  for  the 
little  maid  is  putting  on  less  weight  than  the  boy  (b).  Later  on, 
her  rate  accelerates  (c)  sooner  than  does  his,  but  it  neyer  rises  quite 
so  high  {d).  After  a  first  maximum  at  eleven  or  twelve  her  rate  of 
growth  slows  down  a  Httle,  then  rises  to  a  second  maximum  when 


112 

110 

108 

106 

104 

102 

100 

98 

96 

94 

92 

90 


Matle  weight  /   \y  / 


1. \ L 


J I L 


1  3         5         7         9         11        13 


Fig.  i: 


Relative  weight  of  American  boys  and  girls. 
From  Simmons  and  Todd's  data. 


she  is  sixteen  or  seventeen,  after  the  boy's  phase  of  quickest  growth 
is  over  and  done.  This  second  spurt  of  growth,  this  increase  of 
vigour  and  of  weight  in  the  girl  of  seventeen  or  eighteen,  Quetelet's 
figures  indicate  and  common  observation  confirms.  Last  of  all, 
while  men  stop  adding  to  their  weight  about  the  age  of  thirty  or 
before,  this  does  not  happen  to  women.  They  increase  in  weight, 
though  slowly,  till  much  later  on:  until  there  comes  a  final  phase, 
in  both  sexes  ahke,  when  weight  and  height  and  strength  decline 
together. 


104 


THE  RATE  OF  GROWTH 


[CH. 


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Ill 


OF  GROWTH  IN  CHILDHOOD 


105 


Stature  and  weight  of  American  children  (Ohio) 
(From  Katherine  Simmons  and  T.  Wingate  Todd's  data) 


Stature  (cm.) 

%  ratio 

Weight  (lbs.) 
Boys            Girls 

Age 

Boys 

Girls 

%  rati 

3  months 

61-3 

59-3 

96-7 

14-4 

13-1 

91-1 

1  year 

761 

.  74-2 

97-5 

23-9 

21-9 

91-8 

2     „ 

87-4 

86-2 

98-6 

29-1 

27-5 

94-7 

3    „ 

96-2 

95-5 

99-3 

33-5 

32-5 

96-9 

4    „ 

103-9 

103-2 

99-3 

38-4 

37-1 

96-7 

o     „ 

110-9 

110-3 

99-4 

43-2 

42-3 

98-1 

6     „ 

117-2 

117-4 

1001 

48-5 

48-6 

100-0 

7     „ 

123-9 

123-2 

99-4 

54-7 

54-0 

98-8 

8     „ 

130-1 

129-3 

99-4 

62-2 

61-5 

98-7 

9     „ 

136-0 

135-7 

99-7 

69-5 

70-9 

102-0 

10     „ 

141-4 

140-8 

99-6 

78-5 

77-6 

98-8 

11     » 

146-5 

147-8 

100-7 

86-5 

87-0 

100-6 

12     „ 

151-1 

155-3 

102-8 

92-7 

102-7 

110-7 

13    „ 

156-7 

159-9 

102-0 

102-8 

114-6 

111-4 

Mean  of  observed  incremeyits  of  stature  and  weight  of 
American  children 

Increment  of  stature  (mm.)         Increment  of  weight  (lbs.) 


'' 

% 

r 

/o 

Age 

Boys 

Girls 

ratio 

Boys 

Girls 

ratio 

3  m 

.  -  1  yr.           150-4 

150-1 

99-8 

9-32 

8-07 

93-8 

1  yr.  -  2  ", 

123-9 

132-0 

106-5 

4-97 

5-56 

1120 

2   , 

-  3   , 

88-1 

90-0 

102-2 

4-01 

4-18 

104-3 

3   , 

-  4   , 

73-9 

79-1 

106-9 

4-13 

4-46 

108-0 

4   , 

-  5    , 

69-4 

72-2 

104-0 

4-60 

4-55 

99-8 

5   , 

-  6   , 

67-0 

68-0 

101-5 

4-51 

5-08 

112-8 

6   , 

-  7    , 

64-1 

62-6 

97-6 

5-57 

5-40 

96-9 

7   , 

-  8   , 

61-2 

57-8 

94-4 

6-70 

6-65 

99-4 

8   , 

—9   , 

55-7 

60-1 

108-0 

6-64 

7-38 

1111 

9   , 

-10   , 

54-9 

57-7 

105-1 

7-92 

8-12 

104-8 

10   , 

-11    , 

51-9 

61-3 

118-2 

8-81 

9-58 

108-7 

11    , 

-12   , 

53-2 

66-9 

125-6 

9-54 

11-98 

1331 

12   , 

-13   , 

61-0 

55-1 

89-0 

10-90 

10-29 

94-7 

These  differences  between  the  two  sexes,  which  are  essentially 
phase-differences,  cause  the  ratio  between  their  weights  to  fluctuate 
in  a  somewhat  comphcated  way  (Figs.  10,  11).  At  birth  the  baby 
girl's  weight  is  about  nine-tenths  of  the  boy's.  She  gains  on  him  for 
a  year  or  two,  then  falls  behind  again ;  from  seven  or  eight  onwards 
she  gains  rapidly,  and  the  girl  of  twelve  or  thirteen  is  very  httle 
lighter  than  the  boy;  indeed  in  certain  American  statistics  she  is 
by  a  good  deal  the  heavier  of  the  two.     In  their  teens  the  boy  gains 


106 


THE  RATE  OF  GROWTH 


[CH. 


steadily,  and  the  lad  of  sixteen  is  some  15  per  cent,  heavier  than 
the  lass.  The  disparity  tends  to  diminish  for  a  while,  when  the 
maid  of  seventeen  has  her  second  spurt  of  growth ;  but  it  increases 
again,  though  slowly,  until  at  five-and-twenty  the  young  woman  is 
no  more  than  four-fifths  the  weight  of  the  man.  During  middle  life 
she  gains  on  him,  and  at  sixty  the  difference  stands  at  some  12 


1 1 \ r 


J i \ \ \ \ L 


0       2       4       6       8        10      12      14      16      18 

Age 

Fig.  12.     Annual  increments  of  stature,  in  boys  and  girls. 
From  Quetelet's  data.     (Smoothed  curves.) 


per  cent.,  not  far  from  the  mean  for  all  ages;   but  the  old  woman 
shrinks  and  dwindles,  and  the  difference  tends  to  increase  again. 

The  rate  of  increase  of  stature,  like  stature  itself,  differs  notably 
in  the  two  sexes,  and  the  differences,  as  in  the  case  of  weight,  are 
mostly  a  question  of  phase  (Fig.  12).  The  httle  girl  is  adding  rather 
more  to  her  stature  than  the  boy  at  four  years  old*,  but  she  grows 

*  This  early  spurt  of  growth  in  the  girl  is  shewn  in  Enghsh,  French  and  American 
observations,  but  not  in  Quetelet's. 


Ill 


OF  GROWTH  IN  CHILDHOOD 


107 


slower  than  he  does  for  a  few  years  thereafter  (6).  At  ten  years  old 
the  girl's  growth-rate  begins  to  rise  (c),  a  full  year  before  the  boy's; 
at  twelve  or  thirteen  the  rate  is  much  ahke  for  both,  but  it  has 
reached  its  maximum  for  the  girl.  The  boys'  rate  goes  on  rising, 
and  at  fourteen  or  fifteen  they  are  growing  twice  as  fast  as  the  girls. 
So  much  for  the  annual  increments,  as  a  rough  measure  of  the  rates 
of  growth.  In  actual  stature  the  baby  girl  is  some  2  or  3  per  cent, 
below  the  boy  at  birth;   she  makes  up  the  difference,  and  there  is 


7         9 
Age 
Fig.  13.     Ratio  of  female  stature  to  male. 


II        13        15 


From  Simmons  and  Todd's  data. 


From  R.  M.  Fleming's  data. 


good  evidence  to  shew  that  she  is  by  a  very  little  the  taller  for  a 
while,  at  about  five  years  old  or  six.  At  twelve  or  thirteen  she  is 
very  generally  the  taller  of  the  two,  and  we  call  it  her  "gawky 
age"  (Fig.  13). 

Man  and  woman  differ  in  length  of  life,  just  3,s  they  do  in  weight 
and  stature.  More  baby  boys  are  born  than  girls  by  nearly  5  peF 
cent.     The  numbers  draw  towards  equality  in  their  teens;    after 


108  THE  RATE  OF  GROWTH  '  [ch. 

twenty  the  women  begin  to  outnumber  the  men,  and  at  eighty-five 
there  are  twice  as  many  women  as  men  left  in  the  world*. 

Men  have  pondered  over  the  likeness  and  the  unhkeness  between 
the  short  lifetimes  and  the  long;  and  some  take  it  to  be  fallacious 
to  measure  all  alike  by  the  common  timepiece  of  the  sun.  Life, 
they  say,  has  a  varying  time-scale  of  its  own ;  and  by  this  modulus 
the  sparrow  hves  as  long  as  the  eagle  and  the  day-fly  as  the  manf. 
The  time-scale  of  the  living  has  in  each  case  so  strange  a  property 
of  logarithmic  decrement  that  our  days  and  years  are  long  in 
childhood,  but  an  old  man's  minutes  hasten  to  their  end. 

On  pre-natal  and  post-natal  growth 

The  rates  of  growth  which  we  have  so  far  studied  are  based  on 
annual  increments,  or  "first  differences"  between  yearly  determina- 
tions of  magnitude.  The  first  increment  indicates  the  mean  rate  of 
growth  during  the  first  year  of  the  infant's  life,  or  (on  a  further 
assumption)  the  mean  rate  at  the  mean  epoch  of  six  months  old; 
there  is  a  gap  between  that  epoch  and  the  epoch  of  birth,  of  which 
we  have  learned  nothing;  we  do  not  yet  know  whether  the  very 
high  rate  shewn  within  the  first  year  goes  on  rising,  or  tends  to  fall, 
as  the  date  of  birth  is  approached.  We  are  accustomed  to  inter- 
polate freely,  and  on  the  whole  safely,  between  known  points  on 
a  curve:  "si  timide  que  Ton  soit,  il  faut  bien  que  Ton  interpole," 
says  Henri  Poincare;  but  it  is  much  less  safe  and  seldom  justifiable 
(at  least  until  we  understand  the  physical  principle  involved  and 
its  mathematical  expression)  to  "extrapolate"  beyond  the  limits  of 
our  observations. 

We  must  look  for  more  detailed  observations,  and  we  may  learn 
much  to  begin  with  from  certain  old  tables  of  Russow's  J,  who  gives 

*  Cf.  F.  E.  A.  Crew's  Presidential  Address  to  Section  D  of  the  British  Association, 
1937. 

+  Cf.  Gaston  Backman,  Die  organische  Zeit,  Lunds  Universitets  Arsskrift,  xxxv, 
Nr.  7,  1939. 

J  Quoted  in  Vierordt's  Anatomische . . . Daten  und  Tabellen,  1906,  p.  13.  See 
also,  among  many  others,  Camerer's  data,  in  Pfaundler  and  Schlossman's  Hdb.  d. 
Kinderheilkunde,  i,  pp.  49,  424,  1908;  Variot,  op.  cit.;  for  pre-natal  growth,  R.  E. 
Scammon  and  L.  A.  Calkins,  Growth  in  the  Foetal  Period,  Minneapolis,  1929.  Also, 
on  this  and  many  other  matters,  E.  Faure-Fremiet,  La  cinetique  du  developpement, 
Paris,  1925;  and,  not  least,  J.  Needham,  Chemical  EinJbryology,  1931. 


in] 


OF  GROWTH  IN  CHILDHOOD 


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110 


THE  RATE  OF  GROWTH 


[CH. 


the  stature  of  the  infant,  month  by  month,  during  the  first  year  of 
its  hfe,  as  follows : 

Mean  growth  of  an  infant,  in  its  first  twelve-month 
(After  Russow) 


Age  (months) 

0       12       3       4 

5       6       7       8 

9 

10     11 

12 

Length  (cm.) 
Monthly  incre- 
ment (cm.) 

50     54     58     60     62 
—      4      4      2      2 

64     65     66     67-5 
2       111-5 

68 
0-5 

69     70-5 
1       1-5 

72 
1-5 

From  these  data  of  Russow's  for  German  children,  rough  as 
indeed  they  are,  from  Variot's  for  little  Parisians  (Fig.  14),  and  from 


3  10  12 

Age  in  months 

Fig.  14.     Growth  of  Parisian  children  (boys)  from  birth  to  twelve  months  old. 
From  G.  Variot's  data;  Russow's  German  data  are  also  shewn,  by   x  x  x  . 

many  more,  we  see  that  the  rate  of  growth  rises  steadily  and  even 
rapidly  as  we  pass  backwards  towards  the  date  of  birth.  It  is  never 
anything  like  so  great  again.  It  is  an  impressive  demonstration 
of  the  dynamic  potentiahty,  of  the  store  of  energy,  in  the  newborn 
child. 

But  birth  itself  is  but  an  incident,  an  inconstant  epoch,  in  the 
life  and  growth  of  a  viviparous  animal.     The  foal  and  the  lamb 


Ill]  OF  PRENATAL  GROWTH  111 

are  born  later  than  a  man-child;  the  puppy  and  the  kitten  are  born 
easier,  and  in  more  helpless  case  than  ours;  the  mouse  comes  into 
the  world  still  earher  and  more  inchoate,  so  much  so  that  even  the 
little  marsupial  is  scarcely  more  embryonic  and  unformed*.  We 
must  take  account,  so  far  as  each  case  permits,  of  pre-natal  or  intra- 
uterine growth,  if  we  are  to  study  the  curve  of  growth  in  its  entirety. 
According  to  Hisf,  the  following  are  the  mean  lengths  from  month 
to  month  of  the  imborn  child : 

Months  01  23456789  10 

(Birth) 

Length  (mm.)   0   7-5  40    84   162  275  352  402  443  472  490) 

500) 

Increment  per     —     7-5      32-5      44        78      113        77      '50       41        29        18) 
month  (mm.)  28) 

These  data  hnk  on  very  well  to  those  of  Russow,  which  we  have 
just  considered;  and  (though  His's  measurements  for  the  pre- 
natal months  are  more  detailed  than  are  those  of  Russow  for  the 
first  year  of  post-natal  hfe)  we  may  draw  a  continuous  curve  of 
growth  (Fig.  15)  and  of  increments  of  growth  (Fig.  16)  for  the 
combined  periods.  It  will  be  seen  at  once  that  there  is  a  "point 
of  inflection"  somewhere  about  the  fifth  month  of  intra-uterine  life; 
up  to  that  date  growth  proceeds  with  a  continually  increasing 
velocity.  After  that  date,  though  growth  is  still  rapid,  its  velocity 
tends  to  fall  away;  the  curve,  while  still  ascending,  is  becoming 
an  S-shaped  curve  (Fig.  15).  There  is  a  shght  break  between  our 
two  sets  of  statistics  at  the  date  of  birth,  an  epoch  regarding  which 
we  should  like  to  have  precise  and  continuous  information.  But 
we  can  see  that  there  is  undoubtedly  a  certain  shght  arrest  of  growth, 
or  diminution  of  the  rate  of  growth,  about  this  epoch ;  the  sudden 
change  of  nurture  has  its  inevitable  effect,  but  this  shght  tem- 

*  It  is  part  of  the  story,  though  Ijy  no  mean?  all,  that  (as  Minot  says)  the  larger 
the  litter  the  sooner  does  birth  take  place.  That  the  day-old  foal  or  fawn  can  keep 
pace  with  their  galloping  dams  is  very  remarkable;  it  is  usually  explained 
teleologically,  as  a  provision  of  Nature,  on  which  their  safety  and  their  survival 
depend.  But  the  fact  that  they  come  one  at  a  birth  has  at  least  something  to  do 
with  their  comparative  maturity. 

t  Unsere  Korperform  und  das  physiologische  Problem  ihrer  Entstehung,  Leipzig,  1874. 
On  growth  in  weight  of  the  human  embryo,  see  C.  M.  Jackson,  Amer.  Journ.  Anat. 
XVII,  p.  118,  1909;  also  J.  Needham,  op.  cit.  pp.  379-383. 


112 


THE  RATE  OF  GROWTH 


[CH. 


u 

crns 

70 

- 

^^ 

60 

- 

^^y^^^'^^ 

50 

/ 

40 

/ 

- 

30 

1 

Birth 

20 
10 

Lr-r        ,           ,.',..           1 

1 

8       10      12       14      16       18      20    22 

months 

Fig.  15.     Curve  of  growth  (in  length  or  stature)  of  child,  before  and  after  birth. 
From  His  and  Russow's  data. 


8         10        12        14 


18      20    22 
montha 


Fig.  16.     Mean  monthly  increments  of  length  or  stature  of  child,  in  cm. 
From  His  and  Russow's  data. 


Ill]  OF  GROWTH  IN  INFANCY  113 

porary  set-back  is  immediately  followed  by  a  secondary,  and  equally 
transitory,  acceleration  *. 

Mean  weight  in  grams  of  American  infants  during  ten  days 
after  birth.     (From  Meredith  and  Brown) 

Weight 


Age 

f ^ N 

(days) 

Male 

Femal 

^t  birth 

3491 

3408 

1 

3376 

3283 

2 

3294 

3207 

3 

3274 

3195 

4 

3293 

3213 

5 

3326 

3246 

6 

3366 

3281 

7 

3396 

3315 

8 

3421 

3341 

9 

3440 

3362 

10 

3466 

3387 

Daily 

increment 

A 

Male 

Fema 

-115 

-125 

-  82 

-  76 

-  20 

-  12 

19 

17 

33 

34 

40 

35 

30 

34 

25 

26 

19 

21 

26 

25 

The  set-back  after  birth  of  which  we  have  just  spoken  is  better 
shewn  by  the  child's  weight  than  by  any  linear  measurement.  During 
its  first  three  days  the  infant  loses  weight  visibly,  and  it  is  more  than 
ten  days  old  before  it  has  made  up  the  weight  it  lost  in  those  first 
three  (Fig.  17). 

It  is  worth  our  while  to  illustrate  on  a  larger  scale  His's  careful 
data  for  the  ten  months  of  pre-natal  life  (Fig.  18).  They  give  an 
S-shaped  curve,  beautifully  regular,  and  nearly  symmetrical  on 
either  side  of  its  point  of  inflection;  and  its  differential,  or  curve 
of  monthly  increments,  is  a  bell-shaped  curve  which  indicates  with 
the  utmost  simplicity  a  rise  from  a  minimal  to  a  maximal  rate,  and 
a  fall  to  a  minimum  again.  It  has  a  close  family  Hkeness  to  the 
well-known  "curve  of  probabihty,"  of  which  we  shall  presently 
have  much  more  to  say;  it  is  a  curve  for  which  we  might  well 
hope  to  find  a  simple  mathematical  expression  f- 

These  two  curves,  then,  look  more  "mathematical,"  and  less 
merely  descriptive,  than  any  others  we  have  yet  drawn,  and  much 

*  See  especially,  H.  V.  Meredith  and  A.  W.  Brown,  Growth  in  body-weight 
during  first  ten  days  of  postnatal  life,  Human  Biology,  xi,  pp.  24-77,  1939.  Also 
{int.  al.)  T.  Brailsford  Robertson,  Pre-  and  post-natal  growth,  etc.,  Amer.  Journ. 
Physiol.  XXXVII,  pp.  1^2,  74-85,  1915. 

t  The  same  is  not  less  true  of  Friedenthal's  more  elaborate  measurements,  in  his 
Physiologie  des  Menschenwachstums,  1914;  cf.  Needham,  op.  cit.  p.  1677. 


114 


THE  RATE  OF  GROWTH 

T 


CH. 


4-100 


--100 


Fig.  17. 


0  3  10 

Age  in  days 
Mean  weight  of  American  infants.     From  Meredith  and  Brown's  data. 


500 

^^ 

mm. 

^.-"^"^ 

400 

- 

y  Length 

300 

- 

/ 

1 

200 

J 

1 

100 

- 

leration 

^>^ 

1                      1                      1 

1 

0  2  4  6  8  10 

months 
Fig.  18.     Curve  of  a  child's  pre-natal  growth,  in  length  or  stature;  and  corre- 
sponding curve  of  mean  monthly  increments  (mm.).     (Smoothed  curves.) 


Ill]  OF  GROWTH  IN  INFANCY  115 

the  same  curves  meet  us  again  and  again  in  the  growth  of  other 
organisms.  The  pre-natal  growth  of  the  guinea-pig  is  just  the 
same*.  We  have  the  same  essential  features,  the  same  S-shaped 
curve,  in  the  growth  by  weight  of  an  ear  of  maize  (Fig.  19),  or  the 
growth  in  length  of  the  root  of  a  bean  (Fig.  20);  in  both  we  see 
the  same  slow  beginning,  the  rate  rapidly  increasing  to  a  maximum, 
and  the  subsequent  slowing  down  or  "negative  acceleration "f." 
One  phase  passes  into  another;  so  far  as  these  curves  go,  they 
exhibit  growth  as  a  continuous  process,  with  its  beginning,  its 
middle  and  its  end— a  continuity  which  Sachs  recognised  some 
seventy  years  ago,  and  spoke  of  as  the  "grand  period  of  growth  J." 
But  these  simple  curves  relate  to  simple  instances,  to  the  infant 
sheltered  in  the  womb,  or  to  plant-growth  in  the  sunny  season  of 
the  year.  They  mark  a  favourable  episode,  rather  than  relate  the 
course  of  a  lifetime.  A  curve  of  growth  to  run  all  life  long  is  only 
simple  in  the  simplest  of  organisms,  and  is  usually  a  very  complex 
affair. 

Growth  in  length  of  Vallisneria^,  and  root  ofbean\\ 
and  weight  of  ntaize^ 


VaUisneria 

A 

Vicia 

Zea 

— 

-\ 

Hours           Inches 

Days 

Mm. 

Days 

Gm. 

6                  0-3 

0 

1-0 

6 

1 

16                   1-7 

1 

2-8 

18 

4 

42                 12-6 

2 

6-5 

30 

9 

54                 15-4 

3 

240 

39 

17 

65                 161 

4 

40-5 

46 

26 

77                 16-7 

5 

57-5 

53 

42 

88                 17-1 

6 

72-0 

60 

62 

7 

79-0 

74 

71 

8  790  93  74 

It  would  seem  to  be  a  natural  rule,  that  those  offspring  which 
are  most  highly  organised  at  birth  are  those  which  are  born  largest 

*  See  R.  L.  Draper,  Anat.  Record,  xviii,"p.  369,  1920;  cf.  Needham,  op.  cil., 
p.  1672. 

t  *^f.  R.  Chodat  et  A.  Monnier;  Sur  la  courbe  de  croissance  chez  les  vegetaux. 
Bull.  Herbier  Boissier  (2),  v,  p.  615,  1905. 

X  Arbeiten  a.  d.  bot.  Instit.  Wiirzburg,  i,  p.  569,  1872. 

§  A.  Bennett,  Trans.  Linn.  Soc.  (2),  i  (Bot.),  p.  133,  1880. 

II  Sachs,  I.e. 

^  Stefanowska,  op.  cit. ;  G.  Backman,  Ergebn.  d.  Physiologie,  xxiii,  p.  925,  193j 


116 


THE  RATE  OF  GROWTH 


[CH. 


20  40  60  80  100 

Days 
Fig.  19.     Growth  in  weight  of  maize.     From  Gustav  Backman,  after  Stefanowska. 


0  5  8 

Age  in  days 
Fig.  20.     Growth  in  height  of  a  beanstalk.     From  Sachs's  data. 


Ill]  OF  GROWTH  IN  INFANCY  117 

relatively  to  their  parents'  size.  But  another  rule  comes  in,  which 
is  perhaps  less  to  be  expected,  that  the  offspring  are  born  smaller 
the  larger  the  species  to  which  they  belong.  Here  we  shew,  roughly, 
the  relative  weights  of  the  new-born  animal  and  its  mother*: 


Bear 

1:600 

Sheep 

1:14 

Lion 

160 

Ox 

13 

Hippopotamus 

45 

Horse 

12 

Dog 

45-50 

Rabbit 

40 

Cat 

25 

Mouse  , 

10-25 

Man 

22 

Guinea-pig 

7 

These  differences  at  birth  are  for  the  most  part  made  up  quickly; 
in  other  words,  there  are  great  differences  in  the  rate  of  growth 
during  early  post-natal  life.  Two  Uourcubs,  studied  by  M.  Anthony, 
grew  as  follows: 


Male  Female 


Feb.  23  (born) 

— 

— 

28 

2-0  kilos 

1-7  kilos 

Mar.    8 

30 

2-6 

15 

3-8 

3-3 

22 

4-6 

4-0 

30 

5-3 

4-6 

Apr.    5 

61 

5-2 

12 

7-0 

6-0 

19 

8-0 

7-0 

Thus  the  lion-cub  doubles  its  weight  in  the  first  month,  and 
wellnigh  doubles  it  again  in  the  second;  but  the  newborn  child 
takes  fully  five  months  to  double  its  weight,  and  nearly  two  years 
to  do  so  again. 

The  size  finally  attained  is  a  resultant  of  the  rate  and  of  the 
duration  of  growth;  and  one  or  other  of  these  may  be  the  more 
important,  in  this  case  or  in  that.  It  is  on  the  whole  true,  as  Minot 
said,  that  the  rabbit  is  bigger  than  the  guinea-pig  because  he  grows 
faster,  but  man  is  bigger  than  the  rabbit  because  he  goes  on  growing 
for  a  longer  time. 

A  bantam  and  a  barn-door  fowl  differ  in  their  rate  of  growth, 
which  in  either  case  is  definite  and  specific.  Bantams  have  been 
bred  to  match  almost  every  variety  of  fowl ;  and  large  size  or  small, 
quick  growth  or  slow,  is  inherited  or  transmitted  as  a  Mendehan 

*  Data  from  Variot,  after  Anthony, 


118  OF  THE  RATE  OF  GROWTH  [ch. 

character  in  every  cross  between  a  bantam  and  a  larger  breed. 
The  bantam  is  not  produced  by  selecting  smaller  and  smaller 
specimens  of  a  larger  breed,  as  an  older  school  might  have  supposed ; 
but  always  by  first  crossing  with  bantam  blood,  so  introducing  the 
"character"  of  smallness  or  retarded  growth,  and  then  segregating 
the  desired  types  among  the  dwarfish  offspring.  In  fact.  Darwinian 
selection  plays  a  small  and  unimportant  part  in  the  process*. 

From  the  whole  of  the  foregoing  discussion  we  see  that  rate  of 
growth  is  a  specific  phenomenon,  deep-seated  in  the  nature  of  the 
organism;  wolf  and  dog,  horse  and  ass,  nay  man  and  woman,  grow 
at  different  rates  under  the  same  circumstances,  and  pass  at  different 
epochs  through  like  phases  of  development.  Much  the  same  might 
be  said  of  mental  or  intellectual  growth ;  the  girl's  mind  is  more 
precocious  than  the  boy's,  and  its  development  is  sooner  arrested 
than  the  man's  f. 

On  variability,  and  on  the  curve  of  frequency  or  of  error 

The  magnitudes  which  we  are  dealing  with  in  .this  chapter — 
heights  and  weights  and  rates  of  change — are  (with  few  exceptions) 
mean  values  derived  from  a  large  number  of  individual  cases.  We 
deal  with  what  (to  borrow  a  word  from  atomic  physics)  we  may 
call  an  ensemble;  we  employ  the  equaUsing  powei^  of  averages, 
invoke  the  "law  of  large  numbers  J,"  and  claim  to  obtain  results 
thereby  which  are  more  trustworthy  than  observation  itself  §.  But 
in  ascertaining  a  mean  value  we  must  also  take  account  of  the 
amount  of  variability,  or  departure  from  the  mean,  among  the  cases 
from  which  the  mean  value  is  derived.  This  leads  on  far  beyond 
our  scope,  but  we  must  spare  it  a  passing  word ;  it  was  this  identical 
phenomenon,  in  the  case  of  Man,  which  suggested  to  Quetelet  the 

*  Cf.  Raymond  Pearl,  The  selection  problem,  Amer.  Naturalist,  1917,  p.  82; 
R.  C.  Punnett  and  P.  G.  Bailey,  Journ.  of  Genetics,  iv,  pp.  23-39,  1914. 

t  Cf.  E.  Devaux,  L'p,llure  du  developpement  dans  les  deux  sexes,  Revue  gindr. 
des  Sci.  1926,  p.  598. 

X  S.  D.  Poisson,  following  James  Bernoulli's  Ars  Conjectandi  (op.  posth.  1713), 
was  the  discoverer,  or  inventor,  of  the  law  of  large  numbers.  "Les  chos^s 
de  toute  nature  sont  soumises  a  une  loi  universelle  qu'on  pent  appeler  la  loi  des 
grands  nombres"  {Recherches,  1837,  pp.  7-12). 

§  See  p.  137,  footnote. 


Ill 


OF  VARIABILITY 


119 


statistical  study  of  Variation,  led  Francis  Galton  to  enquire  into 
the  laws  of  Natural  Inheritance,  and  served  Karl  Pearson  as  the 
foundation  of  his  science  of  Biometrics. 

When  Quetelet  tells  us  that  the  mean  stature  of  a  ten-year-old 
boy  is  1-275  metres,  this  is  found  to  imply,  not  only  that  the 
measurements  of  all  his  ten-year-old  boys  group  themselves  about 
this  mean  value  of  1-275  metres,  but  that  they  do  so  in  an  orderly 
way,  many  departing  httle  from  that  mean  value,  and  fewer  and 
fewer  departing  more  and  more.  In  fact,  when  all  the  measure- 
ments are  grouped  and  plotted,  so  as  to  shew  the  number  of 
instances  (y)  ^  each  gradation  of  size  (x),  we  obtain  a  characteristic 


-2cr      —cr 


Fig.  21.     The  normal  curve  of  frequency,  or  of  error. 
a,  -a,  the  "standard  deviation". 

configuration,  mathematically  definable,  called  the  curve  of  frequency, 
or  oi error  (Fig.  21).  This  is  a  very  remarkable  fact.  That  a  "curve 
of  stature"  should  agree  closely  with  the  "normal  curve  of  error" 
amazed  Galton,  and  (as  he  said)  formed  the  mainstay  of  his  long 
and  fruitful  enquiry  into  natural  inheritance*.  The  curve  is  a 
thing  apart,  sui  generis.  It  depicts  no  course  of  events,  it  is  no 
time  or  vector  diagram.  It  merely  deals  with  the  variabihty,  and 
variation,  of  magnitudes;  and  by  magnitudes " we  mean  anything 
which  can  be  counted  or  measured,  a  regiment  of  men,  a  basket  of 


*  Stature  itself,  in  a  homogeneous  population,  is  a  good  instance  of  a  normal 
frequency  distribution,  save  only  that  the  spread  or  range  of  variation  is  unusually 
low;  for  one-half  of  the  population  of  England  differs  by  no  more  than  an  inch 
and  a  half  from  the  average  of  them  all.  Variation  is  said  to  be  greater  among  the 
negroid  than  among  the  white  races,  and  it  is  certainly  very  great  from  one  race 
to  another:  e.g.  from  the  Dinkas  of  the  White  Nile  with  a  mean  height  of  1-8  m. 
to  the  Congo  pygmies  averaging  1-35,  or  say  5  ft.  11  in.  and  4  ft.  6  in.  respectively. 


120  THE  RATE  OF  GROWTH  [ch. 

niits,  the  florets  of  a  daisy,  the  stripes  of  a  zebra,  the- nearness  of 
shots  to  the  bull's  eye*.  It  thereby  illustrates  one  of  the  most 
far-reaching,  some  say  one  of  the  most  fundamental,  of  nature's 
laws. 

We  find  the  curve  of  error  manifesting  itself  in  the  departures 
from  a  mean  value,  which  seems  itself  to  be  merely  accidental — 
as,  for  instance,  the  mean  height  or  weight  of  ten-year-old  English 
boys;  but  we  find  it  no  less  well  displayed  when  a  certain  definite 
or  normal  number  is  indicated  by  the  nature  of  the  case.  For 
instance  the  Medusae,  or  jelly-fishes,  have  a  "radiate  symmetry" 
of  eight  nodes  and  internodes.  But  even  so,  the  number  eight  is 
subject  to  variation,  and  the  instances  of  more  or  less  graup  them- 
selves in  a  Gaussian  curve. 

Number  of  "  tentaculocysts''  in  Medusae  {Ephyra  and  Aurelia) 
[Data  from  E.  T.  Browne,  QJ.M.S.  xxxvii,  p.  245,  1895) 


5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

Ephyra  (1893)      — 
„       (1894)        1 

4 
6 

8 
34 

278 
883 

22 

75 

18 
61 

12 
35 

14 
17 

3 
3 

1 

— 

Aurelia  (1894)      — 

2 

18 

296 

33 

16 

18 

7 

— 

— 

1 

Percentage  numbers: 
Ephyra                  — 

M 

0-5 

2-2 
30 

77-4 
79-0 

61 

6-7 

5-0 
5-4 

3-3 
31 

3-9 
1-4 

0-8 
0-2 

— 

— 

Aurelia                 — 

0-5 

4-7 

77-2 

8-6 

41 

2-6 

1-8 

— 

— 

— 

Mean          — 

0-7 

3-3 

77-9 

7-1 

4-8 

3-0 

2-4 

0-3 

— 

— 

The  curve  of  error  is  a  "bell-shaped  curve,"  a  courbe  en  cloche.  It 
rises  to  a  maximum,  falls  away  on  either  side,  has  neither  beginning 
nor  end.  It  is  (normally)  symmetrical,  for  lack  of  cause  to  make  it 
otherwise;  it  falls  off  faster  and  then  slower  the  farther  it  departs 
from  the  mean  or  middle  line;  it  has  a  "point  of  inflexion,"  of 
necessity  on  either  side,  where  it  changes  its  curvature  and  from 
being  concave  to  the  middle  line  spreads  out  to  become  convex 

*  "I  know  of  scarcely  anything  (says  Galton)  so  apt  to  impress  the  imagination 
as  the  wonderful  form  of  cosmic  order  expressed  by  the  Law  of  Frequency  of 
Error. ...  It  reigns  with  serenity  and  in  complete  self-effacement  amidst  the 
wildest  confusion"  {Natural  Inheritance,  p.  62).  Observe  that  Galton  calls  it  the 
"law  of  frequency  o/  error,"  which  is  indeed  its  older  and  proper  name.  Cf.  (int.  al.) 
P.  G.  Tait,  Trans  R.S.E.  xxiv,  pp.  139-145,  1867. 


i 


Ill]  OF  THE  CURVE  OF  ERROR  121 

thereto.  If  we  pour  a  bushel  of  corn  out  of  a  sack,  the  outhne  or 
profile  of  the  heap  resembles  such  a  curve ;  and  wellnigh  every  hill 
and  mountain  in  the  world  is  analogous  (even  though  remotely)  to 
that  heap  of  corn  *.  Causes  beyond  our  ken  have  cooperated  to  place 
and  allocate  each  grain  or  pebble;  and  we  call  the  result  a  "random 
distribution,"  and  attribute  it  to  fortuity,  or  chance.  Galton 
devised  a  very  beautiful  experiment,  in  which  a  slopmg  tray  is 
beset  with  pins,  and  sand  or  millet-seed  poured  in  at  the  top. 
Every  falhng  grain  has  its  course  deflected  again  and  again;  the 
final  distribution  is  emphatically  a  random  one,  and  the  curve  of 
error  builds  itself  up  before  our  eyes. 

The  curve  as  defined  by  Gauss,  princeps  mathematicorum — who 
in  turn  was  building  on  Laplace  f — is  at  once  empirical  and 
theoretical  J ;  and  Lippmann  is  said  to  have  remarked  to  Poincare : 
*'Les  experimentateurs  s'imaginent  que  c'est  un  theoreme  de 
mathematique,  et  les  mathematiciens  d'etre  un  fait  experimental ! " 
It  is  theoretical  in  so  far  as  its  equation  is  based  on  certam  hypo- 
thetical considerations:  viz.  (1)  that  the  arithmetic  mean  of  a  number 
of  variants  is  their  best  or  likeliest  average,  an  axiom  which  is 
obviously  true  in  simple  cases — but  not  necessarily  in  all;  (2)  that 
"fortuity"  implies  the  absence  of  any  predominant,  decisive  or 
overwhelming  cause,  and  connotes  rather  the  coexistence  and  joint 
effect  of  small,  undefined  but  independent  causes,  many  or  few: 

*  If  we  pour  the  corn  out  carefully  through  a  small  hole  above,  the  heap  becomes 
a  cone,  with  sides  sloping  at  an  "angle  of  repose";  and  the  cone  of  Fujiyama  is  an 
exquisite  illustration  of  the  same  thing.  But  in  these  two  instances  one  predominant 
cause  outweighs  all  the  rest,  and  the  distribution  is  no  longer  a  random  one. 

t  The  Gaussian  curve  of  error  is  really  the  "second  curve  of  error"  of  Laplace. 
Laplace's  first  curve  of  error  (which  has  uses  of  its  own)  consists  of  two  exponential 
curves,  joining  in  a  sharp  peak  at  the  median  value.  Cf.  W.  J.  Luyten,  Proc. 
Nat.  Acad.  Sci.  xvm,  pp.  360-365,  1932. 

I  The  Gaussian  equation  to  the  normal  frequency  distribution  or  "curve  of 
error"  need  not  concern  us  further,  but  let  us  state  it  once  for  all: 

J,        _(xa-x)* 

^        V27T 

where  Xf^  is  the  abscissa  which  gives  the  maximum  ordinate,  and  where  the  maximum 
ordinate,  y^  =  1/^/(27t).  Thus  the  log  of  the  ordinate  is  a  quadratic  function  of  the 
abscissa ;  and  a  simple  property,  fundamental  to  the  curve,  is  that  for  equally  spaced 
ordinates  (starting  anywhere)  the  square  of  any  ordinate  divided  by  the  product  of 
its  neighbours  gives  a  scalar  quantity  which  is  constant  all  along  (G.T.B.). 


122  THE  RATE  OF  GROWTH  [ch. 

producing  their  several  variations,  deviations  or  errors;  and  potent 
in  their  combinations,  permutations  and  interferences*. 

We  begin  to  see  why  bodily  dimensions  lend,  or  submit,  them- 
selves to  this  masterful  law.  Stature  is  no  single,  simple  thing; 
it  is  compounded  of  bones,  cartilages  and  other  elements,  variable 
each  in  its  own  way,  some  lengthening  as  others  shorten,  each 
playing  its  little  part,  hke  a  single  pin  in  Galton's  toy,  towards  a 
''fortuitous"  resultant.  "The  beautiful  regularity  in  the  statures 
of  a  population  (says  Galton)  whenever  they  are  statistically 
marshalled  in  the  order  of  their  heights,  is  due  to  the  number  of 
variable  and  quasi-independent  elements  of  which  stature  is  the 
sum."  In  a  bagful  of  pennies  fresh  from  the  Mint  each  coin  is 
made  by  the  single  stroke  of  an  identical  die,  and  no  ordinary 
weights  and  measures  suffice  to  differentiate  them;  but  in  a  bagful 
of  old-fashioned  hand-made  nails  a  slow  succession  of  repeated 
operations  has  drawn  the  rod  and  cut  the  lengths  and  hammered 
out  head,  shaft  and  point  of  every  single  nail — and  a  curve  of 
error  depicts  the  differences  between  them. 

The  law  of  error  was  formulated  by  Gauss  for  the  sake  of  the 
astronomers,  who  aimed  at  the  highest  possible  accuracy,  and 
strove  so  to  interpret  their  observations  as  to  eliminate  or  minimise 
their  inevitable  personal  and  instrumental  errors.  It  had  its 
roots  also  in  the  luck  of  the  gaming-table,  and  in  the  discovery 
by  eighteenth-century  mathematicians  that  "chance  might  be 
defined  in  terms  of  mathematical  precision,  or  mathematical  'law'." 
It  was  Quetelet  who,  beginning  as  astronomer  and  meteorologist, 
applied  the  "law  of  frequency  of  error"  for  the  first  time  to 
biological  statistics,  with  which  in  name  and  origin  it  had  nothing 
whatsoever  to  do. 

The  intrinsic  significance  of  the  theory  of  probabihties  and  the 
law  of  error  is  hard  to  understand.  It  is  sometimes  said  that  to 
forecast  the  future  is  the  main  purpose  of  statistical  study,  and 
expectation,  or  expectancy,  is  a  common  theme.     But  all  the  theory 

*  "The  curve  of  error  would  seem  to  carry  the  great  lesson  that  the  ultimate 
differences  between  individuals  are*  simple  and  few ;  that  they  depend  on  collisions 
and  arrangements,  on  permutations  and  combinations,  on  groupings  and  inter- 
ferences, of  elementary  qualities  which  are  limited  iyi  variety  and  finite  in  extent'' 
(J.  M.  Keynes).  A  connection  between  this  law  and  Mendelian  inheritance  is 
discussed  by  John  Brownlee,  P.R.S.E.  xxxi,  p.  251,  1910. 


Ill]  bF  THE  LAW  OF  ERROR  123 

in  the  vorld  enables  us  to  foretell  no  single  unknown  thing,  not  even 
the  turn  of  a  card  or  the  fall  of  a  die.  The  theory  of  probabiUties 
is  a  development  of  the  theory  of  combinations,  and  only  deals  with 
what  occurs,  or  has  occurred,  in  the  long  run,  among  large  numbers 
and  many  permutations  thereof.  Large  numbers  simplify  many 
things;  a  million  men  are  easier  to  understand  than  one  man  out 
of  a  million.  As  David  Hume*  said:  "What  depends  on  a  few 
persons  is  in  a  great  measure  to  be  ascribed  to  chance,  or  to  secret 
and  unknown  causes;  what  arises  from  a  great  many  may  often 
be  accounted  for  by  determinate  and  known  causes."  Physics  is, 
or  has  become,  a  comparatively  simple  science,  just  because  its  laws 
are  based  on  the  statistical  averages  of  innumerable  molecular  or 
primordial  elements.  In  that  invisible  world  we  are  sometimes  told 
that  "chance"  reigns,  and  "uncertainty"  is  the  rule;  but  such 
phrases  as  mere  chance,  or  at  random,  have  no  meaning  at  all  except 
with  reference  to  the  knowledge  of  the  observer,  and  a  thing  is  a 
"pure  matter  of  chance"  when  it  depends  on  laws  which  we  do  not 
know,  or  are  not  considering  f.  Ever  since  its  inception  the  merits 
and  significance  of  the  theory  of  probabiUties  have  been  variously 
estimated.  Some  say  it  touches  the  very  foundations  of  know- 
ledge} ;  and  others  remind  us  that  "avec  les  chiffres  on  pent  tout 
demontrer."  It  is  beyond  doubt,  it  is  a  matter  of  common  ex- 
perience, that  probability  plays  its  part  as  a  guide  to  reasoning. 
It  extends,  so  to  speak,  the  theory  of  the  syllogism,  and  has  been 
called  the  "logic  of  uncertain  inference "§. 

In  measuring  a  group  of  natural  objects,  our  measurements  are 
uncertain  on  the  one  hand  and  the  objects  variable  on  the  other; 
and  our  first  care  is  to  measure  in  such  a  way,  and  to  such  a  scale, 
that  our  own  errors  are  small  compared  with  the  natural  variations. 
Then,  having  made  our  careful  measurements  of.  a  group,  we  want 
to  know  more  of  the  distribution  of  the  several  magnitudes,  and 


*  Essay  xiv. 

t  So  Leslie  Ellis  and  G.  B.  Airy,  in  correspondence  with  Sir  J.  D.  Forbes;  see 
his  Life,  p.  480. 

X  Cf.  Hans  Reichenbach,  Les  fondements  logiques  du  calcul  des  probabilit^s, 
Annales  de  Vinst.  Poincare,  vii,  pp.  267,  1937. 

§  Cf.  J.  M.  Keynes,  A  Treatise  on  Probability,  1921;  and  A.  C.  Aitken's  Statistical 
Mathematics,  1939. 


124  THE  RATE  OF  GROWTH  [ch. 

especially  to  know  two  important  things.  We  want  a  mean  value, 
as  a  substitute  for  the  true  value*  if  there  be  such  a  thing;  let  us 
use  the  arithmetic  mean  to  begin  with.  About  this  mean  the  ob- 
served values  are  grouped  like  a  target  hit  by  skilful  or  unskilful 
shots ;  we  want  some  measure  of  their  inaccuracy,  some  measure  of 
their  spread,  or  scatter,  or  dispersion,  and  there  are  more  ways  than 
one  of  measuring  and  of  representing  this.  We  do  it  visibly  and 
graphically  every  time  we  draw  the  curve  (or  polygon)  of  frequency ; 
but  we  want  a  means  of  description  or  tabulation,  in  words  or  in 
numbers.  We  find  it,  according  to  statistical  mathematics,  in  the 
so-called  index  of  variability,  or  standard  deviation  (o),  which  merely 
means  the  average  deviation  from  the  meanf.  But  we  must  take 
some  precautions  in  determining  this  average;  for  in  the  nature  of 
things  these  deviations  err  both  by  excess  and  defect,  they  are 
partly  positive  and  partly  negative,  and  their  mean  value  is  the  mean 
of  the  variants  themselves.  Their  squares,  however,  are  all  positive, 
and  the  mean  of  these  takes  account  of  the  magnitude  of  each 
deviation  with  no  risk  of  cancelling  out  the  positive  and  negative 
terms:  but  the  "dimension  "  of  this  average  of  the  squares  is  wrong. 
The  square  root  of  this  average  of  squares  restores  the  correct 
dimension,  and  the  result  is  the  useful  index  of  variability,  or  of 
deviation,  which  is  called  o-J. 

This  standard  deviation  divides  the  area  under  the  normal  curve 
nearly  into  equal  halves,  and  nearly  coincides  with  the  point  of 
inflexion  on  either  side;  it  is  the  simplest  algebraic  measure  of 
dispersion,  as  the  mean  is  the  simplest  arithmetical  measure  of 
position.     When  we  divide  this  value  by  the  mean,  we  get  a  figure 


*  It  is  not  always  obvious  what  the  "errors"  are,  nor  what  it  is  that  they  depart 
or  deviate  from.  We  are  apt  to  think  of  the  arithmetic  mean,  and  to  leave  it 
at  that.  But  were  we  to  try  to  ascertain  the  ratio  of  circumference  to  diameter 
by  measuring  pennies  or  cartwheels,  our  "errors"  would  be  found  grouped  round 
a  mean  value  which  no  simple  arithmetic  could  define. 

t  a,  the  standard  deviation,  was  chosen  for  its  convenience  in  mathematical 
calculation  and  formulation.  It  has  no  special  biological  significance;  and  a 
simpler  index,  the  "inter-quartile  distance,"  has  its  advantages  for  the  non- 
mathematician,  as  we  shall  see  presently. 

X  That  is  to  say :  Square  the  deviation-from-the-mean  of  each  class  or  ordinate 
(^);    multiply  each  by  the  number  of  instances  (or  "variates")  in  ^that  class  (/); 

divide  by  the  total  number  (N) ;  and  take  the  square-root  of  the  whole :  a^=  ~  "^  *'    . 


I 


Ill]  THE  COEFFICIENT  OF  VARIABILITY  125 

which  is  independent  of  any  particular  units,  and  which  is  called 
the  coefficient  of  variability''^ . 

Karl  Pearson,  measuring  the  amount  of  variability  in  the  weight 
and  height  of  man,  found  this  coefficient  to  run  as  follows :  In  male 
new-born  infants,  for  weight  15-6,  and  for  stature  6-5;  in  male 
adults,  for  weight  10-8,  and  for  stature  3-6.  Here  the  amount  of 
variability  is  thrice  as  great  for  weight  as  for  stature  among  grown 
men,  and  about  2 J  times  as  great  in  infancy  f.  The  same  curious 
fact  is  well  brought  out  in  some  careful  measurements  of  shell-fish, 
as  follows; 

Variability  of  youdg  Clams  (Mactra  sp.)X 


Average  size 

A 

Coefficient  of 
variability 

Age  (years) 
Number  in  sample 

1                   2 
41                 20 

1                    2 
41                  20 

Length  (cm.) 

Height 

Thickness 

3-2                6-3 
2-3                4-7 
1-3                2-8 

15-3                6-3 
140                6-7 

9-6                8-3 

Weight  (gm.) 

6-4              59-8 

35-4               18-5 

The  phenomenon  is  purely  mathematical.  Weight  varies  as  the  product  of 
length,  height  and  depth,  or  (as  we  have  so  often  seen)  as  the  cube  of  any  one 
of  these  dimensions  in  the  case  of  similar  figures.  It  is  then  a  mathematical, 
rather  than  a  biological  fact  that,  for  small  deviations,  the  variability  of  the 
whole  tends  to  be  equal  to  the  sum  of  that  of  the  three  constituent  dimensions. 
For  if  weight,  w,  varies  as  height  x,  breadth  y,  and  depth  z,  we  may  write 

w  =  c.xyz. 

„„  ,.„        ,.  ,.  dw     dx     dy     dz 

Whence,  ditterentiatmg,  = h  -^  H . 

w       X       y       z 

We  see  that  among  the  shell-fish  there  is  much  more  variability 
in  the  younger  than  in  the  older  brood.     This  may  be  due  to 

*  It  is  usually  multiplied  by  100,  to  make  it  of  a  handier  amount;  and  we  may 
then  define  this  coefficient,  C,  AS  —  ajM  x  100. 

t  Cf.  Fr.  Boas,  Growth  of  Toronto  children,-  Rep.  of  U.S.  Cornm.  of  Education, 
1896-7,  1898,  pp.  1541-1599;  Boas  and  Clark  Wissler,  Statist  cs  of  growth, 
Education  Rep.  1904,  1906,  pp.  25-132;  H.  P.  Bowditch  Rep.  Mass.  State  Board 
of  Health,  1877;  K.  Pearson,  On  the  magnitude  of  certain  coefficients  of  correlation 
in  man,  Proc.  R.S.  lxvi,  1900;  S.  Nagai,  Korperkonstitution  der  Japaner,  from 
Brugsch-Levy,  Biologie  d.  Person,  ii,  p.  445,  1928;  R.  M.  Fleming,  A  study  of 
growth  and  development.  Medical  Research  Council,  Special  Report,  No.  190,  1933. 

J  From  F.  W.  Weymouth,  California  Fish  Bulletin,  No.  7,  1923. 


126  THE  RATE  OF  GROWTH  [ch. 

inequality  of  age;  for  in  a  population  only  a  few  weeks  old,  a  few 
days  sooner  or  later  in  the  date  of  birth  would  make  more  diiference 
than  later  on.  But  a  more  important  matter,  to  be  seen  in  man- 
kind (Fig.  22),  is  that  variability  of  stature  runs  j)ari  passu,  or 
nearly  so,  with  the  rate  of  growth,  or  curve  of  annual  increments 
(cf.  Fig.  12).  The  curve  of  variability  descends  when  the  growth- 
rate  slackens,  and  rises  high  when  in  late  boyhood  growth  is  speeded 
up.  In  short,  the  amount  of  variability  in  stature  or  in  weight  is 
correlated  with,  or  is  a  function  of,  the  rate  of  growth  in  these 
magnitudes. 

Judging  from  the  evidence  at  hand,  we  may  say  that  variabihty 
reaches  its  height  in  man  about  the  age  of  thirteen  or  fourteen, 
rather  earlier  in  the  girls  than  in  the  boys,  and  rather  earher  in  the 
case  of  stature  than  of  weight.  The  difference  in  this  respect  between 
the  boys  and  the  girls  is  now  on  one  side,  now  on  the  other.  In 
infancy  variabihty  is  greater  in  the  girls;  the  boys  shew  it  the 
more  at  five  or  six  years  old;  about  ten  years  old  the  girls  have 
it  again.  From  twelve  to  sixteen  the  boys  are  much  the  more 
variable,  but  by  seventeen  the  balance  has  swung  the  other  way 
(Fig.  23). 

Coefficient  of  variability  (ojM  x  100)  in  man,  at  various  ages 
Age  ...     o        6        7        8        9       10       11       12       13       14       15       16       17      18 

ytature 


I 


British  (Fleming): 

Boys 

•51 

5-4 

50 

5-3 

5-4 

5-6 

5-7 

5-6 

5-8 

5-8 

5-8 

5-0 

4-3 

30 

Girls 

5-2/ 

5-2 

50 

5-5 

5-4 

5-6 

5-8 

5-7 

5-6 

4-7 

4-2 

3-9 

3-7 

3-8 

American 

4-8 

4-6 

4-4 

4-5 

4-4 

4-6 

4-7 

4-9 

5-5 

5-8 

5-6 

5-5 

4-6 

3-7 

(Bowditch) 

Japanese  (Nagai): 

Boys 



40 

— 

4-3 

— 

41 

— 

40 

50 

50 

4-2 

3-2 

— 

— 

Girls 

— 

4-3 

— 

41 

— 

4-5 

— 

4-5 

4-6 

3-6 

31 

30 

— 

— 

Mean 

— 

4-7 

— 

4-7 

— 

4-9 

— 

^  5-0 

5-3 

50 

4-6 

41 

— ■ 

— 

Weight 

American 

11-6 

10-3 

Ill 

9-9 

110 

1-6 

1-8 

13-7 

3-6 

6-8 

15-3 

13-3 

130 

10-4 

Japanese : 

Boys 



10-3 



121 

— 

0-8 

— 

7-0 

51 

70 

13-8 

10-9 

— 

— 

Girls 

— 

10-2 

— 

11-2 

— 

21 

— 

150 

5-6 

3-4 

11-4 

11-5 

— 

— 

Mean        —    10-3      —    111      —    11-5      —    11-9   14-8   15-7    13-5    11-9      —      -- 


Ill]  THE  COEFFICIENT  OF  VARIABILITY  127 


61 1 1 1 1 1 r 


1  ^ 


y  r^ 


American.-^         / 
-^  / 


Japanese^ 


I         I         I         I         I \ L 


5      6      7      8      9     10     II     12     13     14     15     16     17     II 
Age 
Fig.  22.     Variability  in  stature  (boys).     After  Fleming,  Bowditch  and  Nagai. 


+  z 

[— 

+  1 

t 

- 

B 

ntish 

/  / 
/  / 

// 

// 

^     \ 
\     \ 

\     \ 
\     \ 
\     \ 
\     \ 

/ 

/ 

/ 
/ 

\ 

-1 

/ 

1 

Japanese 

1 

1 

3 


18 


.  10  15 

Age  in  years 
Fig.  23.     Coefficient  of  variability  in  stature:  excess  or  defect  of  this  coefficient 
in  the  boy  over  the  girl.     Data  from  R.  M.  Fleming,  and  from  Nagai. 


128  THE  RATE  OF  GROWTH  [ch. 

The  amount  of  variability  is  bound  to  differ  from  one  race  or 
nationality  to  another,  and  we  find  big  differences  between  the 
Americans  and  the  Japanese,  both  in  magnitude  and  phase  (Fig.  22). 

If  we  take  not  merely  the  variability  of  stature  or  -weight  at  a 
given  age,  but  the  variability  of  the  yearly  increments,  we  find 
that  this  latter  variabihty  tends  to  increase  steadily,  and  more  and 
more  rapidly,  within  the  ages  for  which  we  have  information;  and 
this  phenomenon  is,  in  the  main,  easy  of  explanation.  For  a  great 
part  of  the  difference  between  one  individual  and  another  in  regard 
to  rate  of  growth  is  a  mere  difference  of  phase — a  difference  in  the 
epochs  of  acceleration  and  retardation,  and  finally  a  difference  as  to 
the  epoch  when  growth  comes  to  an  end ;  it  follows  that  variabihty 
will  be  more  and  more  marked  as  we  approach  and  reach  the  period 
when  some  individuals  still  continue,  and  others  have  already  ceased, 
to  grow.  In  the  following  epitomised  table,  I  have  taken  Boas's 
determinations  *  of  the  standard  deviation  (ct),  converted  them  into 
the  corresponding  coefficients  of  variabihty  {olM  x  100),  and  then 
smoothed  the  resulting  numbers: 

Coefficients  of  variability  in  annual  increments  of  stature 


Age    ... 

7 

8 

9 

10 

11 

12 

13 

14 

15 

Boys 
Girls 

17-3 
171 

15-8 
17-8 

18-6 
19-2 

191 
22-7 

21-0 
25-9 

24-7 
29-3 

29-0 
370 

36-2 
44-8 

461 

The  greater  variabihty  in  the  girls  is  very  marked  f,  and  is 
explained  (in  part  at  least)  by  the  jnore  rapid  rate  at  which  the  girls 
run  through  the  several  phases  of  their  growth  (Fig.  24).  To  say  that 
children  of  a  given  age  vary  in  the  rate  at  which  they  are  growing 
would  seem  to  be  a  more  fundamental  statement  than  that  they 
vary  in  the  size  to  which  they  have  grown. 

Just  as  there  is  a  marked  difference  in  phase  between  the  growth- 
curves  of  the  two  sexes,  that  is  to  say  a  difference  in  the  epochs 
when  growth  is  rapid  or  the  reverse,  so  also,  within  each  sex,  will 
there  be  room  for  similar,  but  individual,  phase-differences.  Thus 
we  may  have  children  of  accelerated  development,  who  at  a  given 

*  Op.  cit.  p.  lo48. 

I  That  women  are  on  the  whole  more  variable  than  men  was  argued  by  Karl 
Pearson  in  one  of  his  earlier  essays:   The  Chances  of  Death  and  other  Studies,  1897. 


in 


THE  COEFFICIENT  OF  VARIABILITY 


129 


epoch  after  birth  are  growing  rapidly  and  are  already  "big  for  their 
age";  and  .others,  of  retarded  development,  who  are  comparatively 
small  and  have  not  reached  the  period  of  acceleration  which,  in 
greater  or  less  degree,  will  come  to  them  in  turn.  In  other  words, 
there  must  under  such  circumstances  be  a  strong  positive  "coefl&cient 
of  correlation"  between  stature  and  rate  of  growth,  and  also  between 


§    20 


7         8         9        10        II        12        13        14       15 

Age  in  years 

Fig.  24.     Coefficients  of  variability,  in  annual  increments  of  stature. 
After  Boas. 

the  rate  of  growth  in  one  year  and  the  next.  But  it  does  not  by 
any  means  follow  that  a  child  who  is  precociously  big  will  continue 
to  grow  rapidly,  and  become  a  man  or  woman  of  exceptional 
stature  *.  On  the  contrary,  when  in  the  case  of  the  precocious  or 
"accelerated"  children  growth  has  begun  to  slow  down,  the  back- 


*  Some  first  attempts  at  analysis  seem  to  shew  that  the  size  of  the  embryo  at 
birth,  or  of  the  seed  at  germination,  has  more,  influence  than  we  were  wont  to 
suppose  on  the  ultimate  size  of  plant  or  animal.  See  (e.g.)  Eric  Ashby,  Heterosis 
and  the  inheritance  of  acquired  characters,  Proc.  R.S.  (B),  No.  833,  pp.  431-441, 
1937;  and  papers  quoted  therein. 


130  THE  RATE  OF  GROWTH  [ch. 

ward  ones  may  still  be  growing  rapidly,  and  so  making  up  (more 
or  less  completely)  on  the  others.  In  other  words,  the  period  of 
high  positive  correlation  between  stature  and  increment  will  tend 
to  be  followed  by  one  of  negative  correlation.  This  interesting  and 
important  point,  due  to  Boas  and  Wissler*,  is  confirmed  by  the 
following  table : 

Correlation  of  stature  and  increment  in  boys  and  girls 
(From  Boas  and  Wissler) 

Age     6  7  8  9  10  11  12  13  14  15 

Stature        (B)  112-7  115-5  123-2     127-4  133-2  136-8  142-7     147-3     155-9  162-2 

(G)  111-4  117-7  121-4     127-9  131-8  136-7  144-6     149-7     153-8  157-2 

Increment  (B)  5-7  5-3  4-9        5-1  5-0  4-7  5-9        7-5        6-2  5-2 

(G)  5-9  5-5  5-5         5-9  6-2  7-2  6-5         5-4        3-3  1-7 

Correlation  (B)  0-25  0-11  0-08      0-25  018  0-18  0-48      0-29    -0-42  -0-44 

(G)  0-44  0-14  0-24      0-47  018  -0-18  -0-42    -0-39    -0-63  0-11 

A  minor  but  very  curious  point  brought  out  by  the  same 
investigators  is  that,  if  instead  of  stature  we  deal  with  height  in 
the  sitting  posture  (or,  practically  speaking,  with  length  of  trunk 
or  back),  then  the  correlations  between  this  height  and  its  annual 
increment  are  throughout  negative.  In  other  words,  there  would 
seem  to  be  a  general  tendency  for  the  long  trunks  to  grow  slowly 
throughout  the  whole  period  under  investigation.  It  is  a  well- 
known  anatomical  fact  that  tallness  is  in  the  main  due  not  to  length 
of  body  but  to  length  of  limb. 

Since  growth  in  height  and  growth  in  weight  have  each,  their  own 
velocities,  and  these  fluctuate,  and  even  the  amount  of  their 
variabihty  alters  with  age,  it  follows  that  the  correlation  between 
height  and  weight  must  not  only  also  vary  but  must  tend  to 
fluctuate  in  a  somewhat  complicated  way.  The  fact  is,  this  corre- 
lation passes  through  alternate  maxima  and  minima,  chief  among 
which  are  a  maximum  at  about  fourteen  years  of  age  and  a  minimum 
about  twenty-one.  Other  intercorrelations,  such  as  those  between 
height  or  weight  and  chest-measurement,  shew  their  periodic 
variations  in  like  manner;    and  it  is  about  the  time  of  puberty 

*  I.e.  p.  42,  and  other  papers  there  quoted.  Cf.  also  T.  B.  Robertson,  Criteria 
of  Normality  in  the  Growth  of  Children,  Sydney,  1922. 


Ill]  THE  CURVE  OF  ERROR  131 

that  correlation  tends  to  be  closest,  or  a  norm  to  be  most  nearly 
approached*. 

The  whole  subject  of  variabiHty,  both  of  magnitude  and  rate  of 
increment,  is  highly  suggestive  and  instructive:  inasmuch  as  it 
helps  further  to  impress  upon  us  that  growth  and  specific  rate  of 
growth  are  the  main  physiological  factors,  of  which  specific  mag- 
nitude, dimensions  and  form  are  the  concrete  and  visible  resultant. 
Nor  may  we  forget  for  a  moment  that  growth-rate,  and  growth 
itself,  are  both  of  them  very  complex  things.  The  increase  of  the 
active  tissues,  the  building  of  the  skeleton  and  the  laying  up 
of  fat  and  other  stores,  all  these  and  more  enter  into  the  complex 
phenomenon  of  growth.  In  the  first  instance  we  may  treat  these 
many  factors  as  though  they  were  all  one.  But  the  breeder  and 
the  geneticist  will  soon  want  to  deal  with  them  apart;  and  the 
mathematician  will  scarce  look  for  a  simple  expression  where 
so  many  factors  are  involved.  But  the  problems  of  variability, 
though  they  are  intimately  related  to  the  general  problem  of 
growth,  carry  us  very  soon  beyond  our  hmitations. 

The  curve  of  error 

To  return  to  the  curve  of  error. 

The  normal  curve  is  a  symmetrical  one.  Its  middle  point,  or 
median  ordinate,  marks  the  arithmetic  mean  of  all  the  measurements ; 
it  is  also  the  mode,  or  class  to  which  the  largest  number  of  individual 
instances  belong.  Mean,  median  and  mode  are  three  diiferent  sorts 
of  average;  but  they  are  one  and  the  same  in  the  normal  curve. 

It  is  easy  to  produce  a  related  curve  which  is  not  symmetrical, 
and  in  which  mean,  median  and  mode  are  no  longer  the  same. 
The  heap  of  corn  will  be  lop-sided  or  "skew"  if  the  wind  be  blowing 
while  the  grain  is  falling:  in  other  words,  if  some  prevailing  cause 
disturb  the  quasi-equihbrium  of  fortuity ;  and  there  are  other  ways, 
some  simple,  some  more  subtle,  by^  which  asymmetry  may  be 
impressed  upon  our  curve. 

The  Gaussian  curve  is  only  one  of  many  similar  bell-shaped  curves ; 
and  the  binomial  coefficients,  the  numerical  coefficients  of  (a  +  by, 
yield  a  curve  so  Hke  it  that  we  may  treat  them  as  the  same.     The 

*  Cf.  Joseph  Bergson,  Growth -changes  in  physical  correlation,  Human  Biology, 
I,  p.  4,  1930. 


132  THE  RATE  OF  GROWTH  [ch. 

Gaussian  curve  extends,  in  theory,  to  infinity  at  either  end;  and 
this  infinite  extension,  or  asymptotism,  has  its  biological  significance. 
We  know  that  this  or  that  athletic  record  is  lowered,  slowly  but 
continually,  as  the  years  go  by.  This  is  due  in  part,  doubtless,  to 
increasing  skill  and  improved  technique ;  but  quite  apart  from  these 
the  record  would  slowly  fall  as  more  and  more  races  are  run,  owing 
to  the  indefinite  extension  of  the  Gaussian  curve*. 

On  the  other  hand,  while  the  Gaussian  curve  extends  in  theory 
to  infinity,  the  fact  that  variation  is  always  limited  and  that  extreme 
v3,riations  are  infinitely  rare  is  one  of  the  chief  lessons  of  the  law 
of  frequency.  If,  in  a  population  of  100,000  men,  170  cm.  be  the 
mean  height  and  6  cm.  the  standard  deviation,  only  11  per  cenl^., 
or  say  130  men,  will  exceed  188  cm.,  only  10  men  will  be  over 
191  cm.,  and  only  one  over  193  cm.,  or  13 J  per  cent,  above  the 
average.  The  chance  is  negligible  of  a  single  one  being  found  over 
210  cm.,  or  7  ft.  high,  or  24  per  cent,  above  the  average. 

Yet,  widely  as  the  law  holds  good,  it  is  hardly  safe  to  count  it 
as  a  universal  law.  Old  Parr  at  150  years  old,  or  the  giant  Chang 
at  more  than  eight  feet  high,  are  not  so  much  extreme  instances  of 
a  law  of  probabihty,  as  exceptional  cases  due  to  some  peculiar  cause 
or  influence  coming  inf.  In  a  somewhat  analogous  way,  one  or  two 
species  in  a  group  grow  far  beyond  the  average  size ;  the  Atlas  moth, 
the  Gohath  beetle,  the  ostrich  and  the  elephant,  are  far-off  outhers 
from  the  groups  to  which  they  belong.  A  reason  is  not  easy  to. find. 
It  looks  as  though  variations  came  at  last  to  be  in  proportion  to  the 
size  attained,  and  so  to  go  on  by  compound  interest  or  geometrical 
progression.  There  may  be  nothing  surprising  in  this ;  nevertheless, 
it  is  in  contradistinction  to  that  summation  of  small  fortuitous 
differences  which  lies  at  the  root  of  the  law  of  error.  If  size  vary  in 
proportion  to  the  magnitude  of  the  variant  individuals,  not  only 


*  This  is  true  up  to  a  certain  extent,  but  would  become  a  mathematical  fiction 
later  on.  There  will  be  physical  limitations  (as  there  are  in  quantum  mechanics) 
both  to  record-breaking,  and  to  the  measurement  of  minute  extensions  of  the 
record. 

t  We  may  indeed  treat  old  Parr's  case  on  the  ordinary  lines  of  actuarial 
probability,  but  it  is  "without  much  actuarial  importance."  The  chance  of  his 
record  being  broken  by  a  modern  centenarian  is  reckoned  at  (5)^°,  by  Major 
Greenwood  and  J.  C.  Irwin,  writing  on  Senility,  in  Human  Biology,  xi,  pp.  1-23, 
1939. 


Ill] 


THE  CURVE  OF  ERROR 


133 


will  the  frequency  curve  be  obviously  skew,  but  the  geometric  mean^ 
not  the  arithmetic,  becomes  the  most  probable  value*.  Now  the 
logarithm  of  the  geometric  mean  of  a  series  of  numbers  is  the 
arithmetic  mean  of  their  logarithms;  and  it  follows  that  in  such 
cases  the  logarithms  of  the  variants,  and  not  the  variants  them- 
selves, will  tend  to  obey  the  Gaussian  law  and  follow  the  normal 
curve  of  frequency  f. 

The  Gaussian  curve,  and  the  standard  deviation  associated  with 
it,  were  (as  we  have  seen)  invented  by  a  mathematician  for  the  use 


21 


33 


36 


24  27  30 

Length  in  mm. 
Fig.  25  A.     Curve  of  frequency  of  a  population  of  minnows. 


39 


of  an  astronomer,  and  their  use  in  biology  has  its  difficulties  and 
disadvantages.  We  may  do  much  in  a  simpler  way.  Choosing  a 
random  example,  I  take  a  catch  of  minnows,  measured  in  3  mm. 
groups,  as  follows  (Fig.  25 A): 

Size  (mm.)  13-15  16^18  19-21  22-24  25-27  28^30  31-33  34-36  37-39 
Number  1  22  52  67  114        257         177  41  2 


*  See  especially  J.  C.  Kapteyn,  Skew  frequency  curves  in  biology  and  statistics, 
Rec.  des  Trav.  Botan.  N4erland.,  Groningen,  xin,  pp.  105-158,  1916.  Also  Axel 
M.  Hemmingsen,  Statistical  analysis  of  the  differences  in  body-size  of  related  species, 
Danske  Vidensk.  Selsk.  Medd.  xcvm,  pp.  125-160,  1934. 

t  This  often  holds  good.  Wealth  breeds  wealth,  hence  the  distribution  of 
wealth  follows  a  skew  curve;  but  logarithmically  this  curve  becomes  a  normal 
one.  Weber's  law,  in  physiology,  is  a  well-known  instance;  on  the  thresholds 
of  sensations,  effects  are  produced  proportional  to  the  magnitudes  of  those 
thresholds,  and  the  logs  of  the  thresholds,  and  not  the  thresholds  themselves, 
are  normaUy  distributed. 


134 


THE  RATE  OF  GROWTH 


[CH. 


Let  us  sum  the  same  figures  up,  so  as  to  show  the  whole  number 
above  or  below  the  respective  sizes. 


Size  (mm.) 

15 

18 

21 

24 

27 

30 

33 

36 

39 

Number  below 

1 

23 

75 

142 

256 

513 

690 

731 

733 

Percentage 

— 

31 

10-2 

19-4 

34-9 

70-0 

941 

99-6 

100 

Our  first  set  of  figures,  the  actual  measurements,  would  give  us 
the  '*courbe  en  cloche,"  in  the  formrof  an  unsymmetrical  (or  "skew") 
Gaussian  curve:  one,  that  is  to  say,  with  a  long  sloping  talus  on 


Extreme 
Decile 

Quartile 


Median 


Quartile 

Decile 
Extreme 


Fig.  25  B. 


18 


33 


100% 
90 

75 


.50 


36       39 


21        24        27       30 
Length  in  mm. 
'Curve  of  distribution"  of  a  population  of  minnows. 


one  side  of  the  hilj.  The  other  gives  us  an  "S-shaped  curve,''  ap- 
parently hmited,  but  really  asymptotic  at  both  ends  (Fig.  25  B) ;  and 
this  S-shaped  curve  is  so  easy  to  work  with  that  we  may  at  once  divide 
it  into  two  halves  (so  finding  the  ''median"  value),  or  into  quarters 
and  tenths  (giving  the  "quartiles"  and  "deciles"),  or  as  we  please. 
In  short,  after  drawing  the  curVe  to  a  larger  scale,  we  shall  find  that 
we  can  safely  read  it  to  thirds  of  a  miUimetre,  and  so  draw  from  it 
the  following  somewhat  rough  but  very  useful  tabular  epitome  of 
our  population  of  minnows,  from  which  the  curve  can  be  recon- 
structed at  aiiy  time: 

mm. 

13 

210 

25-3 

28-6 

30-6 

32-3 


Extreme 
First  decile 
Lower  quartile 
Median 
Upper  quartile 
Last  decile 
Extreme 


39 


Ill] 


OF  Multimodal  curves 


135 


This  S-shaped  "summation-curve"  is  what  Francis  Galton  called  a 
curve  of  distribution,  and  he  "Uked  it  the  better  the  more  he  used  it." 
The  spread  or  "scatter"  is  conveniently  and  immediately  estimated 
by  the  distance  between  the  two  quartiles ;  and  it  happens  that  this 
very  nearly  coincides  with  the  standard  deviation  of  the  normal  curve. 


40 


50  60  70 

Micrometer-scale  units 


80 


90 


100 


Fig.  26.     A  plankton-sample  of  fish-eggs:  North  of  Scotland,  February  1905. 

(Only  eggs  without  oil-globule  are  counted  here.) 

A.  Dab  and  Flounder.     B,  Gadus  Esmarckii  and  G.  luscus. 

C,  Cod  and  Haddock.     D,  Plaice. 

There  are  biological  questions  for  which  we  want  all  the  accuracy 
which  biometric  science  can  give;  but  there  are  many  others  on 
which  such  refinements  are  thrown  away. 

Mathematically  speaking,  we  cannot  integrate  the  Gaussian  curve,, 
save  by  using  an  infinite  series;  but  to  all  intents  and  purposes  we 
are  doing  so,  graphically  and  very  easily,  in  the  illustration  we  have 
just  shewn.  In  any  case,  whatever  may  be  the  precise  character  of 
each,  we  begin  to  see  how  our  two  simplest  curves  of  growth,  the 
bell-shaped  and  the  S-shaped  curve,  form  a  reciprocal  pair,  the 
integral  and  the  differential  of  one  another  "^ — hke  the  distance  travelled 

*  It  is  of  considerable  historical  interest  to  know  that  this  practical  method  of 
summation  was  first  used  by  Edward  Wright,  in  a  Table  of  Latitudes  published  in 
his  Certain  Errors  in  Navigation  corrected,  1599,  as  a  means  of  virtually  integrating 
sec  X.  (On  this,  and  on  Wright's  claim  to  be  the  inventor  of  logarithms,  see  Florian 
Cajori,  in  Napier  Memorial  Volume,  1915,  pp.  94-99.) 


136  THE  RATE  OF  GROWTH  [ch. 

and  the  velocit}^  of  a  moving  body.     If  ^  =  e"^^  be  the  ordinate  of 
the  one,  z  =  le-^'^dx  is  that  of  the  other. 

There  is  one  more  kind  of  frequency-curve  which  we  must  take 
passing  note  of.  We  begin  by.  thinking  of  our  curve,  whether 
symmetrical  or  skew,  as  the  outcome  of  a  single  homogeneous 
group.  But  if  we  happen  to  have  two  distinct  but  intermingled 
groups  to  deal  with,  differing  by  ever  so  little  in  kind,  age,  place  or 
circumstance — leaves  of  both  oak  and  beech,  heights  of  both  men 
and  women — this  heterogeneity  will  tend  to  manifest  itself  in  two 
separate  cusps,  or  modes,  on  the  common  curve:  which  is  then 
indeed  two  curves  rolled  into  one,  each  keeping  something  of  its 
own  individuality.  For  example,  the  floating  eggs  of  the  food-fishes 
are  much  alike,  but  differ  appreciably  in  size.  A  random  gathering, 
netted  at  the  surface  of  the  sea,  will  yield  on  measurement  a  multi- 
modal curve,  each  cusp  of  which  is  recognisable,  more  or  less 
certainly,  as  belonging  to  a  particular  kind  of  fish  (Fig.  26). 

A  further  note  upon  curves 

A  statistical  "curve",  such  as  Quetelet  seems  to  have  been  the 
first  to  use*,  is  a  device  whose  peculiar  and  varied  beauty  we  are 
apt,  through  famiharity,  to  disregard.  The  curve  of  frequency  which 
we  have  been  studying  depicts  (as  a  rule)  the  distribution  of  mag- 
nitudes in  a  material  system  (a  population,  for  instance)  at  a 
certain  epoch  of  time ;  it  represents  a  given  state,  and  we  may  call 
/it  a  diagram  of  configuration  "f.  But  we  oftener  use  our  curves 
to  compare  successive  states,  or  changes  of  magnitude,  as  one 
configuration  gives  place  to  another;  and  such  a  curve  may  be 
called  a  diagram  of  displacement.  An  imaginary  point  moves  in 
imaginary  space,  the  dimensions  of  which  represent  those  of  the 
phenomenon  in  question,  dimensions  which  we  may  further  define 
and  measure  by  a  system  of  "coordinates";  the  movements  of  our 
point  through  its  figurative  space  are  thus  analogous  to,  and  illus- 
trative of,  the  events  which  constitute  the  phenomenon.  Time  is 
often  represented,  and  measured,  on  one  of  the  coordinate  axes,  and 
our  diagram  of  "displacement"  then  becomes  a  diagram  of  velocity. 

*  In  his  Theorie  des  probabilites,  1846. 

t  See  Clerk  Maxwell's  article  "Diagrams,"  in  the  Encyclopaedia  Britannica, 
9th  edition. 


Ill]  OF  STATISTICAL  CURVES  137 

This  simple  method  (said  Kelvin)  of  shewing  to  the  eye  the  law  of 
variation,  however  complicated,  of  an  independent  variable,  is  one 
of  the  most  beautiful  results  of  mathematics*. 

We  make  and  use  our  curves  in  various  ways.  We  set  down  on 
the  coordinate  network  of  our  chart  the  points  givQ^i  by  a  series  of 
observations,  and  connect  them  up  into  a  continuous  series  as  we 
chart  the  voyage  of  a  ship  from  her  positions  day  by  day ;  we  may 
"smooth"  the  line,  if  we  so  desire.  Sometimes  we  find  our  points 
so  crowded,  or  otherwise  so  dispersed  and  distributed,  that  a  line 
can  be  drawn  not  from  one  to  another  but  among  them  all — a  method 
first  used  by  Sir  John  Herschel  f,  when  he  studied  the  orbits  of  the 
double  stars.  His  dehcate  observations  were  affected  by  errors,  at 
first  sight  without  rhyme  or  reason,  but  a  curve  drawn  where  the 
points  lay  thickest  embodied  the  common  lesson  of  them  all;  any 
one  pair  of  observations  would  have  sufiiced,  whether  better  or 
worse,  for  the  calculation  of  an  orbit,  but  Herschel's  dot-diagram 
obtained  "from  the  whole  assemblage  of  observations  taken  together, 
and  regarded  as  a  single  set  of  data,  a  single  result  in  whose  favour 
they  all  conspire."  It  put  us  in  possession,  said  Herschel,  of 
something  truer  than  the  observations  themselves  % ;  and  Whewell 
remarked  that  it  enabled  us  to  obtain  laws  of  Nature  not  only  from 
good  but  from  very  imperfect  observations  §.  These  are  some 
advantages  of  the  use  of  "curves,"  which  have  made  them  essential 
to  research  and  discovery. 

It  is  often  helpful  and  sometimes  necessary  to  smooth  our  curves, 

*  Kelvin,  Nature,  xxix,  p.  440,  1884. 

t  Mem.  Astron.  Soc.  v,  p.  171,  1830;  Nautical  Almanack,  1835,  p.  495;  etc. 

X  Here  a  certain  distinction  may  be  observed.  We  take  the  average  height  of  a 
regiment,  because  the  men  actually  vary  about  a  mean.  But  in  estimating  the  place 
of  a  star,  or  the  height  of  Mont  Blanc,  we  average  results  which  only  differ  by 
I)ersonal  or  instrumental  error.  It  is  this  latter  process  of  averaging  which  leads, 
in  Herschel's  phrase,  to  results  more  trustworthy  than  observation  itself.  Laplace 
had  made  a  similar  remark  long  before  {Oeuvres,yii,  Theorie  des  probabilites) :  that 
we  may  ascertain  the  very  small  effect  of  a  constant  cause,  by  means  of  a  long  series 
of  observations  the  errors  of  which  exceed  the  effect  itself.  He  instances  the  small 
deviation  to  the  eastward  which  the  rotation  of  the  earth  imposes  on  a  falling  body. 
In  like  manner  the  mean  level  of  the  sea  may  be  determined  to  the  second  decimal 
of  an  inch  by  observations  of  high  and  low  water  taken  roughly  to  the  nearest  inch, 
provided  these  are  faithfxilly  carried  out  at  every  tide,  for  say  a  hundred  years. 
Cf.  my  paper  on  Mean  Sea  Level,  in  Scottish  Fishery  Board's  Sci.  Report  for  1915. 

§  Novum  Organum  Renovatum  (3rd  ed.),  1858,  p.  20. 


138  THE  RATE  OF  GROWTH  [ch. 

whether  at  free  hand  or  by  help  of  m^athematical  rules;  it  is  one 
way  of  getting  rid  of  non-essentials — and  to  do  so  has  been  called 
the  very  key-note  of  mathematics*.  A  simple  rule,  first  used  by 
Gauss,  is  to  replace  each  point  by  a  mean  between  it  and  its  two 
or  more  neighbours,  and  so  to  take  a  "floating"  or  "running 
average."  In  so  doing  we  trade  once  more  on  the  "principle  of 
continuity";  and  recognise  that  in  a  series  of  observations  each 
one  is  related  to  another,  and  is  part  of  the  contributory  evidence 
on  which  our  knowledge  of  all  the  rest  depends.  But  all  the  while 
we  feel  that  Gaussian  smoothing  gives  us  a  practical  or  descriptive 
result,  rather  than  a  mathematical  one. 

Some  curves  are  more  elegant  than  others.  We  may  have  to  rest 
content  with  points  in  which  no. order  is  apparent,  as  when  we  plot 
the  daily  rainfall  for  a  month  or  two;  for  this  phenomenon  is  one 
whose  regularity  only  becomes  apparent  over  long  periods,  when 
average  values  lead  at  last  to  "statistical  uniformity."  But  the 
most  irregular  of  curves  may  be  instructive  if  it  coincide  with  another 
not  less  irregular :  as  when  the  curve  of  a  nation's  birth-rate,  in  its 
ups  and  downs,  follows  or  seems  to  follow  the  price  of  wheat  or  the 
spots  upon  the  sun. 

It  seldom  happens,  outside  of  the  exact  sciences,  that  we  com- 
prehend the  mathematical  aspect  of  a  phenomenon  enough  to  define 
(by  formulae  and  constants)  the  curve  which  illustrates  it.  But, 
failing  such  thorough  comprehension,  we  can  at  least  speak  of  the 
trend  of  our  curves  and  put  into  words  the  character  and  the  course 
of  the  phenomena  they  indicate.  We  see  how  this  curve  or  that 
indicates  a  uniform  velocity,  a  tendency  towards  acceleration  or 
retardation,  a  periodic  or  non-periodic  fluctuation,  a  start  from  or  an 
approach  to  a  limit.  When  the  curve  becomes,  or  approximates  to, 
a  mathematical  one,  the  types  are  few  to  which  it  is  Kkely  to 
belong f.  A  straight  line,  a  parabola,  or  hyperbola,  an  exponential 
or  a  logarithmic  curve  (like  x'=ay^),  a  sine-curve  or  sinusoid,  damped 
or  no,  suffice  for  a  wide  range  of  phenomena;  we  merely  modify  our 
scale,  and  change  the  names  of  our  coordinates. 

*  Cf.  W.  H.  Young,  The  mathematic  method  and  its  limitations,  Atti  del  Congresso 
dei  Matematici,  Bologna,  192/8,  i,  p.  203. 

t  Hence  the  engineer  usually  begins,  for  his  first  tentative  construction,  by 
drawing  one  of  the  familiar  curves,  catenary,  parabola,  arc  of  a  circle,  or  curve  of, 
sines. 


Ill]  OF  STATISTICAL  CURVES  139 

The  curves  we  mostly  use,  other  than  the  Gaussian  curve,  are 
time-diagrams.  Each  has  a  beginning  and  an  end;  and  one  and 
the  same  curve  may  illustrate  the  life  of  a  man,  the  economic  history 
of  a  kingdom,  the  schedule  of  a  train  between  one  st^^tion  and 
another.  What  it  then  shews  is  a  velocity,  an  acceleration,  and 
a  subsequent  negative  acceleration  or  retardation.  It  depicts  a 
"mechanism"  at  work,  and  helps  us  to  see  analogous ' mechanisms 
in  different  fields;  for  Nature  rings  her  many  changes  on  a  few 
simple  themes.  The  same  expressions  serve  for  different  orders  of 
phenomena.  The  swing  of  a  pendulum,  the  flow  of  a  current,  the 
attraction  of  a  magnet,  the  shock  of  a  blow,  have  their  analogues  in 
a  fluctuation  of  trade,  a  wave  of  prosperity,  a  blow  to  credit,  a  tide 
in  the  affairs  of  men. 

The  same  exponential  curve  may  illustrate  a  rate  of  cooHng,  a  loss 
of  electric  charge,  the  chemical  action  of  a  ferment  or  a  catalyst. 
The  S-shaped  population-curve  or  "logistic  curve"  of  Verhulst  (to 
which  we  are  soon  coming)  is  the  hysteresis-curve  by  which  Ewing 
represented  self-induction  in  a  magnetic  field ;  it  is  akin  to  the  path 
of  a  falhng  body  under  the  influence  of  friction;  and  Lotka  has 
drawn  a  curve  of  the  growing  mileage  of  American  railways,  and 
found  it  to  be  a  typical  logistic  curve.  A  few  bars  of  music  plotted 
in  wave-lengths  of  the  notes  might  be  mistaken  for  a  tidal  record. 
The  periodicity  of  a  wave,  the  acceleration  of  gravity,  retardation 
by  friction,  the  role  of  inertia,  the  explosive  action  of  a  spark  or 
an  electric  contact — these  are  some  of  the  modes  of  action  or  "forms 
of  mechanism"  which  recur  in  Hmited  number,  but  in  endless  shapes 
and  circumstances*.  The  way  in  which  one  curve  fits  many 
phenomena  is  characteristic  of  mathematics  itself,  which  does  not  deal 
with  the  specific  or  individual  case,  but  generalises  all  the  while,  and 
is  fond  (as  Henri  Poincare  said)  of  giving  the  same  name  to  different 
things. 

Our  curves,  as  we  have  said,  are  mostly  time-diagrams,  and 
represent  a  change  in  time  from  one  magnitude  to  another;  they  are 
diagrams  of  displacement,  in  Maxwell's  phrase.  We  may  consider 
four  different  cases,  not  equally  simple   mathematically,  but  all 

*  See  an  admirable  little  book  by  Michael  Petrovich,  Les  micanismes  communs 
aux  phenomenes  disparates,  Paris,  1921. 


140  THE  RATE  OF  GROWTH  [ch. 

capable   of  explanation,   up   to   a  certain  point,  without   mathe- 
matics. 

(1)  If  in  our  coordinate  diagram  we  have  merely  to  pass  from 
one  isolated  jpoint  to  another,  a  straight  line  joining  the  two  points 
is  the  shortest — and  the  hkeliest  way. 

(2)  To  rise  and  fall  alternately,  going  to  and  fro  from  maximum 
to  minimum,  a  zig-zag  rectilinear  path  would  still  be,  geometrically, 
the  shortest  way;  but  it  would  be  sharply  discontinuous  at  every 
turn,  it  would  run  counter  to  the  "principle  of  continuity,"  it  is  not 
likely  to  be  nature's  way.  A  wavy  course,  with  no  more  change  of 
curvature  than  is  absolutely  necessary,  is  the  path  which  nature 
follows.     We  call  it  a  simple  harmonic  motion,  and  the  simplest  of 


The  Sine -curve 


The  S- shaped  curve 


The  bell-shaped  curve 
Fig.  27.     Simple  curves,  representing  a  change  from  one  magnitude  to  another. 

all  such  wavy  curves  we  call  a  sine-curve.  If  there  be  but  one 
maximum  and  one  minimum,  which  our  variant  alternates  between, 
the  vector  pathway  may  be  translated  into  jpolar  coordinates;  the 
vector  does-  what  the  hands  of  the  clock  do,  and  a  circle  takes  the 
place  of  the  sine-curve. 

(3)  To  pass  from  a  zero-line  to  a  maximum  once  for  all  is  a  very 
different  thing ;  for  now  minimum  and  maximum  are  both  of  them 
continuous  states,  and  the  principle  of  continuity  will  cause  our 
vector-variant  to  leave  the  one  gradually,  and  arrive  gradually  at 
the  other.  The  problem  is  how  to  go  uphill  from  one  level  road  to 
another,  with  the  least  possible  interruption  or  discontinuity.  The 
path  follows  an  S-shaped  course;  it  has  an  inflection  midway;  and 
the  first  phase  and  the  last  are  represented  by  horizontal  asymptotes. 
This  is  an  important  curve,  and  a  common  one.     It  so  far  resembles 


Ill]  OF  CURVES  IN  GENERAL  141 

an  "elastic  curve"  (though  it  is  not  mathematically  identical  with  it) 
that  it  may  be  roughly  simulated  by  a  watchspring,  lying  between 
two  parallel  straight  lines  and  touching  both  of  them.  It  has  its 
kinetic  analogue  in  the  motion  of  a  pendulum,  which  starts  from 
rest  and  comes  to  rest  again,  after  passing  midway  through  its 
maximal  velocity.  It  indicates  a  balance  between  production  and 
waste,  between  growth  and  decay:  an  approach  on  either  side  to 
a  state  of  rest  and  equilibrium.  It  shows  the  speed  of  a  train 
between  two  stations ;  it  illustrates  the  growth  of  a  simple  organism, 
or  even  of  a  population  of  men.  A  certain  simple  and  symmetrical 
case  is  called  the  Verhulst- Pearl  curve,  or  the  logistic  curve. 

(4)  Lastly,  in  order  to  leave  a  certain  minimum,  or  zero-hne,  and 
return  to  it  again,  the  simplest  way  will  be  by  a  curve  asymptotic 
to  the  base-line  at  both  ends — or  rather  in  both  directions;  it  will 
be  a  bell-shaped  curve,  having  a  maximum  midway,  and  of  necessity 
a  point  of  inflection  on  either  side ;  it  is  akin  to,  and  under  certain 
precise  conditions  it  becomes,  the  curve  of  error  or  Gaussian  curve. 

Besides  the  ordinary  curve  of  growth,  which  is  a  summation- 
curve,  and  the  curve  of  growth-rates,  which  is  its  derivative,  there 
are  yet  others  which  we  may  employ.  One  of  these  was  introduced 
by  Minot*,  from 'a  feeling  that  the  rate  of  growth,  or  the  amount 
of  increment,  ought  in  some  way  to  be  equated  with  the  growing 
structure.  Minot's  method  is  to  deal,  not  with  the  actual  increments 
added  in  successive  periods,  but  with  these  successive  increments 
represented  as  percentages  of  the  amount  already  reached.  For 
instance,  taking  Quetelet's  values  for  the  height  (in  centimetres)  of 
a  male  infant,  we  have  as  follows: 


Years 

0 

1 

2 

3 

4 

era. 

500 

69-8 

791 

86-4 

92-7 

But  Minot  would  state  the  percentage-growth  in  each  of  these 
four  annual  periods  at  39-6,  13-3,  9-2  and  7-3  per  cent,  respectively: 

Years       0  1  2  3  4 

Height  (cm.)  50-0  69-8  79-1  86-4  92-7 

Increments  (cm.)  —  19-8  9-3  7-3  6-3 

(per  cent.)  ~  39-6  13-3  9-2  7-3 

*  C.  S.  Minot,  On  certain  phenomena  of  growing  old,  Proc.  Amer.  Assoc,  xxxix, 
1890,  21  pp.;  Senescence  and  rejuvenation,  Journ.  Physiol,  xii,  pp.  97-153, 
1891;   etc.     Criticised  by  S.  Brody  and  J.  Needham,  ojp,  cit.  pp.  401  seq. 


142  THE  RATE  OF  GROWTH  [ch. 

Now,  in  our  first  curve  of  growth  we  plotted  length  against  time, 
a  very  simple  thing  to  do.  When  we  differentiate  L  with  respect  to  T, 
we  have  dL/dT,  which  is  rate  or  velocity,  again  a  very  simple  thing ; 
and  from  this,  by  a  second  differentiation,  we  obtain,  if  necessary, 
d^L/dT^,  that  is  to  say,  the  acceleration. 

But  when  you  take  percentages  of  y,  you  are  determining  dyjy, 
and  w^hen  you  plot  this  against  dx,  you  have 

^,     or  -^,     or  -.^. 
dx  '  y.dx'  y' dx' 

That  is  to  say,  you  are  multiplying  the  thing  whose  variations 
you  are  studying  by  another  quantity  which  is  itself  continually 
varying;  and  are  dealing  with  something  more  complex  than  the 
original  factors*.  Minot's  method  deals  with  a  perfectly  legitimate 
function  of  x  and  y,  and  is  tantamount  to  plotting  log  y  against  x, 
that  is  to  say,  the  logarithm  of  the  increment  against  the  time. 
This  would  be  all  to  the  good  if  it  led  to  some  simple  result,  a  straight 
line  for  instance ;  but  it  is  seldom  if  ever,  as  it  seems  to  me,  that  it 
does  anything  of  the  kind.  It  has  also  been  pointed  out  as  a  grave 
fault  in  his  method  that,  whereas  growth  is  a  continuous  process, 
Minot  chooses  an  arbitrary  time-interval  as  his  basis  of  comparison, 
and  uses  the  same  interval  in  all  stages  of  development.  There  is 
little  use  in  comparing  the  percentage  increase  fer  week  of  a  week- 
old  chick,  with  that  of  the  same  bird  at  six  months  old  or  at  six 
years. 


The  growth  of  a  population 

After  dealing  with  Man's  growth  and  stature,  Quetelet  turned  to 
the  analogous  problem  of  the  growth  of  a  populatfon — all  the  more 
analogous  in  our  eyes  since  we  know  man  himself  to  be  a  "statistical 
unit,"  an  assemblage  of  organs,  a  population  of  cells.     He  had  read 

*  Schmalhausen,  among  others,  uses  the  same  measure  of  rate  of  growth,  in  the 
form 

log  F-logF      dvl^ 
^~      k{t-t)      ~^  dt    v' 

Arch.  f.  Entw.  Mech.  cxm,  pp.  462-519,  1928. 


m]  MALTHUS  ON  POPULATION  143 

Malthus's  Essay  on  Population*  in  a  French  translation,  and  was 
impressed  like  all  the  world  by  the  importance  of  the  theme.  He 
saw  that  poverty  and  misery  ensue  when  a  population  outgrows  its 
means  of  support,  and  believed  that  multiphcation  is  checked  both 
.  by  lack  of  food  and  fear  of  poverty.  He  knew  that  there  were, 
and  must  be,  obstacles  of  one  kind  or  another  to  the  unrestricted 
increase  of  a  population;  and  he  knew  the  more  subtle  fact  that 
a  population,  after  growing  to  a  certain  height,  oscillates  about  an 
unstable  level  of  equilibrium  f. 

Malthus  had  said  that  a  population  grows  by  geometrical  pro- 
gression (as  1,  2,  4,  8)  while  its  means  of  subsistence  tend  rather  to 
grow  by  arithmetical  (as  1,  2,  3,  4) — that  one  adds  up  while  the 
other  multiphesj.     A  geometrical  progression  is  a  natural  and  a 

*  T.  R.  Malthus,  An  Essay  on  the  Principle  of  Population,  as  it  affects  the  Future 
Improvement  of  Society,  etc.,  1798  (6th  ed.  1826;  transl.  by  P.  and  G.  Prevost, 
Geneva,  1830,  1845).  Among  the  books  to  which  Malthus  was  most  indebted  was 
A  Dissertation  on  the  Numbers  of  Mankind  in  ancient  and  modern  Times,  published 
anonymously  in  Edinburgh  in  1753,  but  known  to  be  by  Robert  Wallace  and  read 
by  him  some  years  before  to  the  Philosophical  Society  at  Edinburgh.  In  this 
remarkable  work  the  writer  says  (after  the  manner  of  Malthus)  that  mankind 
naturally  increase  by  successive  doubling,  and  tend  to  do  so  thrice  in  a  hundred 
years.  He  explains,  on  the  other  hand,  that  "mankind  do  not  actually  propagate 
according  to  the  rule  in  our  tables,  or  any  other  constant  rule;  ^et  tables  of  this 
nature  are  not  entirely  useless,  but  may  serve  to  shew,  how  much  the  increase  of 
mankind  is  prevented  by  the  various  causes  which  confine  their  number  within 
such  narrow  limits."  Malthus  was  also  indebted  to  David  Hume's  Political 
Discourse,  Of  the  Populousness  of  ancient  Nations,  1752,  a  work  criticised  by  Wallace. 
See  also  McCulloch's  notes  to  Adam  Smith's  Wealth  of  Nations,  1828. 

■f"  That  the  nearest  approach  to  equilibrium  in  a  population  is  long- continued 
ebb  and  flow,  a  mean  level  and  a  tide,  was  known  to  Herbert  Spencer,  and  was 
stated  mathematically  long  afterwards  by  Vito  Volterra.  See  also  Spencer's  First 
Principles,  ch. ^22,  sect.  173:  "Every  species  of  plant  or  animal  is  perpetually 
undergoing  a  rhythmical  variation  in  number — now  from  abundance  of  food  and 
absence  of  enemies  rising  above  its  average,  and  then  by  a  consequent  scarcity 
of  food  and  abundance  of  enemies  being  depressed  below  its  average.. .  .Amid 
these  oscillations  produced  by  their  conflict,  lies  that  average  number  of  the  species 
at  which  its  expansive  tendency  ie  in  equilibrium  with  surrounding  repressive 
tendencies."  Cf.  A.  J.  Lotka,  Analytical  note  on  certain  rhythmic  relations  in 
organic  systems,  Proc.  Nat.  Acad.  Sci.  vi,  pp.  ^10-415,  1920;  but  cf.  also  his  Elements 
of  Physical  Biology,  1915,  p.  90.  An  analogy,  and  perhaps  a  close  one,  may  be  found 
on  the  Bourse  or  money  market. 

X  That  a  population  will  soon  oiitrun  its  means  of  subsistence  was  a  natural 
assumption  in  Malthus's  day,  and  in  his  own  thickly  populated  land.  The  danger 
may  be  postponed  and  the  assumption  apparently  falsified,  as  by  an  Argentine 
cattle-ranch  /or  prairie  wheat-farm — but  only  so  long  as  we  enjoy  world-wide 
freedom  of  import  and  exchange. 


144  THE  RATE  OF  GROWTH  [ch. 

common  thing,  and,  apart  from  the  free  growth  of  a  population  or 
an  organism,  we  find  it  in  many  biological  phenomena.  An  epidemic 
dechnes,  or  tends  to  decline,  at  a  rate  corresponding  to  a  geometrical 
progression;  the  mortahty  from  zymotic  diseases  declines  in  geo- 
metrical progression  among  children  from  one  to  ten  years  old; 
and  the  chances  of  death  increase  in  geometrical  progression  after 
a  certain  time  of  Hfe  for  us  all*. 

But  in  the  ascending  scale,  the  story  of  the  horseshoe  nails  tells 
us  how  formidable  a  thing  successive  multipHcation  becomes  f. 
Enghsh  law  forbids  the  protracted  accumulation  of  compound 
interest;  and  hkewise  Nature  deals  after  her  own  fashion  with  the  case, 
and  provides  her  automatic  remedies.  A  fungus  is  growing  on  an 
oaktree — it  sheds  more  spores  in  a  night  than  the  tree  drops  acorns 
in  a  hundred  years.  A  certain  bacillus  grows  up  and  multiphes  by 
two  in  two  hoTlrs'  time;  its  descendants,  did  they  all  survive,  would 
number  four  thousand  in  a  day,  as  a  man's  might  in  three  hundred 
years.  A  codfish  lays  a  million  eggs  and  more — all  in  order  that 
one  pair  may  survive  to  take  their  parents'  places  in  the  world. 
On  the  other  hand,  the  humming-birds  lay  only  two  eggs,  the  auks 
and  guillemots  only  one;  yet  the  former  are  multitudinous  in  their 
haunts,  and  some  say  that  the  Arctic  auks  and  auklets  outnumber 
all  other  birds  in  the  world.  Linnaeus  {  shewed  that  an  annual 
plant  would  have  a  miUion  offspring  in  twenty  years,  if  only  two 
seeds  grew  up  to  maturity  in  a  year. 

But  multiply  as  they  will,  these  vast  populations  have  their 
limits.  They  reach  the  end  of  their  tether,  the  pace  slows  down,  and 
at  last  they  increase  no  more.  Their  world  is  fully  peopled,  whether 
it  be  an  island  with  its  swarms  of  humming-birds,  a  test-tube  with 
its  myriads  of  yeast-cells,  or  a  continent  with  its  miUions  of  mankind. 
Growth,  whether  of  a  population  or  an  individual,  draws  to  its 
natural  end;  and  Quetelet  compares  it,  by  a  bold  metaphor,  to  the 
motion  of  a  body  in  a  resistant  medium.  A  typical  population 
grows  slowly  from  an  asymptotic  minimum;  it  multiphes  quickly; 

*  According  to  the  Law  of  Gompertz ;  cf.  John  Brownlee,  in  Proc.  R.S.E.  xxxi, 
pp.  627-634,  1"911. 

t  Herbert  Spencer,  A  theory  of  population  deduced  from  the  general  law  of 
animal  fertility,  Westminster  Review,  April  1852. 

X  In  his  essay  De  Tellure,  1740. 


Ill]  VERHULST'S  LAW  145 

it  draws  slowly  to  an  ill-defined  and  asymptotic  maximum.  The 
two  ends  of  the  population-curve  define,  in  a  general  way,  the 
whole  curve  between;  for  so  beginning  and  so  ending  the  curve 
must  pass  through  a  point  of  inflection,  it  must  be  an  S-shaped 
curve.  It  is  just  such  a  curve  as  we  have  seen  imder  simple 
conditions  of  growth  in  an  individual  organism. 

This  general  and  all  but  obvious  trend  of  a  population-curve  has 
been  recognised,  with  more  or  less  precision,  by  many  writers.  It 
is  imphcit  in  Quetelet's  own  words,  as  follows:  "Quand  une 
population  pent  se  developper  hbrement  et  sans  obstacles,  elle  croit 
selon  une  progression  geometrique;  si  le  developpement  a  heu  au 
miheu  d'obstacles  de  toute  espece  qui  tendent  a  Farreter,  et  qui 
agissent  d'une  maniere  uniforme,  c'est  a  dire  si  I'etat  sociale  ne 
change  point,  la  population  n'augmente  pas  d'une  maniere  indefinie, 
mais  elle  tend  de  plus  en  plus  a  devenir  stationnaire*.''  P.  F.  Verhulst, 
a  mathematical  colleague  of  Quetelet's,  was  interested  in  the  same 
things,  and  tried  to  give  a  mathematical  shape  to  the  same  general 
conclusions;  that  is  to  say,  he  looked  for  a  "fonction  retardatrice" 
which  should  turn  the  Malthusian  curve  of  geometrical  progression 
into  the  S-shaped,  or  as  he  called  it,  the  logistic  curve,  which  should 
thus  constitute  the  true  "law  of  population,"  and  thereby  indicate 
(among  other  things)  the  hmit  above  which  the  population^  was  not 
likely  to  grow  f . 

Verhulst  soon  saw  that  he  could  only  solve  his  problem  in  a 
prehminary  and  tentative  way;  ''la  hi  de  la  population  nous  est 
inconnue,  parcequ'on  ignore  la  nature  de  la  fonction  qui  sert  de 
mesure  aux  obstacles  qui  s'opposent  a  la  multiphcation  indefini  de 
I'espece  humaine."  The  materials  at  hand  were  almost  unbeUevably 
scanty  and  poor.  The  French  statistics  were  taken  from  documents 
"qui  ont  ete  reconnus  entierement  fictifs";  in  England  the  growth 

*  Physique  Sociale,  i,  p.  27,  1835.  But  Quetelet's  brief  account  is  somewhat 
ambiguous,  and  he  had  in  mind  a  body  falling  through  a  resistant  medium — which 
suggests  a  limiting  velocity,  or  limiting  annual  increment,  rather  than  a  terminal 
value.  See  Sir  G.  Udny  Yule,  The  growth  of  population,  Journ.  R.  Statist.  Soc. 
Lxxxvm,  p.  42,  1925. 

t  P.  F.  Verhulst,  Notice  sur  la  loi  que  la  population  suit  dans  son  accroissement. 
Correspondence  math.  etc.  public  par  M.  A.  Quetelet,  x,  pp.  113-121,  1838;  Rech. 
math,  sur  la  loi  etc.,  Nozw.  Mem.  de  VAcad.  R.  de  Bruxelles,  xviii,  38  pp.,  1845; 
deuxieme  Mem.,  ihid.  xx,  32  pp.,  1847.  The  term  logistic  curve  had  already  been 
used  by  Edward  Wright;  see  antea,  p.  135.  footnote. 


146  THE  RATE  OF  GROWTH  [ch. 

of  the  population  was  estimated  by  the  number  of  births,  and  the 
births  by  the  baptisms  in  the  Church  of  England,  "de  maniere  que 
les  enfants  des  dissidents  ne  sont  point  portes  sur  les  registres 
officiels."  A  law  of  population,  or  "loi  d'affaibhssement"  became 
a  mere  matter  of  conjecture,  and  the  simplest  hypothesis  seemed  to 
Verhulst  to  be,  to  regard  "cet  affaibhssement  comme  proportionnel 
a  I'accroissement  de  la  population,  depuis  le  moment  ou  la  difficulty 
de  trouver  de  bonnes  terres  a  commence  a  se  faire  sentir*." 

Verhulst  was  making  two  assumptions.  The  first,  which  is  beyond 
question,  is  that  the  rate  of  increase  cannot  be,  and  indeed  is  not, 
a  constant;  and  the  second  is  that  the  rate  must  somehow  depend 
on  (or  be  some  function  of)  the  population  for  the  time  being. 
A  third  assumption,  again  beyond  question,  is  that  the  simplest 
possible  function  is  a  linear  function.  He  suggested  as  the  simplest 
possible  case  that,  once  the  rate  begins  to  fall  (or  once  the  struggle 
for  existence  sets  in),  it  will  fall  the  more  as  the  population  continues 
to  grow;  we  shall  have  a  growth-factor  and  a  retardation-factor  in 
proportion  to  one  another.  He  was  making  early  use  of  a  simple 
differential  equation  such  as  Vito  Volterra  and  others  now  employ 
freely  in  the  general  study  of  natural  selec.tionf. 

The  point  wher^  a  struggle  for  existence  first  sets  in,  and  where 
ipso  facto  the  rate  of  increase  begins  to  diminish,  is  called  by  Verhulst 
the  normal  level  of  the  population ;  he  chooses  it  for  the  origin  of  his 
curve,  which  is  so  defined  as  to  be  symmetrical  on  either  side  of 
this  origin.  Thus  Verhulst's  law,  and  his  logistic  curve,  owe  their 
form  and  their  precision  and  all  their  power  to  forecast  the  future 
to  certain  hypothetical  assumptions;  and  the  tentative  solution 
arrived  at  is  one  "sous  le  point  de  vue  mathematiquej." 

♦  Op.  cit.  p.  8. 

t  Besides  many  well-known  papers  by  Volterra,  see  V.  A,  Kostitzin,  Biologie 
mathematique,  Paris,  1937.  Cf.  also,  for  the  so-called  "Malaria  equations,"  Ronald 
Ross,  Prevention  of  Malaria,  2nd  ed.  1911,  p.  679;  Martini,  Zur  Epidemiologie  d. 
Malaria,  Hamburg,  1921;  W.  R.  Thompson,  C.R.  clxxiv,  p.  1443,  1922,  C.  N. 
Watson,  Nature,  cxi,  p.  88,  1923. 

J  Verhulst  goes  on  to  say  that  "  une  longue  serie  d 'observations,  non  interrompues 
par  de  grandes  catastrophes  sociales  ou  des  revolutions  du  globe,  fera  probablement 
decouvrir  la  fonction  retardatrice  dont  il  vient  d'etre  fait  mention."  Verhulst 
simplified  his  problem  to  the  utmost,  but  it  is  more  complicated  today  than  ever; 
he  thought  it  impossible  that  a  country  should  draw  its  bread  and  meat  from 
overseas:    "lors  meme  qu'une  partie  considerable  de  la  population  pourrait  etre 


Ill]  THE  LOGISTIC  CURVE  147 

The  mathematics  of  the  Verhulst-Pearl  curve  need  hardly  concern 
us;  they  are  fully  dealt  with  in  Raymond  Pearl's,  Lotka's  and  other 
books.  Verhulst  starts,  as  Malthus  does,  with  a  population  growing 
in  geometrical  progression,  and  so  giving  a  logarithmic  curve: 

dp 

He  then  assumes,  as  his  "loi  d'affaibUssement,"  a  coefficient  of 
retardation  {n)  which  increases  as  the  population  increases: 


dp  2 


Integrating,  p  =  ^^_^_^_. 

If  the  point  of  inflection  be  taken  as  the  origin,  k  =  0;  and  again 
for  <  =  00,  jt?  =  —  =  L.     We  may  write  accordingly: 


1  +  e-^ 


Malthus  had  reckoned  on  a  population  doubling  itself,  if  unchecked 
by  want  or  "accident,"  every  twenty-five  years*;  but  fifty  years 
after,  Verhulst  shewed  that  this  "grande  vitesse  d'accroissement" 
was  no  longer  to  be  found  in  France  or  Belgium  or  other  of  the 
older  countries!,  but  wa^  still  being  reaUsed  in  the  United  States 
(Fig.  28).  All  over  Europe,  "le  rapport  de  I'exces  annuel  des  nais- 
sances  sur  les  deces,  a  la  population  qui  I'a  fourni,  va  sans  cesse  en 
s'affaiblissant ;  de  maniere  que  Faccroissement  annuel,  dont  la  valeur 
absolue  augmente  continuellement  lorsqu'il  y  a  progression  geo- 
metrique,  parait  suivre  une  progression  tout  au  plus  arithmetique." 

nourrie  de  bles  etrangers,  jamais  un  gouvernement  sage  ne  consentira  a  faire 
dependre  I'existence  de  milliers  de  citoyens  du  bon  vouloir  des  souverains  etrangers." 
On  this  and  other  problems  in  the  growth  of  a  human  population,  see  L.  Hogben's 
Genetic  Problems,  etc.,  1937,  chap.  vii.  See^also  [int.  al.)  Warren  S.  Thompson  and 
P.  K.  Whelpton,  Population  Trends  in  the  United  States,  1933;  F.  Lorimer  and 
F.  Osborn,  Dynamics  of  Population,  1934,  etc. 

*  An  estimate  based,  like  the  rest  of  Malthus's  arithmetic,  on  very  slender 
evidence. 

t  In  Quetelet's  time  the  European  countries,  far  from  doubling  in  twenty-five 
years,  were  estimated  to  do  so  in  from  sixty  years  (Norway)  to  four  hundred  years 
(France);  see  M.  Haushofer,  Lehrbuch  der  Statiatik,  1882. 


148 


THE  RATE  OF  GROWTH 


[CH. 


The  "celebrated  aphorism"  of  Malthus  was  thus,  and  to  this  extent, 
confirmed*.  In  the  United  States,  the  Malthusian  estimate  of 
unrestricted  increase  continued  to  be  reahsed  for  a  hundred  years 
after  Malthus  wrote;  for  the  3-93  millions  of  the  U.S.  census  of 
1790  were  doubled  three  times  over  in  the  census  of  1860,  and  four 
times  over  in  that  of  1890.  A  capital  which  doubles  in  twenty-five 
years  has  grown  at  2-85  per  cent,  per  annum,  compound  interest; 
the  U.S.  population  did  rather  more,  for  it  grew  at  fully  3  per  cent, 
for  fifty  of  those  hundred  years  f. 


1790  1800 


1850 


1900 


1930 


Fig.  28.     Population  of  the  United  States,  1790-1930. 

The  population  of  the  whole  world  and  of  every  continent  has 
increased  during  modem  times,  and  the  increase  is  large  though 
the  rate  is  low.  The  rate  of  increase  has  been  put  at  about  half-a- 
per-cent  per  annum  for  the  last  three  hundred  years — a  shade  more 
in  Europe  and  a  shade  less  in  the  rest  of  the  world  { : 

*  Op.  cit.  1845,  p.  7. 

t  Verhulst  foretold  forty  millions  as  the  "extreme  limit"  of  the  population 
of  France,  and  6^  millions  as  that  of  Belgium.  The  latter  estimate  he  increased 
to  8  millions  later  on.  The  actual  populations  of  France  and  Belgium  at  the 
present  time  are  a  little  more  than  the  ultimate  limit  which  Verhulst  foretold. 

+  From  A.  M.  Carr-Saunders'  World  Population,  1936,  p.  30. 


in]  RAYMOND  PEARL  149 

An  estimate  of  the  population  of  the  world 
(After  W.  F.  Willcox) 

Mean  rate  of 
1650         1750        1800         1850         1900  increase 

Europe  100  140  187  266  401         0-52  %  per  annum 

World  total        545  728  906         1171         1608         0-49%    „ 

Verhulst  was  before  his  time,  and  his  work  was  neglected  and 
presently  forgotten.  Only  some  twenty  years  ago,  Raymond  Pearl 
and  L.  J.  Reed  of  Baltimore,  studying  the  U.S.  population  as 
Verhulst  had  done,  approached  the  subject  in  the  same  way,  and 
came  to  an  identical  result;  then,  soon  afterwards  (about  1924), 
Raymond  Pearl  came  across  Verhulst's  papers,  and  drew  attention 
to  what  we  now  speak  of  as  the  Verhulst-Pearl  law.  Pearl  and 
Reed  saw,  as  Verhulst  had  dope,  that  a  "law  of  population"  which 
should  cover  all  the  ups  and  downs  of  human  affairs  was  not  to  be 
found;  and  yet  the  general  form  which  such  a  law  must  take  was 
plain  to  see.  There  must  be  a  limit  to  the  population  of  a  region, 
great  or  small;  and  the  curve  of  growth  must  sooner  or  later  "turn 
over,"  approach  the  limit,  and  resolve  itself  into  an  S-shaped  curve. 
The  rate  of  growth  (or  annual  increment)  will  depend  (1)  on  the 
population  at  the  time,  and  (2)  on  "the  still  unutiUsed  reserves  of 
population-support  existing"  in  the  available  land.  Here  we  have, 
to  all  intents  and  purposes,  the  growth-factor  and  retardation-factor 
of  Verhulst,  and  they  lead  to  the  same  formula,  or  the  same 
differential  equation,  as  his*. 

A  hundred  years  have  passed  since  Verhulst  dealt  with  the  first 
U.S.  census  returns,  and  found  them  verifying  the  Malthusian 
expectation  of  a  doubhng  every  twenty-five  years.  That  "grande 
vitesse  d'accroissement "  continued  through  five  decennia;  but  it 
ceased  some  seventy  years  ago,  and  a  retarding  influence  has  been 
manifest  through  all  these  seventy  years  (Fig.  29).  It  is  more 
recently,  only  after  the  census  of  1910,  that  the  curve  seemed  to  be 

*  Raymond  Pearl  and  L.  J.  Reed,  on  the  Rate  of  growth  of  the  population  of 
the  U.S.  since  1790,  and  its  mathematical  representation,  Proc.  Nat.  Acad.  Sci. 
VI,  pp.  275-288,  1920;  ibid,  vin,  pp.  365-368,  1922;  Metron,  m,  1923.  In  the 
first  edition  of  Pearl's  Medical  Biometry  and  Statistics,  1923  (2nd  ed.  1930),  Verhulst 
is  not  mentioned.  See  also  his  Studies  in  Human  Biology,  Baltimore,  1924,  Natural 
History  of  Population,  1939,  and  other  works. 


150 


THE  RATE  OF  GROWTH 


[CH. 


1800  1850  1900  1940 

Fig.  29.     Decennial  increments  of  the  population  of  the  United  States. 
*   The  Civil  War.     *  *   The  "slump". 


zuu 

— 1 — 1 

-1 — r- 

1      1      1      1      1 

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- 

1 

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_ 

y 



y 

- 

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/ 

~ 

• 

150 

— 

/                           _ 

/ 

_ 

1 

/ 
/ 

¥            ^ 

- 

t 

/ 

§ 

- 

.  ^ 

/ 

1    '^^ 

/ 

f 

1 

- 

0 

- 

^ 

- 

- 

P 

- 

50 

- 

- 

"""i 

1      1 

1      1      1      1      1 

1  1  1  1  1   1  r  1 

1     1     1 

1800  1850  1900  1950  2000 

Fig.  30.     Conjectural  population  of  the  United  States, 
according  to  the  Verhulst- Pearl  Law. 


Ill]       POPULATION   OF  THE  UNITED   STATES      151 

finding  its  turning-point,  or  point  of  inflection;  and  only  now,  since 
1940,  can  we  say  with  full  confidence  that  it  has  done  so. 

A  hundred  years  ago  the  conditions  were  still  relatively'  simple, 
but  they  are  far  from  simple  now.  Immigration  was  only  beginning 
to  be  an  important  factor;  but  immigrants  made  a  quarter  of  the 
whole  increase  of  the  population  of  the  United  States  during  eighty 
of  these  hundred  years*.  Wars  and  financial  crises  have  made  their 
mark  upon  the  curve;  manners  and  customs,  means  and  standards 
of  living,  have  changed  prodigiously.  But  the  S-shaped  curve 
makes  its  appearance  through  all  of  these,  and  the  Verhulst-Pearl 
formula  meets  the  case  with  surprising  accuracy. 

Population  of  the  United  States 

I  In  ten  years 


Calculated 

No.  of 

A 

Total 

■\ 

Increase 
by  multi- 

by logistic 

immigrants 

increase  of 

plication 

Population 

curve 

landed 

population  Percentage 

in  25 

Year 

xlOOO 

(Udny  Yule) 

xlOOO 

xlOOO 

increase 

years 

1790 

3,929 

3,929 

— 

— 

— 

— 

1800 

5,308 

5,336 

— 

1,379 

351 

— 

1810 

7,240 

7,223 

— 

1,932 

36-4 



1820 

9,638 

9,757 

250 

2,398 

331 

208 

1830 

12,866 

13,109 

228 

3,228 

33-5 

205 

1840 

17,069 

17,506 

538 

4.203 

32-7 

2-02 

1850 

23,192 

23,192 

1,427 

6,123 

35-9 

206 

1860 

31,443 

30,418 

2,748 

8,251 

35-6 

210 

1870 

38,558 

39,372 

2,123 

7,115 

32-6 

1-91 

1880 

50,156 

50,177 

2,741 

11,598 

301 

1-83 

1890 

62.948 

62,769 

5,249 

12,792 

25-5 

1-80 

1900 

75,995 

76,870 

3,694 

13,047 

20-7 

1-71 

1910 

91,972 

91,972 

8,201 

15,977 

210 

1-63 

1920 

105,711 

— 

6,347 

13,739 

14-9 

1-52 

1930 

122,975 

— 

— 

17,264 

161 

1-46 

1940 

131,669 

— 

— 

8,694 

71 

1-33 

A  colony  of  yeast  or  of  bacteria  is  a  population  in  its  simplest 
terms,  and  Verhulst's  law  was  rediscovered  in  the  growth  of  a 
bacterial  colony  some  years  before  Raymond  Pearl  found  it  in  a 
population  of  men,  by  Colonel  M'Kendrick  and  Dr  Kesava  Pai,  who  put 
their  case  very  simply  indeed  f.    The  bacillus  grows  by  geometrical 

*  Without  counting  the  children  born  to  those  immigrants  after  landing,  and 
before  the  next  census  return. 

t  A.  G.  M'Kendrick  and  M.  K.  Pai,  The  rate  of  multiplication  of  micro-organisms : 
a  mathematical  study,  Proc.  R.S.E.  xxxi,  pp.  649-655,  1911.  (The  period  of 
generation  in  B.  coli,  answering  to  Malthus's  twenty-five  years  for  men,  was  found 
to  be  22^  minutes.)  Cf.  also  Myer  Coplans,  Journ.  of  Pathol,  and  Bacleriol.  xiv, 
p.  1,  1910  and  H.  G.  Thornton,  Ana.  of  Applied  Biology,  1922,  p.  265. 


152  THE  RATE  OF  GROWTH  [ch. 

progression  so  long  as  nutriment  is  enough  and  to  spare;  that  is  to 
say,  the  rate  of  growth  is  proportional  to  the  number  present: 

dt        ^ 

But  in  a  test-tube  colony  the  supply  of  nourishment  is  hmited, 
and  the  rate  of  multiphcation  is  bound  to  fall  off.  If  a  be  the 
original  concentration  of  food-stuff,  it  will  have  dwindled  by  time  t 
to  (a  —  y).     The  rate  of  growth  will  now  be 

%  =  by{<^-y), 

which  means  that  the  rate  of  increase  iS  proportional  to  the  number 
of  organisms  present,  and  to  the  concentration  of  the  food-supply. 
It  is  Verhulst's  case  in  a  nutshell;  the  differential  equation  so 
indicated  leads  to  an  S-shaped  curve  which  further  experiment 
confirms;  and  Sach's  "grand  period  of  growth"  is  seen  to  accom- 
phsh  itself*. 

The  growth  of  yeast  is  studied  in  the  everyday  routine  of  a 
brewery.  But  the  brewer  is  concerned  only  with  the  phase  of 
unrestricted  growth,  and  the  rules  of  compound  interest  are  all  he 
needs,  to  find  its  rate  or  test  its  constancy.  A  population  of  1360 
yeast-cells  grew  to  3,550,000  in  35  hours:  it  had  multiplied -2610 
times.    Accordingly, 

1^8^51^^^^  =  0-098  =  log  1-254. 

That  is  to  say,  the  population  had  increased  at  the  rate  of  25-4  per 
cent,  per  hour,  during  the  35  hours. 

The  time  (^2)  required  to  double  the  population  is  easily  found : 

log  2         0-301      ^  ^^  , 

*  The  sigmoid  curve  illustrates  a  theorem  which,  obvious  as  it  may  seem,  is  of  no 
small  philosophical  importance,  to  wit,  that  a  body  starting  from  rest  must,  in  order 
to  attain  a  certain  velocity,  pass  through  all  intermediate  velocities  on  its  way. 
Galileo  discusses  this  theorem,  and  attributes  it  to  Plato:  "Platone  avendo  per 
avventura  avuto  concetto  non  potere  alcun  mobil  passare  daUa  quiete  ad  alcun 

determinate  grado  di  velocita se  non  col  passare  per  tutti  gli  altri  gradi  di 

velocita  minori,  etc.";  Discorsi  e  dimostrazioni,  ed.  1638,  p.  254. 


Ill]  A  POPULATION  OF  YEAST  153 

The  duplication-period  thus  determined  is  known  to  brewers  as 
the  generation-time. 

Much  care  is  taken  to  ensure  the  maximal  growth.  If  the  yeast 
sink  to  the  bottom  of  the  vat  only  its  upper  layers  enjoy  unstinted 
nutriment;  a  potent  retardation-factor  sets  in,  and  the  exponential 
phase  of  the  growth-curve  degenerates  into  a  premature  horizontal 
asymptote.  Moreover,  both  the  yeast  and  the  bacteria  differ  in 
this  respect  from  the  typical  (or  perhaps  only  simpHfied)  case  of 
man,  that  they  not  only  begin  to  suffer  want  as  soon  as  there  comes 
to  be  a  deficiency  of  any  one  essential  constituent  of  their  food*,  but 
they  also  produce  things  which  are  injurious  to  their  own  growth 
and  in  time  fatal  to  their  existence.  Growth  stops  long  before  the 
food-supply  is  exhausted ;  for  it  does  so  as  soon  as  a  certain  balance 
is  reached,  depending  on  the  kind  or  quahty  of  the  yeast,  between 
the  alcohol  and  the  sugar  in  the  cellf. 

If  we  use  the  compound-interest  law  at  all,  we  had  better  think 
of  Nature's  interest  as  being  paid,  not  once  a  year  nor  once  an  hour 
as  our  elementary  treatment  of  the  yeast-population  assumed,  but 
continuously;  and  then  we  learn  (in  elementary  algebra)  that  in 
time  t,  at  rate  r,  a  sum  P  increases  to  Pe'^\  or  P<  ^  ^o^^*. 

Applying  this  to  the  growth  of  our  sample  of  1360  yeast  cells, 
we  have 

PtIPo  =  2610,  log  2610  =  3-417,  which,  multipHed  by  the  modulus 
2-303  =  7-868.     Dividing  by  n  =  35,  the  number  of  hours, 

7-868/35  =  0-225  =  r. 

The  rate,  that  is  to  say,  is  22-5  per  cent,  per  hour,  continuous 
compound  interest.    It  becomes  a  well-defined  physiological  constant, 
and  we  may  call  it,  with  V.  H.  Blackman,  an  index  of  efficiency. 
Our  former  result,  for  interest  at  hourly  intervals,  was  25-4  per 

*  According  to  Liebig's  "law  of  the  minimum." 

t  T.  Carlson,  Geschwindigkeit  und  Grosse'der  Hefevermehrung,  Biochem.  Ztschr. 
Lvn,  pp.  313-334,  1913;  A.  Slator,  Journ.  Chem.  Soc.  cxix,  pp.  128-142,  1906; 
Biochem.  Journ.  vii,  p.  198, 1913;  0.  W.  Richards,  Ann.  of  Botany,  xlh,  pp.  271-283, 
1928;  Alf  Klem,  Hvalradets  Skrifter,  nr.  7,  pp.  55-91,  Oslo,  1933;  Per  Ottestad, 
ibid.  pp.  30-54.  For  optimum  conditions  of  temperature,  nutriment,  pB.,  etc.  see 
Oscar  W.  Richards,  Analysis  of  growth  as  illustrated  by  yeast.  Cold  Spring  Harbour 
Symposia,  ii,  pp.  157-166,  1934. 


154 


THE  RATE  OF  GROWTH 


[CH. 


cent. ;  there  is  no  great  difference  between  such  short  intervals  and 
actual  continuity,  but  there  is  a  deal  of  difference  between  continuous 
payment  and  payment  (say)  once  a  year*.  Certain  sunflowers 
(Helianihus)  were  found  to  grow  as  follows,  in  thirty-seven  days: 


Compound  interest  rate  (%) 


Weight  (gm.) 


Continuous 


Discontinuous 


Giant  sunflower 
Dwarf  sunflower 


Seedling 

0033 
0035 


Plant 

17-33 
14-81 


Per  day   Per  wk.    Per  day      Per  wk. 

170  119  18-5  228% 

16-4  114  17-7  214% 


When  the  yeast  population  is  allowed  to  run  its  course,  it  yields 
a  simple  S-shaped  curve ;   and  the  curve  of  first  differences  derived 


rTi*-2*5<rm-2(r 

m-o" 

m 

m-fcr 

m+2(r   m+2*5(r 

( 

1 
3             1 

2 

3 

4 

5             6             7 

Fig.  31.     The  growth  of  a  yeast-population.     After  Per  Ottestad. 

from  this  is,  necessarily,  a  bell-shaped  curve,  so  closely  resembling 
the  Gaussian  curve  that  any  difference  between  them  becomes  a 
deUcate  matter.  Taking  the  numbers  of  the  population  at  equal 
intervals  of  time  from  asymptotic  start  to  asymptotic  finish,  we 
may  treat  this  series  of  numbers  like  any  other  frequency  distribu- 
tion.   Finding  in  the  usual  way  the  mode  and  standard  deviation, 

*  Cf.  V.  H.  Blackman,  The  compound  interest  law  and  plant  growth,  Ann.  of 
Botany,  xxxiii,  pp.  353-360,  1919.  The  first  papers  on  growth  by  compound 
interest  in  plants  were  by  pupils  of  Noll  in  Bonn:  e.g.  von  Kreusler,  Wachstum  der 
Maispflanze,  Landw.  JB.  1877-79;  P.  Gressler,  Suhstanz-quotienten  von  Helianthus, 
Diss.  Bonn,  1907  etc. 


Ill]  A  POPULATION  OF  FLIES  155 

we  draw  the  corresponding  Gaussian  curve;  and  the  close  "fit" 
between  the  observed  population-curve  and  the  calculated  Gaussian 
curve  is  sufficiently  shewn  by  Mr  Per  Ottestad's  figure  (Fig.  31). 
This  is  a  very  remarkable  thing.  We  began  to  think  of  the  curve 
of  error  as  a  function  with  which  time  had  nothing  to  do,  but  here 
we  have  the  same  curve  (or  to  all  intents  and  purposes  the  same) 
with  time  for  one  of  its  coordinates.  We  might  (I  think)  add  one 
more  to  the  names  of  the  curve  of  error,  and  call  it  the  curve 
of  optimum ;  it  represents  on  either  hand  the  natural  passage  from 
best  to  worst,  from  Ukehest  to  least  likely. 

A  few  flies  (Drosophila)  in  a  bottle  illustrate  the  rise  and  fall  of 
a  population  more  complex  than  yeast,  as  Raymond  Pearl  has 
shewn*  The  colony  dwindles  to  extinction  if  food  be  v/ithheld; 
if  it  be  sufficient,  the  numbers  rise  in  a  smooth  S-shaped  curve; 
if  it  be  plentiful  and  of  the  best,  they  end  by  fluctuating  about  an 
unstable  maximum.  "The  population  waves  up  and  down  about 
an  average  size,"  as  Raymond  Pearl  says,  as  Herbert  Spencer  had 
foreseen t,  and  as  Vito  Volterra's  differential  equations  explain. 
The  growth-rate  slackens  long  before  the  hunger  hne  is  reached; 
crowding  affects  the  birth-rate  as  well  as  the  death-rate,  and  a 
bottleful  of  flies  produces  fewer  and  fewer  offspring  per  pair  the 
more  flies  we  put  into  the  bottle  {.  It  is  true  also  of  mankind,  as 
Dr  WiUiam  Farr  was  the  first  to  shew,  that  overcrowding  diminishes 
the  birth-rate  and  shortens  the  "expectation  of  Hfe§ ."  It  happened 
so  in  the  United  States,  pari  passu  with  the  growth  of  immigration, 
incipient  congestion  acting  (or  so  it  seemed)  as  an  obstacle,  or  a 
deterrent,  to  the  large  families  of  former  days.  Nevertheless,  children 
still  pullulate  in  the  slums.  The  struggle  for  existence  is  no  simple 
affair,  and  things  happen  which  no  mathematics  can  foretell. 

*  Raymond  Pearl  and  S.  L.  Parker,  in  Proc.  Nat.  Acad.  Sci.  vni,  pp.  212-219, 
1922;   Pearl,  Journ.  Exper.  Zool.  Lxm,  pp.  57-84,  1932. 

t  "Wherever  antagonistic  forces  are  in^  action,  there  tends  to  be  alternate 
predominance." 

X  In  certain  insects  an  optimum  density  has  been  observed;  a  certain  amount 
of  crowding  accelerates,  and  a  greater  amount  retards,  the  rate  of  reproduction. 
Cf.  D.  Stewart  Maclagan,  Effect  of  population-density  on  rate  of  reproduction, 
Proc.  R,  S.  (B),  CXI,  p.  437,  1932;  W.  Goetsch,  Ueber  wachstumhemmende  Factoren, 
Zool.  Jakrb.  (Allg.  Zool.),  xlv,  pp.  799-840,  1928. 

§  Dr  W.  Farr,  Fifth  Report  of  the  Registrar-General,  1843,  p.  406  (2nd  ed.). 


156  THE  RATE  OF  GROWTH  [ch. 

An  analogous  S-shaped  curve,  given  by  the  formula  L^  =  kg^, 
was  introduced  by  Benjamin  Gompertz  in  1825* ;  it  is  well  known  to 
actuaries,  and  has  been  used  as  a  curve  of  growth  by  several  writers 
in  preference  to  the  logistic  curve.  It  was  devised,  and  well  devised, 
to  express  a  "law  of  human  mortality",  and  to  signify  the  number 
surviving  at  any  given  age  (x),  "if  the  average  exhaustions  of  a 
man's  power  to  avoid  death  were  such  that  at  the  end  of  infinitely 
small  intervals  of  time  he  lost  equal  portions  (i.e.  equal  proportions) 
of  his  remaining  power 'to  oppose  destruction."  The  principle 
involved  is  very  important.  Death  comes  by  two  roads.  One  is 
by  cha^ce  or  accident,  the  other  by  a  steady  deterioration,  or 
exhaustion,  or  growing  inability  to  withstand  destruction;  and 
exhaustion  comes  (roughly  speaking)  as  by  the  repeated  strokes  of 
an  air-pump,  for  the  life-tables  shew  mortality  increasing  in  geo- 
metrical progression,  at  least  to  a  first  approximation  and  over 
considerable  periods  of  years.  Gompertz  relied  wholly  on  the 
experience  of  "life-contingencies,"  but  the  same  deterioration  of 
bodily  energies  is  plainly  visible  as  growth  itself  slows  down;  for 
we  have  seen  how  growth-rate  in  infancy  is  such  as  is  never  after- 
wards attained,  and  we  may  speak  of  growth-energy  and  its  gradual 
loss  or  decrement,  by  an  easy  but  significant  alteration  of  phrase. 
To  deal  with  the  declining  growth-rate,  as  Gompertz  did  with  the 
falling  expectation  of  life,  and  so  to  measure  the  remaining  energy 
available  from  time  to  time,  would  be  a  greater  thing  than  to  record 
mere  weights  and  sizes;  it  raises  the  problem  from  mere  change  of 
physical  magnitudes  to  an  estimation  of  the  falling  or  fluctuating 
physiological  energies  of  the  bodyf.     We  have  seen  how  in  only 

*  Benjamin  Gompertz,  On  the  nature  of  the  function  expressive  of  the  law  of 
human  mortality,  Phil.  Trans,  xxxvi,  pp.  513-585,  1825.  First  suggested  for  use 
in  growth-problems  by  Sewall  Wright,  Journ.  Amer.  Statist,  Soc.  xxi,  p.  493, 
1926.  See  also  C.  P.  Winsor,  The  Gompertz  curve  as  a  growth  curve,  Proc.  Nat. 
Acad.  Sci.  XVIII,  pp.  1-8,  1932;  cf.  {int.  al.)  G.  R.  Da  vies.  The  growth  curve, 
Journ.  Amer.  Statist.  Soc.  xxii,  pp.  370-374,  1927;  F.  W.  Weymouth  and  S.  H. 
Thompson,  Age  and  growth  of  the  Pacific  cockle,  Bull.  Bureau  Fisheries,  xlvi, 
pp.  63f3-641,  1930-31 ;  also  Weymouth,  McMillen  and  Rich,  in  Journ.  Exp.  Biol. 
vni,  p.  228,  1931. 

t  A  bold  attempt  to  treat  the  question  from  the  physiological  side,  and  on 
Gompertz's  lines,  was  made  only  the  other  day  by  P.  B.  Medawar,  The  growth, 
growth-energy  and  ageing  of  the  chicken's  heart,  Proc.  R.S.  (B),  cxxix,  pp.  332- 
355,  1940.  Cf.  James  Gray,  The  kinetics  of  growth,  Journ.  Exp.  Biol,  vi,  pp.  248- 
274,  1929. 


Ill]  THE   GOMPERTZ   CURVE  157 

few  and  simple  cases  can  a  simple  curve  or  single  formula  be  found 
to  represent  the  growth-rate  of  an  organism;  and  how  our  curves 
mostly  suggest  cycles  of  growth,  each  spurt  or  cycle  enduring  for 
a  time,  and  one  following  another.  Nothing  can  be  more  natural 
from  the  physiological  point  of  view  than  that  energy  should  be 
now  added  and  now  withheld,  whether  with  the  return  of  the 
seasons  or  at  other  stages  on  the  eventful  journey  from  childhood 
to  manhood  and  old  age. 

The  symmetry,  or  lack  of  skewness,  in  the  Verhulst-Pearl  logistic 
curve  is  a  weak  point  rather  than  a  strong;  the  Gompertz  curve 
is  a  skew  curve,  with  its  point  of  inflexion  not  half-way,  but  about 
one- third  of  the  way  between  the  asymptotes.  But  whether  in 
this  or  in  the  logistic  or  any  other  equation  of  growth,  the  precise 
point  of  inflexion  has  no  biological  significance  whatsoever.  What 
we  want,  in  the  first  instance,  is  an  S-shaped  curve  with  a  variable, 
or  modifiable,  degree  of  skewness.  After  all,  the  same  difficulty 
arises  in  all  the  use  we  make  of  the  Gaussian  curve :  which  has  to 
be  eked  out  by  a  whole  family  of  skew  curves,  more  or  less  easily 
derived  from  it.  We  are  far  from  being  confined  to  the  Gaussian 
curve  {sensu  stricto)  in  our  studies  of  biological  probabihty,  or  to  the 
logistic  curve  in  the  study  of  population. 

Yet  another  equation  has  been  proposed  to  the  S-shaped  curve 
of  growth,  by  Gaston  Backman,  a  very  dihgent  student  of  the 
whole  subject.  The  rate  of  growth  is  made  up,  he  says,  of  three 
components:  a  constant  velocity,  an  acceleration  varying  with  the 
time,  and  a  retardation  which  we  may  suppose  to  vary  with  the 
square  of  the  time.  Acceleration  would  then  tend  to  prevail  in  the 
earlier  part  of  the  curve,  and  retardation  in  the  latter,  as  in  fact 
they  do;   and  the  equation  to  the  curve  might  be  written: 

log  H  =  ko  +  k^  log  T-k^  log2  T. 

The  formula  is  an  elastic  one,  and  can  be  made  to  fit  many  an 
S-shaped  curve;   but  again  it  is  empirical. 

The  logistic  curve,  as  defined  by  Verhulst  and  by  Pearl,  has 
doubtless  an  interest  of  its  own  for  the  mathematician,  the  statistician 
and  the  actuary.  But  putting  aside  all  its  mathematical  details  and 
all  arbitrary  assumptions,  the  generalised  S-shaped  curve  is  a  very 
symbol  of  childhood,  maturity  and  age,  of  activity  which  rises  to 


158  THE  RATE  OF  GROWTH  [ch. 

fall  again,  of  growth  which  has  its  sequel  in  decay.  The  growth 
of  a  child  or  of  a  nation;  the  history  of  a  railway*,  or  the  speed 
between  stations  of  a  train;  the  spread  of  an  epidemic]",  or  the 
evolutionary  survival  of  a  favoured  type  J — all  these  things  run 
their  course,  in  its  beginning,  its  middle  and  its  end,  after  the  fashion 
of  the  S-shaped  curve.  That  curve  represents  a  certain  common 
pattern  among  Nature's  "mechanisms,"  and  is  (as  we  have  said 
before),  a  "mecanisme  commun  aux  phenomenes  disparates§." 

At  the  same  time — and  this  is  a  very  interesting  part  of  the  story 
— the  S-shaped  curve  is  no  other  than  what  Galton  called  a  curve  of 
distribution,  that  is  to  say  a  curve  of  integration  or  summation- 
curve,  whose  differential  is  closely  akin  to  the  Gaussian  curve  of 
error. 

Such,  to  a  first  approximation,  is  our  S-shaped  population-curve, 
and  such  are  the  many  phenomena  which,  to  a  first  approximation, 
it  helps  us  to  compare.  But  it  is  only  to  a  first  approximation  that 
we  compare  the  growth  of  a  population  with  that  of  an  organism, 
or  for  that  matter  of  one  organism  or  one  pbpulation  with  another. 
There  are  immense  differences  between  a  simple  and  a  complex 
organism,  between  a  primitive  and  a  civihsed  population.  The 
yeast-plant  gives  a  growth-curve  which  we  can  analyse;  but  we 
must  fain  be  content  with  a  qualitative  description  of  the  growth 
of  a  complex  organism  in  its  complex  world ||. 

There  is  a  simphcity  in  a  colony  of  protozoa  and  a  complexity  in 
a  warm-blooded  animal,  a  uniformity  in  a  primitive  tribe  and  a 
heterogeneity  in  a  modern  state  or  town,  which  affect  all  their 
economies  and  interchanges,  all  the  relations  between  milieu  interne 
and  ex^rne.  and  all  the  coefficients  in  any  but  the  simplest  equations  of 
growth  which  we  can  ever  attempt  to  frame.  Every  growth-problem 
becomes  at  last  a  specific  one,  running  its  own  course  for  its  own 
reasons.     Our  curves  of  growth  are  all  alike — but  no  two  are  ever 


*  Raymond  Pearl,  Amer.  Nat.  lxi,  pp.  289-318,  1927. 

t  Ronald  Ross,  Prevention  of  Malaria  (2nd  ed.),  1911,  p.  679. 

j  J.  B.  S.  Haldane,  Trans.  Camb.  Phil.  Soc.  xxm,  pp.  19^1,  1924. 

§  Cf.  {int.  al.)  J.  R.  Miner's  Note  on  birth-rate  and  density  in  a  logistic  population, 
Human  Biology,  iv,  p.  119,  1932;  and  cf.  Lotka,  ibid,  in,  p.  458,  1931. 

II  Cf.  {int.  al.)  C.  E.  Briggs,  Attempts  to  analyse  growth- curves,  Proc.  E.S.  (B), 
en,  pp.  280-285,  1928. 


Ill]  IN  VARIOUS  ORGANISMS  159 

the  same.  Growth  keeps  caUing  our  attention  to  its  own  com- 
plexity. We  see  it  in  the  rates  of  growth  which  change  with  age 
or  season,  which  vary  from  one  hmb  to  another,;  in  the  influence 
of  peace  and  plenty,  of  war  and  famine ;  not  least  in  those  composite 
populations  whose  own  parts  aid  or  hamper  one  another,  in  any 
form  or  aspect  of  the  struggle  for  existence.  So  we  come  to  the 
differential  equations,  easy  to  frame,  more  difficult  to  solve,  easy  in 
their  first  steps,  hard  and  very  powerful  later  on,  by  which  Lotka 
and  Volterra  have  shewn  how  to  apply  mathematics  to  evolutionary 
biology,  but  which  he  just  outside  the  scope  of  this  book*. 

An  important  element  in  a  population,  and  one  seldom  easy  to 
define,  is  its  age-composition.  It  may  vary  one  way  or  the  other; 
for  the  diminution  of  a  population  may  be  due  to  a  decrease  in  the 
birth-rate,  or  to  an  increasing  mortality  among  the  old.  A  remark- 
able instance  is  that  of  the  food-fishes  of  the  North  Sea.  Their 
birth-rate  is  so  high  that  the  very  young  fishes  remain,  to  all 
appearance,  as  numerous  as  ever;  those  somewhat  older  are  fewer 
than  before,  and  the  old  dwindle  to  a  fraction  of  what  they  were 
wont  to  be. 

The  rate  of  growth  in  other  organisms 

The  rise  and  fall  of  growth-rate,  the  acceleration  followed  by 
retardation  which  finds  expression  in  the  S^shaped  curve,  are  seen 
alike  in  the  growth  of  a  population  and  of  an  individual,  and  in 
most  things  which  have  a  beginning  and  an  end.  But  the  law  of 
large  numbers  smooths  the  population-curve;  the  individual  Hfe 
draws  attention  to  its  own  ups  and  downs;  and  the  characteristic 
sigmoid  curve  is  only  seen  in  the  simpler  organisms,  or  in  parts  or 
"phases"  of  the  more  complex  lives.  We  see  it  at  its  simplest  in 
the  simple  growth-cycle,  or  single  season,  of  an  annual  plant,  which 
cycle  draws  to  its  end  at  flowering;  and  here  not  only  is  the  curve 
simple,  but  its  amplitude  may  sometimes  be  very  large.  The  giant 
Heracleum  and  certain  tall  varieties  of  Indian  corn  grow  to  twelve  feet 

*  See  (int.  al.)  A.  J.  Lotka,  Elements  of  Physical  Biology,  Baltimore,  1925; 
Theorie  analytique  des  associations  biologiques,  Paris,  1934;  Vito  Volterra,  Lemons 
sur  la  theorie  mathematique  de  la  lutte  pour  la  vie,  1931;  Volterra  et  U.  d'Ancona, 
Les  associations  biologiques  au  point  de  vue  mathematique,  1935;  V.  A.  Kostitzin, 
op.  ci7.;\  etc 


160 


THE  RATE  OF  GROWTH 


[CH. 


high  in  a  summer;  the  kudzu  vine  (Pucraria)  may  grow  twelve  inches 
in  twenty-fouE  hours,  and  some  bamboos  are  said  to  have  grown 
twenty  feet  in  three  days  (Figs.  32,  33). 


10 
Days 

Fig.  32.     Growth  of  Lupine.     After  Pfeffer. 


Growth  of  Lupinus  albus.     (From  G.  Backman,  after  Pfeffer) 


Length 

Length 

Day 

(mm.) 

DiflFerence 

Day 

(mm.) 

DiflFerence 

4 

10-5 

— 

14 

132-3 

12-2 

5 

16-3' 

5-8 

15 

140-6 

8-3 

6 

23-3 

7-0 

16 

149-7 

9-1 

7 

32-5 

9-2 

17 

155-6 

5-9 

8 

42-2 

9-7 

18 

158-1 

2-5 

9 

58-7 

14:5 

19 

160-6 

2-5 

10 

77-9 

19-2 

20 

161-4 

0-8 

n 

93-7 

15-8 

21 

161-6 

0-2 

12 

107-4 

13-7 

13 

1201 

12-7 

In  the  pre-natal  growth  of  an  infant  the  S-shaped  curve  is  clearly 
seen  (Fig.  18);  but  immediately  after  birth  another  phase  begins, 
and  a  third  is  imphcit  in  the  spurt  of  growth  which  precedes  puberty. 
In  short,  it  is  a  common  thing  for  one  wave  of  growth  (or  cycle,  as 


Ill]  IN  VARIOUS  ORGANISMS  161 

some  call  it)  to  succeed  another,  whether  at  special  epochs  in  a 
lifetime,  or  as  often  as  winter  gives  place  to  spring*. 
20 


Days 
Fig.  33.     Growth  of  Lupine:  daily  increments. 


45       50 
days 

Fig.  34.     Growth  in  weight  of  a  mouse.     After  W.  Ostwald. 

In  the  accompanying  curve  of  weight  of  the  mouge  (Fig.  34)  we 
see  a  slackening  of  the  rate  of  growth  when  the  mouse  is  about  a 
fortnight  old,  at  which  epoch  it  opens  its  eyes,  and  is  weaned  soon 

*  W.  PfefiFer,  Pflanzenphysiologie,  1881,  Bd.  ii,  p.  78;  A.  Bennett,  On  the  rate 
of  growth  of  the  flower-stalk  of  Vallisneria  spiralis  and  of  Hyacinthus,  Trans.  Linn. 
Soc.  (2),  I,  Botany,  pp.  133,  139,  1880;    cited  by  G.  Backman,  Das  Wachstums- 


162  THE  EATE  OF  GROWTH  [ch. 

after.  At  six  weeks  old  there  is  another  well-marked  retardation; 
it  follows  on  a  rapid  spurt,  and  coincides  with  the  epoch  of  puberty  *. 

In  arthropod  animals  growth  is  apt  to  be  especially  discontinuous, 
for  their  bodies  are  more  or  less  closely  confined  until  released  by 
the  casting  of  the  skin.  The  blowfly  has  its  striking  metamorphoses, 
yet  its  growth  is  wellnigh  continuous;  for  its  larval  skin  is  too  thin 
and  dehcate  to  impede  growth  in  the  usual  arthropod  way.  But 
in  a  thick-skinned  grasshopper  or  hard-shelled  crab  growth  goes  by 
fits  and  starts,  by  steps  and  stairs,  as  Reaumur  was  the  first  to  shew ; 
for,  speaking  of  insects f,  he  says:  "Peut-etre  est-il  vrai  generale- 
ment  que  leur  accroissement,  ou  au  moins  leur  plus  considerable 
accroissement,  ne  se  fait  que  dans  le  temps  qu'ils  muent,  ou  pendant 
im  temps  assez  court  apres  la  mue.  lis  ne  sont  obHges  de  quitter 
leur  enveloppe  que  parce  qu'elle  ne  prend  pas  un  accroissement 
proportionne  a  celui  que  prennent  les  parties  qu'elle  couvre." 
All  the  visible  growth  of  the  lobster  takes  place  once  a  year  at 
moulting-time,  but  he  is  growing  in  weight,  more  or  less,  all  along. 
He  stores  up  material  for  months  together;  then  comes  a  sudden 
rush  of  water  to  the  tissues,  the  carapace  sphts  asunder,  the  lobster 
issues  forth,  devours  his  own  exuviae,  and  lies,  low  for  a  month  while 
his  new  shell  hardens. 

The  silkworm  moults  four 'times,  about  once  a  week,  beginning 
on  the  sixth  or  seventh  day  after  hatching.  There  is  an  arrest  or 
retardation  of  growth  before  each  moult,  but  our  diagram  (i^ig.  35) 
is  too  small  to  shew  the  sHght  ones  which  precede  the  first  and 

problem,  in  Ergebnisse  d.  Physiologie,  xxxiii,  pp.  883-973,  1931.  These  two  cases 
of  Lupinus  and  Vallisneria,  a^e  among  the  many  which  lend  themselves  easily  to 
Backman's  growth -formula,  viz.  Lupinus,  \ogp=  -  2-40  + 1-48  log  T  -  6-61  log^  T  and 
Vallisneria,  log  p  =  +  1-28  +  4-51  log  T  -  2-62  log^  T.  See  for  an  admirable  resume 
of  facts,  Wolfgang  Ostwald,  Ueher  die  zeitliche  Eigenschaften  der  Entwicklungsvorgdnge 
(71  pp.),  1908  (in  Roux's  Vortrdge,  Heft  v);  and  many  later  works. 

*  Cf.  R.  Robertson,  Analysis  of  the  growth  of  the  white  mouse  into  its  con- 
stituent processes,  Journ.  Gen.  Physiology,  vin,  p.  463,  1926.  Also  Gustav 
Backman,  Wachstum  d.  w.  Maus,  Li^nds  Univ.  Arsskrift,  xxxv,  Nr.  12,  1939, 
with  copious  bibliography.  Backman  analyses  the  complicated  growth- curve  of 
the  mouse  into  one  main  and  three  subordinate  cycles,  two  df  which  are  embryonic. 
Cf.  St  Loup,  Vitesse  de  croissance  chez  les  souris.  Bull.  Soc.  Zool.  Fr.  xviii, 
p.  242,  1893;  E.  Le  Breton  and  G.  Schafer,  Trav.  Inst.  Physiol.  Strasburg,  1923; 
E.  C.  MacDowell,  Growth-curve  of  the  suckling  mouse.  Science,  Lxvin,  p.  650, 
1928;  cf.  Journ.  Gen.  Physiol,  xi,  p.  57,  1927;  Ph.  THeritier,  Croissance. .  .dfeins  les 
souris,  Ann.  Physiol,  et  Phys.  Chemie,  v,  p.  i,  1929. 

I  Memoires,  iv,  p.  191. 


i 


Ill] 


OF  CERTAIN  INSECTS 


163 


second.     Before  entering  on  the  pupal  or  chrysalis  stage,  when  the 
worm  is  about  seven  weeks  old,  a  remarkable  process  of  purgation 


e 

mgms 

4000 

1 

3000 

- 

• 

/ 

r 

2000 

- 

^ 

1 

\ 

1000 

- 

- 

IV     / 

I 

II 

III      / 

1     ,  .1 

10 


15 


20 


25 


30 


35 


40 
days 


Fig.  35.     Growth  in  weight  of  silkworm.     From  Ostwald,  after  Luciani 
and  Lo  Monaco. 


takes  place,  with  a  sudden  loss  of  water,  and  of  weight,  which 
becomes  the  most  marked  feature  of  the  curve*    That  the  meta- 

*  Luciani  e  Lo  Monaco,  Arch.  Ital.  de  Biologie,  xxvii,  p.  340,  1897;  see  also 
Z.  Kuwana,  Statistics  of  the  body-weight  of  the  silkworm,  Japan.  Journ.  Zool. 
VII,  pp.  311-346,  1937.  Westwood,  in  1838,  quoted  similar  data  from  Count 
Dandolo:  according  to  whom  100  silkworms  weigh  on  hatching  1  grain;  after 
the  first  four  moults,  15^  94,  270  and  1085  grains;  and  9500  grains  when  full-grown. 


164  THE  RATE  OF  GROWTH  [ch. 

morphoses  of  an  insect  are  but  phases  in  a  process  of  growth  was 
clearly  recognised  by  Swammerdam,  in  the  Bihlid  Naturae*. 

A  stick-insect  (Dexippus)  moults  six  or  seven  times  in  as  many 
months ;  it  lengthens  at  every  ilioult,  and  keeps  of  the  same  length 
until  the  next.  Weight  is  gained  more  evenly;  but  before  each 
moult  the  creature  stops  feeding  for  a  day  or  two,  and  a  little  weight 
is  lost  in  the  casting  pf  the  skin.  After  its  last  moult  the  stick- 
insect  puts  on  more  weight  for  a  while;  but  growth  soon  draws  to 
an  end,  and  the  bodily  energies  turn  towards  reproduction. 

We  have  careful  measurements  of  the  locust  from  moult  to  moult, 
and  know  from  these  the  relative  growth-rates  of  its  parts,  though 
we  cannot  plot  these  dimensions  against  time.  Unlike  the  meta- 
morphosis of  the  silkworm,  the  locust  passes  through  five  larval 
stages  (or  "instars")  all  much  ahke, '  until  in  a  final  moult  the 
"hoppers"  become  winged.  Here  are  three  sets  of  measurements, 
of  Hmbs  and  head,  from  stage  to  stage  f. 

Growth  of  locust,  from  one  moult  to  another 

Length  (mm.)  Percentage-grpwth  Ratios 


Anterior 

Median 

^ 

r 

Anterior 

Median 

"* 

Anterior 

Median 

"* 

Stage 

femur 

femur 

Head 

femur 

femur 

Head 

femur 

femur 

Heac 

I 

1-44 

3-98 

1-44 

— 

— 

•^- 

2-76 

1-00 

II 

2-06 

5-69 

1-94 

1-44 

1-43 

1-35 

2-76 

0-94 

III 

308 

8-22 

2-70 

1-40 

1-44 

1-39 

2-67 

0-88 

IV 

4-53 

11-94 

3-71 

1-47 

1-45 

1'37 

2-76 

0-82 

V 

6-40 

17-22 

4-89 

1-41 

1-44 

1-32 

2-69 

0-76 

Adult 

8-03 

22-85 

5-59 

1-25 

1-33 

V14 

2-84 

0-70 

As  a  matter  of  fact  the  several  parts  tend  to  grow,  for  a  time,  at 
a  steady  rate  of  compound  interest,  which  rate  is  not  identical  for 
head  and  hmbs,  and  tends  in  each  case  to  fall  off  in  the  final  moult, 
when  material  has  to  be  found  for  the  wings.  Some  fifty  years  ago, 
W.  K.  Brooks  found  the  larva  of  a  certain  crab  (Squilla)  increasing 
at  each  moult  by  a  quarter  of  its  own  length ;  and  soon  after 
H.  G.  Dyar  declared  that  caterpillars  grow  hkewise,  from  moult  to 
moult,  by  geometrical  progression  % .     This  tendency  to  a  compoimd- 

*  1737,  pp.  6,  579,  etc. 

t  A.  J.  Duarte,  Growth  of  the  migratory  locust.  Bull.  Ent.  Res.  xxix,  pp.  425-456, 
1938. 

%  W.  K.  Brooks,  Challenger  Report  on  the  Stomatopoda,  1886;  H.  G.  Dyar; 
Number  of  moult^  in  lepidopterous  larvae,  Psyche,  v,  p.  424,  1896. 


Ill]  OF  BROOKS'S  LAW  165 

interest  rate  in  the  growth  and  metamorphosis  of  insects  is  known 
as  Dyar's,  sometimes  as  Brooks's,  law.  According  to  Przibram,  an 
insect  moults  as  soon  (roughly  speaking)  as  cell-division  has  doubled 
the  number  of  cells  throughout  the  larval  body.  That  being  so,  each 
stage  or  instar  should  weigh  twice  as  much  as  the  one  before,  and 
each  linear  dimension  should  increase  by  ^2,  or  1-26  times — a 
Ineasure  identical,  to  all  intents  and  purposes,  with  Brooks's  first 
estimate.  As  sl  first  rough  approximation  the  rule  has  a  certain  value. 
According  to  Duarte's  measurements  the  locust's  total  weight  in- 
creases from  moult  to  moult  by  2-31,  2-16,  242,  2-35,  2-21,  or  a 
mean  increase  of  2-29,  the  cube-root  of  which  is  1-32.  Each  phase 
is  doubled  and  more  than  doubled,  in  passing  to  the  next*,  but 
Przibram's  estimate  is  not  far  departed  from. 

Whatever  truth  Przibram's  law  may  have  in  insects,  or  (as  Fowler 
asserted)  in  the  Ostracods,  it  would  seem  to  have  none  in  the 
Cladocera :  and  this  for  the  sufficient  reason  that  the  shell  (on  which 
the  form  of  the  creature  depends)  goes  on  growing  all  through  post- 
embryonic  life  without  further  division  or  multiphcation  of  its  cells, 
but  only  by  their  individual,  and  therefore  collective,  enlargementf. 

Shells  are  easily  weighed  and  measured  and  their  various  dimen- 
sions have  been  often  studied;  only  in  oysters,  pearl-oysters  and 
the  Hke,  have  they  been  so  kept  under  observation  that  their  actual 
age  is  known.  The  oyster-shell  grows  for  a  few  weeks  in  spring  just 
before  spawning  time,  and  again  in  autumn  when  spawning  is  over ; 
its  growth  is  imperceptible  at  other  times  J. 

*  Cf.  H.  Przibram  and  F.  Megusar,  Wachstummessungen  an  Sphodromantis, 
Arch.  f.  Entw.  Mech.  xxxiv,  pp.  680-741,  1912;  etc.  How  the  discrepancy  is 
accounted  for,  by  Bodenheimer  and  others,  need  not  concern  us  here.  But  cf. 
P.*  P.  Calvert,  On  rates  of  growth  among.  .  .the  Odonata,  Proc.  Amer.  Phil.  Soc. 
Lxviii,  pp.  227-274,  1929,  who  finds  growth  faster  in  nine  cases  out  of  ten  than 
Przibram's  rule  lays  down. 

Millet  asserts,  in  support  of  Przibram's  law,  that  in  spiders  mitotic  cell-division 
is  confined  to  the  epoch  of  the  moult,  and  is  then  manifested  throughout  most  of 
the  tissues  (Bull,  de  Biologie  {SuppL),  viii,  p.  1,  1926).  On  the  other  hand,  the 
rule  is  rejected  by  R.  Gurney,  Rate,  of  growth  in  Copepoda,  Int.  Rev.  Hydrohiol. 
XXI,  pp.  189-27,  1929;  Nobumasa  Kagi,  Growth-curves*  of  insect-larvae,  Mem. 
Coll.  Agric.  Kyoto,  No.  1,  1926;   and  others. 

t  Cf.  W.  Rammer,  Ueber  die  Giiltigkeit  des  Brooksschen  Wachsturasgesetzes 
bei  den  Cladoceren,  Arch.  f.  Entw.  Mech.  cxxi,  pp.  111-127,  1930. 

X  Cf.  J.  H.  Orton,  Rhythmic  periods... in  Ostrea,  Jonrn.  Mar.  Biol.  Assoc. 
XV,  pp.  365^27,  1928;  Nature,  March  2,  1935,  p.  340. 


166 


THE  RATE  OF  GROWTH 


[CH. 


The  window-pane  oyster  in  Ceylon  (Placuna  placenta)  has  been 
kept  under  observation  for  eight  years,  during  which  it  grows  from 
two  inches  long  to  six  (Fig.  36).  The  young  grow  quickly,  and  slow 
down  asymptotically  towards  the  end;  an  S-shaped  beginning  to  the 
growth-curve  has  not  been  seen,  but  would  probably  be  found  in 
the  growth  of  the  first  year.  Changes  of  shape  as  growth  goes  on 
are  hard  to  see  in  this  and  other  shells ;  rather  is  it  characteristic  oi 


Fig.  36. 


12  3  4  5  6  7 

Age  in  years 

Growth  of  the  window-pane  oyster;  short  diameter  of  the  shell. 
From  Pearson's  data. 


them  to  keep  their  shape  from  first  to  last  unchanged.  Nevertheless, 
shght  changes  are  there;  in  the  window-pane  oyster  the  shell  grows 
somewhat  rounder;  in  seven  or  eight  years  the  one  diameter  multi- 
plies (roughly  speaking)  by  eleven,  and  the  other  by  ten*. 

Window-pane  oysters  (Placuna) 


Ratio 

117 
109 
107 
1-06 
105 

The  American  slipper-limpet  has  lately  and  quickly  become  a  pest 
on  English  oyster-beds.     Its  mode  of  growth  is  interesting,  though 

*  Joseph  Pearson,  The  growth-rate. .  .oi Placuna  placenta,  Ceylon  Bulletin,  1928. 


Short 

Long 

diameter 

diameter 

(mm.) 

(mm.) 

150 

17-6 

650 

70-5 

102-5 

109-7 

132-5 

139-9 

167-5 

175-2 

Ill 


OF  THE  GROWTH  OF  SHELLFISH 


167 


the  actual  rate  remains  unknown.  It  grows  a  little  longer  and 
narrower  with  age.  Its  weight-length  coefficient  (of  which  we  shall 
have  more  to  say  presently)  increases  as  time  goes  on,  and  appears 
to  follow  a  wavy  course  which  might  be  accounted  for  if  the 
shell  grew  thinner  and  then  thicker  again,  as  if  ever  so  little  more 
lime  were  secreted  at  one  season  than  another.  The  growth  of  a 
shell,  or  the  deposition  of  its  calcium  carbonate,  is  much  influenced 
by  temperature;  clams  and  oysters  enlarge  their  shells  only  so  long 
as  the  temperature  stands  above  a  certain  specific  minimum,  and 
the  mean  size  of  the  same  hmpet  is  very  different  in  Essex  and  in 
the  United  States*.  Curious  peculiarities  of  growth  have  been 
discovered  in  slipper-hmpets.  Young  limpets  clustered  roimd  an 
old  female  grow  slower  than  others  which  Uve  sohtary  and  apart. 
The  solitary  forms  become  in  turn  male,  hermaphrodite  and  at  last 
female,  but  the  gregarious  or  clustered  forms .  develop  into  males, 
and  so  remain;  development  of  male  characters  and  duration  of 
the  male  phase  depend  on  the  presence  or  absence  of  a  female  in 
the  near  neighbourhood. 


Measurements  of  slipper-limpets 
{From  J.  H.  Eraser's  data,  epitomised) 


No. 

Mean  length 

Breadth 

Ratio 

Weight 

measured 

(mm.) 

(mm.) 

L/B 

(gra-) 

WjD 

3 

15-3 

8-8 

1-74 

0-33 

92 

8 

17-6 

9-8 

1-80 

0-46 

84 

9 

19-4 

10-5 

1-85 

.0-63 

88 

16 

21-5 

11-5 

1-87 

0-77 

77 

18 

23-5 

125 

1-88 

104 

80 

41 

25-5 

13-7 

1-86 

1-37 

85 

91 

27-4 

14-5 

1-89 

1-81 

88 

125 

39-4 

15-4 

1-91 

2-33 

92 

98 

31-4 

16-5 

1-90 

3-22 

104 

70 

33-6 

17-8 

1-89 

3-61 

95 

38 

35-5 

18-6 

1-90 

4-28 

95 

10 

37-3 

19-5 

1-91 

4-95 

95 

1 

321 

19-4 

201 

5-35 

90 

Mean        1-87 


89-3 


*  Cf.  J.  H.  Fraser,  On  the  size  of  Urosalpinx  etc.,  Proc.  Malacol.  Soc.  xix, 
pp.  243-254,  1931.  Much  else  is  known  about  the  growth  of  various  limpets, 
their  seasonal  periodicities,  the  change  of  shape  in  certain  species,  and  other 
matters;  cf.  E.  S.  Russell,  Growth  of  Patella,  P.Z.S.  cxcix,  pp.  235-253;  J.  H. 
Orton,  Journ.  Mar.  BioL  Assoc,  xv,  pp.  277-288,  1929;  Noboru  Abe,  Sci.  Rep. 
Tohoku  Imp.  Univ.  Biol,  vi,  pp.  347-363,  1932,  and  Okuso  Hamai,  ibid,  xii, 
pp.  71-95,  1937. 


168 


THE  RATE  OF  GROWTH 


[CH. 


The  growth  of  the  tadpole  *  is  Hkewise  marked  by  epochs  of 
retardation,  and  finally  by  a  sudden  and  drastic  change  (Fig.  37). 
There  is  a  shght  diminuti9n  in  weight  immediately  after  the  little 
larva  frees  itself  from  what  remains  of  the  egg ;  there  is  a  retardation 


mgms. 

1 

6000 

' 

/ 

') 

5000 
4000 

~  • 

/       ^ 

/    1 

j 

:  1 
■j  j 

3000 

- 

/ 

i 

i 

2000 

- 

1 

/ 

1000 

-  1 

u3 

J 

0  10        20        30        40        50        60         70        80       90 

days 

Fig.  37.     Growth  in  weight  of  tadpole.     From  Ostwald,  after  Schaper. 

of  growth  about  ten  days  later,  when  the  external  gills  disappear;  and 
finally  the  complete  metamorphosis,  with  the  loss  of  the  tail,  the  growth 
of  the  legs  and  the  end  of  branchial  respiration,  brings  about  a  loss 
of  weight  amounting  to  wellnigh  half  the  weight  of  the  full-grown 


*  Cf.   (int.   at.)   Barfurth,    Versiiche  iiber  die   Verwandlung  der  Froschlarven 
Arch.  f.  mikroak.  Anat.  xxix,  1887.* 


Ill] 


OF  LARVAL  EELS 


169 


Fig.  38. 


Development  of  eel :  from  Leptocephalus  larvae  to  young  elver. 
After  Johannes  Schmidt. 


170  THE  RATE  OF  GROWTH  [ch- 

larva.  At  the  root  of  the  matter  hes  the  simple  fact  that  meta- 
morphosis involves  wastage  of  tissue,  increase  of  oxidation,  expendi- 
ture of  energy  and  the  doing  of  work.  While  as  a  general  rule  the 
better  the  animals  be  fed  the  quicker  they  grow  and  the  sooner  they 
metamorphose,  Barfurth  has  pointed  out  the  curious  fact  that  a 
short  spell  of  starvation,  just  before  metamorphosis  is  due,  appears 
to  hasten  the  change. 

The  negative  growth,  or  actual  loss  of  bulk  and  weight  which 
often,  and  perhaps  always,  accompanies  metamorphosis,  is  well 
shewn  in  the  case  of  the  eel  *.  The  contrast  of  size  is  great  between 
the  flattened,  lancet-shaped  Leptocephalus  larva  and  the  httle  black, 
cylindrical,  almost  thread-Uke  elver,  whose  magnitude  is  less  than 
that  of  the  Leptocephalus  in  every  dimension,  even  at  first  in  length 
(Fig.  38),  as  Grassi  was  the  first  to  shew. 

The  lamprey's  case  is  hardly  less  remarkable.  The  larval  or 
Ammocoete  stage  lasts  for  three  years  or  more,  and  metamorphosis, 
though  preceded  by  a  spurt  of  growth,  is  followed  by  an  actual 
decrease  in  size.  The  little  brook  lamprey  neither  feeds  nor  grows 
after  metamorphosis,  but  spawns  a  few  months  later  and  then  dies; 
but  the  big  sea-lampreys  become  semi-parasitic  on  other  fishes,  and 
live  and  grow  to  an  unknown  agef. 

Such  fluctuations  as  these  are  part  and  parcel  of  the  general  flux 
of  physiological  activity,  and  suggest  a  finite  stock  of  energy  to  be 
spent,  now  more  now  less,  on  growth  and  other  modes  of  expenditure. 
The  larger  fluctuations  are  special  interruptions  in  a  process  which 
is  never  continuous,  but  is  perpetually  varied  by  rhythms  of  various 
kinds  and  orders.  Hofmeister  shewed  long  ago,  for  instance,  that 
Spirogyra  grows  by  fits  and  starts,  in  periods  of  activity  and  rest 
alternating  with  one  another  at  intervals  of  so  many  minutes  { 
(Fig.  39).  And  Bose  tells  us  that  plant-growth  proceeds  by  tiny  and 
perfectly  rhythmical  pulsations,  at  intervals  of  a  few  seconds  of  time. 

*  Johannes  Schmidt,  Contributions  to  the  life-history  of  the  eel,  Rapports 
du  Conseil  Intern,  pour  V exploration  de  la  mer,y,  pp.  137-274,  Copenhagen,  1906; 
and  other  papers. 

t  Cf.  {int.  al.)  A.  Meek,  The  lampreys  of  the  Tyne,  Rep.  Dove  Marine  Laboratory 
(N.S.),  VI,  p.  49,  1917;  cf.  L.  Hubbs,  in  Papers  of  the  Michigan  Academy,  iv, 
p.  587,  1924. 

%  Die  Lehre  der  Pflanzenzelle,  1867.  Cf.  W.  J.  Koningsberger,  Tropismus  und 
Wachstum  (Thesis),  Utrecht,  1922. 


Ill] 


OF  PERIODIC  GROWTH 


171 


A  crocus  grows,  he  says,  by  little  jerks,  each  with  an  amphtude  of 
about  0-002  mm.,  every  twenty  seconds  or  so,  each  increment  being 
followed  by  a  partial  recoil*  (Fig.  40).     If  this  be  so  we  have  come 


20   40    60    80   100   120   140   160   180   200   220   240  260 

minutes 

Fig.  39.     Growth  in  length  (mm.)  of  Spirogyra.     From  Ostwald,  after  Hofmeister. 


Fig.  40. 


seconds. 

Pulsations  of  growth  in  Crocus,  in  micro-millimeter? 
After  Bose. 


down,  so  to  speak,  from  a  principle  of  continuity  to  a  principle  of  dis- 
continuity, and  are  face  to  face  with  what  we  might  call,  by  rough 
analogy,  "quanta  of  growth."  We  seem  to  be  in  touch  with  things 
of  another  order  than  the  subject  of  this  bookt. 

*  J.  C.  Bose,  Plant  Response,  1906,  p.  417;  Growth  and  Tropic  Movements  of 
Plants,  1929. 

t  There  is  an  apparent  and  perhaps  a  real  analogy  between  these  periodic 
phenomena  of  growth  and  the  well-known  phenomenon  of  periodic,  or  oscillatory, 
chemical  change,  as  described  by  W.  Ostwald  and  others;  cf.  (e.g.)  Zeitschr.  f. 
phys.  Chem.  xxxv,  pp.  33,  204,  1900. 


172  THE  RATE  OF  GROWTH  [ch. 

We  may  want  now  and  then  to  make  use  of  scanty  data,  and  find 
a  rougli  estimate  better  than  none.  The  giant  tortoises  of  the 
Galapagos  and  the  Seychelles  grow  to  a  great  age,  and  some  have 
weighed  5001b.  and  more;  but  the  scanty  records  of  captive 
tortoises  shew  much  variation,  depending  on  food  and  climate  as 
well  as  age,  Ninety  young  tortoises  brought  from  the  Galapagos 
in  1928  to  the  southern  United  States  weighed  on  the  average 
18J  lb.,  and  grew  to  44-3  lb.  in  two  years.  Six  taken  to  Honolulu 
weighed  26 J  lb.  each  in  1929,  and  63  lb.  each  the  following  year. 
Another,  kept  in  CaHfornia,  weighed  29  lb.  and  360  lb.  seven  years 
later,  but  only  gained  65  lb.  more  in  the  next  seven  years.     Growth, 


1899  1906  1913 

Fjg.  41.     Approximate  growth  in  weight  of  Galapagos  tortoise. 

as  usual,  is  quick  to  begin  with,  slower  lat^r  on,  and  in  the  old  giants 
must  be  slow  indeed.  If  we  plot  (Fig.  41 )  the  three  successive  weights 
of  the  CaHfornian  specimen,  at  first  they  help  us  little;  but  we  can 
fit  an  S-shaped  curve  to  the  three  points  as  a  first  approximation, 
and  it  suggests,  with  some  plausibility,  that,  at  29  lb.  weight  the 
tortoise  was  from  two  to  three  years  old.  A  loggerhead  turtle, 
which  reaches  a  great  size,  was  found  to  grow  from  a  few  grammes 
to  42  lb.  in  three  years,  and  to  double  that  weight  in  another  year 
and  a  half;  these  scanty  data  are  in  fair  accord,  ^o  far  as  they  go, 
with  those  for  the  giant  tortoises*. 

*  For  these  and  other  data,  see  C.  H.  Townsend,  Growth  and  age  in  the  giant 
tortoises  of  the  Galapagos,  Zoologica,  ix,  pp.  459-466,  1931;  G.  H.  Parker,  Growth 
of  the  loggerhead  turtle,  Amer.  Naturalist,  lxvii,  pp.  367-373,  1929;  Stanley  F. 
Flower,  Duration  of  Life  in  Animals,  in,  Reptiles,  P.Z.S.  (A),  1937,  pp.  1-39. 


Ill] 


OF  WHALES  AND  TORTOISES 


173 


The  horny  plates  of  the  tortoise  grow,  to  begin  with,  a  trifle  faster 
than  the  bony  carapace  below,  and  are  consequently  wrinkled  into 
folds.  There  is  some  evidence,  at  least  in  the  young  tortoises,  that 
these  folds  come  once  a  year,  which  is  as  much  as  to  say  that  there 
is  one  season  of  the  year  when  the  growth-rates  of  bony  and  horny 
carapace  are  especially  discrepant.  This  would  give  an  easy  estimate 
of  age;  but  it  is  plainer  in  some  species  than  in  others,  and  it  never 
lasts  for  long. 


re  m  years 
Fig.  42.     Growth-rate  (approximate)  of  blue  and  finner  whales. 

The  blue  whale,  or '  Sibbald's  rorqual,  largest  of  all  animals, 
grows  to  100  ft.  long  or  thereby,  the  females  being  a  httle  bigger 
than  the  males.  The  mother  goes  with  young  eleven  months.  The 
calf  measures  22  to  25  ft.  at  birth,  and  weighs  between  three  and 
four  tons;  it  is  bom  big,  were  it  smaller  it  might  lose  heat  too 
quickly.  It  is  weaned  about  nine  months  later,  and  is  said  to  be 
some  16  metres,  or  say  53  ft.,  long  by  then.  It  is  believed  to  be 
mature  at  two  years  old,  by  which  time  it  is  variously  stated  to  be 
60  or  even  75  ft.  long ;  the  modal  size  of  pregnant  females  is  about 
80  ft.  or  rather  more.  How  long,  the  whale  takes  to  grow  the 
further  15  or  20  feet  which  bring  it  to  its  full  size  is  not  known; 
but,  even  so  far,  the  rapid  growth  and  early  maturity  seem  very 
remarkable  (Fig.  42).     The  Norwegian  whalers  give  us  statistics, 


174 


THE  RATE  OF  GROWTH 


[CH. 


month  by  month  during  the  Antarctic  season,  of  the  sizes  of  pregnant 
females  and  the  foetuses  they  contain;  and  from  these  I  draw  the 
following  averages: 

Antarctic  blue  whales;  length  of  mother  and  of  foetus 
(Season  1938-39) 


Number 
measured 

Mother 

Foetus 

Nov. 

1,  1938 
16 

59 
86 

84-0  ft. 
83-0 

4-2  ft 
4-6 

Dec. 

1 
16 

359 
522 

84-0 
83-6 

61 

7-0 

Jan. 

1,  1939 
16 

403 
317 

83-7 
84-8 

8-4 
9-3 

Feb. 

1 
16 

184 
125 

83-9 
83-9 

11-2' 
12-3 

Mar. 

1 

71 

83-6 

14-5 

2126 


83-8 


^       .- 


NOV 


DEC 


JAN 


FEB 


MAR 


Fig.  43.     Pre-natal  growth  of  blue  whale.     Average  monthly  sizes, 
from  data  in  International  Whaling  Statistics,  xiv,  1940. 

The  observations  are  rough  but  numerous.  At  the  lower  end  of 
the  scale  measurements  are  few,  and  the  value  indicated  is  probably 
too  high;  but  on  the  whole  the  curve  of  growth  taUies  with  other 
estimates,  and  points  to  birth  about  June  or  July,  and  to  conception 
about  the  same  time  last  year  (Fig.  43).  The  mean  size  of  the 
mother- whales  does  not  alter  during  the  five  months  in  question; 


Ill]  THE  GROWTH  OF  WHALES  175 

they  do  not  seem  to  be  increasing,  though  at  84  ft.  they  still  have 
another  10  feet  or  more  to  grow.  They  may  grow  slower,  and  live 
longer,  than  is  often  supposed*. 

On  the  other  hand,  if  we  draw  from  the  same  official  statistics 
the  mean  size  of  mother-whale  and  foetus  at  some  given  epoch  of 
the  year  (e.g.  March  1934),  there  appears  to  be  a  marked  correlation 
between  them,  such  as  would  indicate  very  considerable  growth 
of  the  mother  during  the  months  of  pregnancy.  The  matter  deserves 
further  study,  and  the  data  need  confirmation. 

Blue  whales;  length  of  mother  and  foetus  {March  1934) 


Size  of 

Size  (ft.) 

foetus  (ft.) 

Number 
observed 

X 

smoothed 
in  threes 

Mother 

Foetus 

1 

74 

10 

' — 

1 

75 

7-0 

4-4 

5 

76 

5-2 

6-4 

7 

77 

7-3 

6-4 

9 

78 

6-7 

6-9 

10 

79 

6-7 

6-8 

21 

80 

71 

7-2 

27 

81 

7-7 

7-5 

28 

,    82 

7-6 

7-9 

33 

83 

'8-4 

8-3 

38 

84 

8-9 

8-6 

46 

85 

8-5 

8-6 

37 

86 

8-5 

8-8 

19 

87 

9-5 

9-3 

18 

88 

9-9 

10-3 

12 

89 

11-4 

10-9 

18 

90 

11-5 

111 

9 

91 

10-4 

111 

2 

92 

11-5 

— 

341 

On  the  growth  of  fishes,  and  the  determination  of  their  age 

We  may.  keep  a  child  under  observation,  and  weigh  and  measure 
him  every  day;  but  more  roundabout  ways  are  needed  to  determine 
the  age  and  growth  of  the  fish  in  the  sea.  A  few  fish  may  be  caught 
and  marked,  on  the  chance  of  their  being  caught  again;    or  a  few 

*  The  growth  of  the  finner  whale,  or  common  rorqual,  is  estimated  as  follows 
(Hamburg  Museum) :  at  birth,  6  m.;  at  6  months,  12  m. ;  at  one  and  two  years  old, 
15  and  19  m. ;  when  full-grown,  at  6-8  ( ?)  years  old,  21  m.  For  data,  see  Hvalradets 
Skrifter  and  Jhtemational  Whaling  Statistics,  passim ;  also  N.  Mackintosh  and  others 
in  Discovery  Reports;  also  Sigmund  Rusting,  Statistics  of  whales  and  whale- 
foetuses,  Rapports  du  Conseil  Int.  1928;   etc. 


176  THE  RATE  OF  GROWTH  [ch. 

more  may  be  kept  in  a  tank  or  pond  and  watched  as  they  grow. 
Both  ways  are  slow  and  difficult.  The  advantage  of  large  numbers 
is  not  obtained ;  and  it  is  needed  all  the  more  because  the  rate  of 
growth  turns  out  to  be  very  variable  in  fishes,  as  it  doubtless  is 
in  all  cold-blooded  or  " poecilothermic "  animals:  changing  and 
fluctuating  not  only  with  age  and  season,  but  with  food-supply, 
temperature  and  other  known  and  unknown  conditions.  Trout  in 
a  chalk-stream  so  differ  from  those  in  the  peaty  water  of  a  highland 
burn  that  the  former  may  grow  to  three  pounds  weight  while  the 
latter  only  reach  four  ounces,  at  three  years  old  or  four*. 

It  is  found  (and  easily  verified)  that  shells  on  the  seashore,  kind 
for  kind,  do  not  follow  normal  curves  of  frequency  in  respect  of 
magnitude,  but  fall  into  size-groups  with  intervals  between,  so 
constituting  a  multimodal  curve.  The  reason  is  that  they  are  not 
born  all  the  year  round,  as  we  are,  but  each  at  a  certain  annual 
breeding-season ;  so  that  the  whole  population  consists  of  so  many 
"groups,"  each  one  year  older,  and  bigger  in  proportion,  than 
another.  In  short  we  find  size-groups,  and  recognise  them  as  age- 
groups.  Each  group  has  its  own  spread  or  scatter,  which  increases 
with  ^ize  and  age;  even  from  the  first  one  group  tends  to  overlap 
another,  but  the  older  groups  do  so  more  and  more,  for  they  have 
had  more  time  and  chance  to  vary.  Hence  this  way  of  determining 
age  gets  harder  and  less  certain  as  the  years  go  by;  but  it  is  a  safe 
and  useful  method  for  short-lived  animals,  or  in  the  early  lifetime 
of  the  rest.  Aristotle's  fishermen  used  it  when  they  recognised 
three  sorts  or  sizes  of  tunnies,  the  auxids,  pelamyds  and  full-grown 
fi^h;  and  when  they  found  a  scarcity  of  pelamyds  in  one  year  to 
be  followed  by  a  failure  of  the  tunny-fishery  in  the  nextf. 

Shells  lend  themselves  to  this  method,  as  Louis  Agassiz  found  when 
he  gathered  periwinkles  on  the  New  England  shore.  Winckworth 
found  the  Paphiae  in  Madras  harbour  "of  two  sizes,  one  group  just 
under  15  mm.  in  length,  the  other  nearly  all  over  30  mm.  A  small 
sample,  dredged  ^ve  months  earlier  hora  the  same  ground,  was  inter- 
mediate between  the  other  two."  When  the  mean  sizes  of  the  two 
groups  were  plotted  against  time,  the  lesser  group  being  shifted 

*  Cf.  C.  A.  Wingfield,  Effect  of  environmental  factors  on  the  growth  of  brown 
trout,  Journ.  Exp.  Biol,  xvii,  pp.  435-448,  1939. 
I  Aristotle,  Hist.  Anim.  vi,  571  a. 


Ill 


THE  GROWTH  OF  FISHES  177 


back  a  year,  a  growth-curve  extending  over  two  seasons  was  obtained ; 
when  extrapolated,  it  seemed  to  start  from  zero  about  May  or  June, 
and  this  date,  at  the  beginning  of  the  hot  season,  was  in  all  proba- 
bihty  the  actual  spawning  time.  Growth  stopped  in  winter,  a 
common  thing  in  our  northern  climate  but  surprising  at  Madras, 
where  the  sea-temperature  seldom  falls  below  24°  C.  Shells  over 
40  mm.  long  were  rare,  and  over  50  mm.  hardly  to  be  found — an 
indication  that  Paphia  seldom  lives  over  a  third  season.  Here  then, 
though  the  numbers  studied  were  all  too  few,  the  method  tells  us 
with  httle  doubt  or  ambiguity  the  age  of  a  sample  and  the  growth- 
rate  of  the  species  to  which  it  belongs*. 

Dr  C.  J.  G.  Petersen  of  Copenhagen  brought  this  method  into  use 
for  the  study  of  fishes,  and  up  to  a  certain  point  it  is  safe  and 
trustworthy  though  seldom  easy.  For  one  thing,  it  is  hard  to  get 
a  "random  sample"  of  fish,  for  one  net  catches  the  big  and  another 
the  small.  The  trawl-net  takes  all  the  big,  but  lets  more  and  more 
of  the  small  ones  through.  The  drift-net  catches  herring  by  their 
heads;  if  too  big,  the  head  fails  to  catch  and  the  fish  goes  free,  if 
too  small  the  fish  shps  through;  so  the  net  selects  a  certain  modal 
size  according  to  its  mesh,  and  with  no  great  spread  or  scatter. 
When  we  use  Petersen's  method  and  plot  the  sizes  of  our  catch  of 
fish,  the  younger  age-groups  are  easily  recognised,  even  though  they 
tend  to  overlap;  but  the  older  fish  are  few,  each  size-group  has  a 
wider  spread,  and  soon  the  groups  merge  together  and  the  modal 
cusps  cease  to  be  recognisable.  There  is  no  way,  save  a  rough 
conjectural  one,  of  analysing  the  composite  curve  into  the  several 
groups  of  which  it  is  composed;  in  short,  this  method  works  well 
for  the  younger,  but  fails  for  the  older  fish. 

Fig.  44  is  drawn  from  a  catch  of  some  500  small  cod,  or  codhng, 
caught  one  November  in  the  Firth  of  Forth,  in  a  small-meshed 
experimental  trawl-net.  They  are  too  few  for  the  law  of  large 
numbers  to  take  full  effect;  but  after  smoothing  the  curve,  three 
peaks  are  clearly  seen,  with  some  sign  of  a  fourth,  indicating  about 


*  R.  Winckworth,  Growth  of  Paphia  undulata,  Proc.  Malacolog.  Soc.  xix, 
pp.  171-174,  1931.  Cf.  {int.  al.)  Weymouth,  on  Mactra  stultorum.  Bull.  Calif. 
Fish  Comm.  vii,  1923;  Orton,  on  Cardium,  Journ.  Mar.  Biol.  Assoc,  xiv,  1927, 
on  Ostrea,  and  on  Patella,  ibid,  xv,  1928;  Ikuso  Hamai,  on  Limpets,  Sci.  Rep. 
Tohoku  Imp.  Univ.  (4),  xn,  1937. 


178  THE  RATE  OF  GROWTH  [ch. 

11cm.,  26,  44  and  60  cm.,  as  the  mean  or  modal  sizes  of  four 
successive  broods.  The  dwindhng  heights  of  the  successive  cusps 
are  a  first  approximation  to  a  "curve  of  mortahty,"  shewing  how 
the  young  are  many  and  the  old  are  few.  Again,  plotting  the  several 
sizes  against  time,  we  should  get  our  curve  of  growth  for  four  years, 
or  a  first  rough  approximation  to  it.  Thus  we  learn  from  a  random 
sample,  caught  in  a  single  haul,  the  mean  (or  modal)  sizes  of  a  fish 
at  several  epochs  of  its  fife,  say  at  two,  three  or  even  more  successive 
intervals  of  a  year ;  and  we  learn  (to  a  first  approximation)  its  rate 
of  growth  and  its  actual  age,  for  the  slope  of  the  growth-curve, 
drawing  to  the  base-fine,  points  to  the  time  when  growth  began. 


^,  Cod.  Nov.  1906.  F. of  F 


20  JO  40 

Length,  in  centimetres 
Fig.  44.     A  catch  of  cod,  shewing  a  multimodal  curve  of  frequency. 

Another  haul,  soon  after,  will  add  new  points  to  the  curve,  and 
confirm  our  first  rough  approximation. 

An  experiment  in  the  Moray  Firth,  a  month  or  two  later,  shewed 
the  first  three  annual  groups  in  much  the  same  way;  but  it  also 
shewed  another  group,  of  about  90  cm.  long,  and  others  larger  still. 
At  first  sight  these  did  not  seem  to  fit  on  to  our  four  successive 
year-groups,  of  11,  26,  44  and  60 .cm.;  but  they  did  so  after  all, 
only  with  a  gap  between.  They  were  older  fish,  six  and  seven  years 
old,  which  had  come  back  to  the  Moray  Firth  to  breed  after  spending 
a  couple  of  years  elsewhere. 

It  was  thought  at  first  that  every  such  experiment  should  tally 
with  another,  and  bring  us  to  a  more  and  more  accurate  knowledge 
of  the  growth-rate  of  this  fish  or  that;  but  there  were  continual 
discrepancies,  and  it  was  soon  found  that  the  rate  varied  from  place 


Ill] 


THE  GROAVTH  OF  FISHES 


179 


to  place,  from  month  to  month,  and  from  one  year  to  another. 
The  growth-rate  of  a  fish  varies  far  more  than  does  that  of  a  warm- 
blooded animal.  The  general  character  of  the  curve  remains*, 
save  that  the  fish  continues  to  grow  even  in  extreme  old  age,  but 
it  draws  towards  its  upper  asymptote  with  exceeding  slowness. 


Fig.  45.     Growth  of  cod  (after  Michael  Graham);  and  of 
mullet  (after  C.  D.  Serbetis). 

The  following  estimate  of  the  mean  growth  of  North  Sea  cod  is 
based,  by  Michael  Graham,  on  a  great  mass  of  various  evidence; 
and  beside  it,  for  comparison,  is  an  estimate  for  the  grey  mullet, 
by  C.  D.  Serbetis.  The  shape  of  the  curve  (Fig.  45)  is  enough  to 
indicate  that  at  six  years  old  the  cod  is  still  growing  vigorously  f, 
while  the  grey  mullet  has  all  but  ceased  to  grow.     As  a  matter  of 

*  It  is  essentially  an  S-shaped  curve,  as  usual;  but  the  conditions  of  larval  life 
obscure  the  first  beginnings  of  the  S. 

t  Norwegian  results,  based  largely  on  otoliths,  are  different.  Gunner  Rollefsen 
holds  that  the  spawning  cod,  or  skrei,  do  not  reach  maturity,  for  the  most  part, 
till  10  or  11  years  old,  and  grow  by  no  more  than  1  to  3  cms.  a  year  (Fiskeriskrifter, 
Bergen,  1933). 


180  THE  RATE  OF  GROWTH  [ch. 

fact,  90  cm.  is,  or  was  till  lately,  the  median  size  of  cod*  in  our 
Scottish  trawl-fishery;  one-tenth  are  over  a  metre  long  and  the 
largest  are  in  the  neighbourhood  of  120  cm.,  with  an  occasional 
giant  of  150  cm.  or  even  more.  But  it  has  come  to  pass  that  fish 
of  outstanding  size  are  seen  no  more  save  on  the  virgin  fishing 
grounds;  a  Greenland  halibut,  brought  home  to  Hull  in  1938, 
weighed  four  hundredweight,  was  nearly  two  feet  thick,  and  must 
have  been  of  prodigious  age. 


Age  (years)     

1 

2 

3 

4 

5 

6 

Length  of  cod  (cm.) 

18 

36 

55 

68 

79 

89 

Length  of  grey  mullet 

21 

36 

46 

51 

53 

55 

There  are  other  ways  of  determining,  or  estimating,  a  fish's  age. 
The  Greek  fishermen  shewed  Aristotle  f  how  to  tell  the  age  of  the 
purple  Murex,  up  to  six  years  old,  by  counting  the  whorls  and 
sculptured  ridges  of  the  shell,  and  also  how  to  estimate  the  age  of 
a  scaly  fish  by  the  size  and  hardness  of  its  scales ;  and  Leeuwenhoek 
saw  that  a  carp's  scales  J  bear  concentric  rings,  which  increase  in 
number  as  the  fish  grows  old.  In  these  and  other  cases,  as  in  the 
woody  rings  of  a  tree,  some  part  of  plant  or  animal  carries  a  record 
of  its  own  age;  and  this  record  may.  be  plain  and  certain,  or  may 
too  often  be  dubious  and  equivocal. 

The  scales  of  most  fishes  shew  concentric  rings,  sometimes  (as  in 
the  herring)  of  a  simple  kind,  sometimes  (as  in  the  cod)  in  a  more 
complex  pattern;  and  the  ear-bones,  or  otohths,  shew  opaque 
concentric  zones  in  their  translucent  structure.  The  scales  are 
''read"  with  apparent  ease  in  herring,  haddock,  salmon,  the  otohths 
in  plaice  and  hake;  but  the  whole  matter. is  beset  with  difficulties, 
and  every  result  deserves  to  be  checked  and  scrutinised  §. 

*  As  distinguished  from  "codling." 

t  Hist.  Animalium,  5476,  10;   6076,  30. 

X  The  carp-breeder  is  especially  interested  in  the  age  of  his  fish;  for,  like  the 
brewer  with  his  yeast,  his  profit  depends  on  the  rate  at  which  thej'^  grow. 
Leeuwenhoek 's  and  other  early  observations  were  brought  to  light  by  C.  HoflFbauer, 
Die  Alterbestimmung  der  Karpfen  an  seiner  Schuppen,  Jahresber.  d.  schles. 
Fischerei-Vereins,  Breslau,  1899. 

§  Thus,  for  instance,  Mr  A.  Dannevig  says  (On  the  age  and  growth  of  the  cod, 
Fiakeridirektorets  Skrifter,  1933,  p.  82):  "as  to  the  problem  of  the  determination 
of  the  age  of  the  cod  by  means  of  scales  and  otohths,  all  workers  agree  that  the 
method  is  useful.  But  on  a  number  of  fundamental  points  there  are  just  as  many 
divergences  of  opinion  as  there  are  investigators." 


Ill]         OF  THE  SIZE  AND  AGE  OF  HERRING         181 

In  the  following  table,  we  see  (a)  the  sizes,  and  (6)  the  number 

of  scale-rings,  in  a  sample  of  some  550  herring  from  the  autumn 
fishery  off  the  east  of  Scotland. 

Rings  3  4  5  6  7  8  9  10         11         12      Total    Mean 

cm.  rings 

31         —        —        —  1  1         _  1  _        _  1  4         8-5 

30        —        —        —  7  5  6  4         —        —        —        22         7-3 

29        —        —  5         18         13  6  6  1  1  1  51         70 

28        —  3         29         38         11  3  3  1  —        —         88         5-9 

27  2  13        41         34  5  5  2         _         _        _       102         5-1 

26  7         43         64         29         —        —  1  _        _        _       144         5.0 

25  4         36        41         11         —        —  —        —        —        —        92        4-6 

24  2  17         15  4         —        —  —        —        —        —        38        4-8 

23        —  5        —        —        —        —  —        —        —        —  5 4K) 

Total   15   117   195   142    35    20  17    2     1    2 

Mean   25-6  25-4  26-5  27-4  28-6  28-7  28-8  28-5  —        — 


In  this  sample,  the  sizes  of  the  550  fish  are  grouped  in  a  somewhat 
skew  curve,  about  a  mode  at  26  cm. ;  and  the  numbers  of  scale-rings 
group  themselves  in  like  manner,  but  with  rather  more  skewness, 
about  a  modal  number  of  five  rings.  Either  way  we  look  at  it, 
there  is  only  one  "group"  of  fish;  and  it  is  highly  characteristic 
of  the  herring  that  a  single  sample,  taken  from  a  single  shoal, 
exhibits  a  unimodal  curve.  Accepting  in  principle  the  view  that 
scale-rings  tend  to  synchronise  with  age  in  years,  we  may  draw  this 
first  deduction  that  our  sample  consists  in  part  (if  not  in  whole) 
of  five-year  old  fish,  whose  average  length  is  about  26  cm. ;  and 
this  length,  of  26  cm.  for  5-ringed,  or  5-year-old  herring,  agrees  well 
with  many  other  determinations  from  the  same  region.  We  shall 
be  on  the  safe  side  if  we  deal,  after  this  fashion,  with  the  one 
predominant  group,  or  mode,  in  each  sample  of  fish;  and  Fig.  46 
shews  an  approximate  curve  of  growth  for  our  East  Coast  herring 
drawn  in  this  way. 

But  the  further  assumption  is  commonly  and  all  but  universally 
made  that  each  individual  herring  carries  the  record  of  its  age  on  its 
scale-rings.  If  this  be  so,  then  our  sample  of  550  fish  is  a  com- 
posite population  of  some  ten  separate  broods  or  successive  ages, 
all  mixed  up  in  a  shoal.  And  again,  if  so,  the  5-year-olds  in  the 
said  population  average  26-5  cm.  in  length,  the  3-year-olds  25-6  cm., 
the  10-year-olds  28-5  cm. ;  but  these  values  do  not  fit  into  a  normal 


182 


THE  RATE  OF  GROWTH 


[CH. 


curve  of  growth  by  any  means.  Still  more  obvious  is  it  that  the 
several  year-classes  (if  such  they  be)  do  not  tally  with  the  age- 
composition  of  any  ordinary  population,  nor  agree  with  any  ordinary 
curve  of  mortality.  But  even  if  we  had  ten  separate  year-groups 
represented  here,  which  I  most  gravely  doubt,  all  that  we  know  of 
the  selective  action  of  the  drift-net  forbids  us  to  assume  that  we 
are  deahng  with  a  fair  random  sample  of  the  herring  population; 
so  that,  even  though  the  number  of  rings  did  enable  us  to  distinguish 
the  successive  broods,  we  should  still  have  no  right  to  assume  that 


Fig.  46. 


0  1  2  3  4  5  6 

Years 
Mean  curve  of  growth  of  Scottish  (East  Coast)  herring. 


these  annual  broods  actually  combine  in  the  proportions  shewn, 
to  form  the  composite  population. 

It  is  held  by  many  (in  the  first"  instance  by  Einar  Lea)  that  we 
may  deduce  the  dimensions  of  a  herring  at  each  stage  of  its  past 
life  from  the  corresponding  dimensions  of  the  rings  upon  its  scales. 
Some  such  relation  nmst  obviously  exist,  but  it  is  an  approximation 
of  the  roughest  kind.  For  it  involves  the  assumption  not  only  that 
the  scales  add  ring  to  ring  regularly  year  to  year,  and  that  fish 
and  scale  grow  all  the  while  at  corresponding  rates  or  in  direct 
proportion  to  one  another,  but  also  that  the  scale  grows  by  mere 


Ill]         OF  THE  SIZE  AND  AGE  OF  HERRING        183 

accretion,  each  annual  increment  persisting  without  further  change 
after  it  is  once  laid  down.  This  is  what  happens  in  a  moUuscan 
shell,  which  is  secreted  or  deposited  as  mere  dead  substance  or 
"formed  material";  but  it  is  by  no  means  the  case  in  bone,  and 
we  have  httle  reason  to  expect  it  bf  the  bony  mesoblastic  tissue  of 
a  fish's  scale.  It  is  much  more  likely  (though  we  do  not  know  for 
sure)  that  "osteoblasts"  and  "osteoclasts"  continue  (as  in  bone)  to 
play  their  part  in  the  scale's  growth  and  maintenance,  and  that 
some  sort  of  give  and  take  goes  on.     In  any  case,  it  is  a  matter  of 

Mean  ajyparent  length  of  one-year-old  herring,  as  deduced  by 
scale-reading  from  herring  of  various  ages  or  ''year-classes'^'' 


Year-class  (or  number 
of  rings) 

.> 

3 

4 

.■) 

6 

7 

8 

9 

Estimated    length    at 
1  year  old 

140 

13-2 

12-7 

I2-0 

12- 1 

11-8 

11-9 

11-8 

fact  and  observation  that  the  rings  alter  in  breadth  as  the  fish  goes 
on  growing  f ;  that  the  oldest  or  innermost  rings  grow  steadily 
narrower,  while  the  outermost  hardly  change  or  even  widen  a  little ; 
that-  the  relative  breadths  of  successive  rings  alter  accordingly; 
and  it  follows  that  when  we  try  to  trace  the  growth  of  a  herring 
through  its  lifetime  from  its  scales  when  it  is  old,  the  result  is  more 
or  less  misleading,  and  the  values  for  the  earher  years  are  apt  to 
be  much  too  small.  The  whole  subject  is  very  difficult,  as  we  might 
well  expect  it  to  be;  and,  I  am  only  concerned  to  shew  some 
small  part  of  its  difficulty  J. 

While  careful  observations  on  the  rate  of  growth  of  the  higher 
animals  are  scanty,  they  shew  so  fax  as  they  go  that  the  general 
features  of  the  phenomenon  are  much  the  same.  Whether  the 
animal  be  long-lived,  as  man  or  elephant,  or  short-lived  hke  horse  § 

*  From  T.  Emrys  Watkin,  The  Drift  Herring  of  the  S.E.  of  Ireland,  Rapports  du 
Conseil  pour  V Exploration  de  la  Mer,  lxxxiv,  p.  85,  1933. 

t  Cf.  {int.  al.)  Rosa  M.  Lee,  Methods  of  age  and  growth  determination  in  fishes 
by  means  of  scales,  Fishery  Investigations,  Dept.  of  Agr.  and  Fisheries,  1^20. 

X  The  copious  literature  of  the  subject  is  epitomised,  so  far,  by  Michael  Graham, 
in  Fishery  lyivestigations  (2),  xi,  No.  3,  1928. 

§  There  is  a  famous  passage  in  Lucretius  (v,  883)  where  he  compares  the  course 
of  life,  or  rate  of  growth,  in  the  horse  and  his  boyish  master:  Principio  circum 
tribus  actis  impiger  annis  Floret  equus,  puer  hautquaquam,  etc. 


184  THE  RATE  OF  GROWTH  [ch. 

or  dog,  it  passes  through  the  same  phases  of  growth ;  and,  to  quote 
Dr  Johnson  again,  "whatsoever  is  formed  for  long  duration  arrives 
slowly  to  its  maturity*."  In  all  cases  growth  begins  slowly;  it 
attains  a  maximum  velocity  somewhat  early  in  its  course,  and 
afterwards  slows  down  (subject  to  temporary  accelerations)  towards 
a  point  where  growth  ceases  altogether.  But  in  cold-blooded 
animals,  as  fish  or  tortoises,  the  slowing  down  is  greatly  protracted, 
and  the  size  of  the  creature  would  seem  never  to  reach,  but  only 
to  approach  asymptotically,  to  a  maximal  limit.  This,  after  all, 
is  an  important  difference.  Among  certain  still  lower  animals 
growth  ceases  early  but  Hfe  goes  on,  and  draws  (apparently)  to  no 
predetermined  end.  So  sea-anemones  have  been  kept  in  captivity 
for  sixty  or  even  eighty  years,  have  fed,  flourished  and  borne 
offspring  all  the  while,  but  have  shewn  no  growth  at  all. 

The  rate  of  growth  of  various  parts  or  organs  f 

That  the  several  parts  and  organs  of  the  body,  within  and 
without,  have  their  own  rates  of  growth  can  be  amply  demonstrated 
in  the  case  of  man,  and  illustrated  also,  but  chiefly  in  regard  to 
external  form,  in  other  animals.  There  lies  herein  an  endless 
field  for  the  study  of  correlation  and  of  variability  J. 

In  the  accompanying  table  I  show,  from  some  of  Vierordt's  data, 
the  relative  weights  at  various  ages,  compared  with  the  weight  at 
birth,  of  the  entire  body,  and  of  brain,  heart  and  liver;  also  the 
changing  relation  which  each  of  these  organs  consequently  bears, 
as  time  goes  on,  to  the  weight  of  the  whole  body  (Fig.  47) §. 

*  All  of  which  is  tantamount  to  a  mere  change  of  scale  of  the  time-curve. 

f  This  phenomenon,  of  incrementum  inequale,  as  opposed  to  incrementum  in 
universum,  was  most  carefully  studied  by  Haller:  "Incrementum  inequale  multis 
modis  fit,  ut  aliae  partes  corporis  aliis  celerius  increscant.  Diximus  hepar  minus 
fieri,  majorem  pulmonem,  minimum  thymum,  etc."     (Elem.  viii  (2),  p.  34.) 

X  See  {int.  al.)  A.  Fischel,  Variabilitat  und  Wachsthum  des  embryonalen 
Korpers,  Morphol.  Jahrb.  xxiv,  pp.  369-404,  1896;  Oppel,  Vergleickung  des 
Entwickelungsgrades  der  Organe  zu  verschiedenen  Entwickelungszeiten  hei  Wirhel- 
thieren,  Jena,  1891;  C.  M.  Jackson,  Pre-natal  growth  of  the  human  body  and  the 
relative  growth  of  the  various  organs  and  parts,  Amer.  Journ.  of  Anat.  ix,  1909; 
and  of  the  albino  rat,  ibid,  xv,  1913;  L.  A.  Calkins,  Growth  of  the  human  body  in 
the  foetal  period.  Rep.  Amer.  Assoc.  Anat.  1921.  For  still  more  detailed  measure- 
ments, see  A.  Arnold,  Korperuntersuchungen  an  1656  Leipziger  Studenten,  Ztschr. 
f.  Konstitutionslehre,  xv,  pp.  43-113,  1929. 

§  From  Vierordt's  Anatomische  Tabellen,  pp.  38,  39,  much  abbreviated. 


Ill] 


OF  PARTS  OR  ORGANS 


185 


Weight  of  various  organs,  compared  with  the  total  weight  of  the  human 
body' (male).     (From  Vierordfs  Anatomische  Tahellen) 


Percentage  increase 


Percentage  of  body-wt. 


Wt. 

r 

-A.        ,        ,        . 

^ 

, 

A 

s 

Age 

(kgm.) 

Body 

Brain 

Heart 

Liver 

Brain 

Heart 

Live 

0 

31 

10 

10 

10 

10 

12-3 

0-76 

4-6 

1 

9-0 

2-9 

2-5 

1-8 

2-4 

10-5 

0-46 

3-7 

2 

110 

3-6 

2-7 

2-2 

30 

9-3 

0-47 

3-9 

3 

125 

40 

2-9 

2-8 

3-4 

8-9 

0-52 

3-9 

4 

140 

4-5 

3-5 

31 

4-2 

9-5 

0-53 

4-2 

5 

15-9 

51 

3-3 

3-9 

3-8 

7-9 

0-51 

3-4 

6 

17-8 

5-7 

3-6 

3-6 

4-3 

7-6 

0-48 

3-5 

7 

19-7 

6-4 

3-5 

3-9 

4-9 

6-8 

0-47 

3-5 

8 

21-6 

70 

3-6 

40 

4-6 

6-4 

0-44 

30 

9 

23-5 

7-6 

3-7 

4-6 

50 

61 

0-46 

30 

10 

25-2 

8-1 

3-7 

5-4 

5-9 

5-6 

0-51 

3-3 

11 

27-0 

8-7 

3-6 

6-0 

61 

50 

0-52 

3-2 

12 

290 

9-4 

3-8 

(4.1) 

6-2 

4-9 

(0-34) 

30 

13 

331 

10-7 

3-9 

7-0 

7-3 

45 

0-50 

31 

14 

371 

120 

3-4 

9-2 

8-4 

3-5 

0-58 

3-2 

15 

41-2 

13-3 

3-9 

8-5 

9-2 

3-6 

0-48 

32 

16 

45-9 

14-8 

3-8 

9-8 

9-5 

3-2 

0-51 

30 

17 

49-7 

160 

3-7 

10-6 

10-5 

2-8 

0-51 

30 

18 

53-9 

17-4 

3-7 

10-3 

10-7 

2-6 

0-46 

2-8 

19 

57-6 

18-6 

3-7 

11-4 

11-6 

2-4 

0-51 

2-9 

20 

59-5 

19-2 

3-8 

12-9 

110 

2-4 

0-51 

2-6 

21 

61-2 

19-7 

3-7 

12-5 

11-5 

2-3 

0-49 

2-7 

22 

62-9 

20-3 

3o 

13-2 

11-8 

2-2 

0-50 

2-7 

23 

64-5 

20-8 

3-6 

12-4 

10-8 

2-2 

0-46 

2-4 

24 





3-7 

131 

13-0 

— 

— 

— 

25 

66-2 

21-4 

3-8 

12-7 

12-8 

2-2 

0-46 

2-8 

Fig.  47. 


10 

Age  in  years 
Relative  growth  in  weight  of  brain,  heart  and  body  of  man. 
From  Quetelet's  data  (smoothed  curves). 


186  THE  RATE  OF  GROWTH  [ch. 

We  see  that  neither  brain,  heart  nor  Hver  keeps  pace  by  any 
means  with  the  growing  weight  of  the  whole;  there  must  then 
be  other  parts  of  the  fabric,  probably  the  muscles  and  the  bones, 
which  increase  more  rapidly  than  the  general  average.  Heart  and 
liver  grow  nearly  at  the  same  rate,  the  liver  keeping  a  little  ahead 
to  begin  with,  and  the  heart  making  up  on  it  in  the  end;  by  the 
age  of  twenty-five  both  have  multiplied  their  original  weight  at 
birth  about  thirteen  times,  but  the  body  as  a  whole  has  multiplied 
by  twenty-one.  In  contrast  to  these  the  brain  has  only  multiplied 
its  weight  about  three  and  three-quarter  times,  and  shews  but  little 
increase  since  the  child  was  four  or  five,  and  hardly  any  since  it 
was  eight  years  old.  Man  and  the  gorilla  are  born  with  brains  much 
of  a  size;  but  the  gorilla's  brain  stops  growing  very  soon  indeed, 
while  the  child's  has  four  years  of  steady  increase.  The  child's 
brain  grows  quicker  than  the  gorilla's,  but  the  great  ape's  body 
grows  much  quicker  than  the  child's;  at  four  years  old  the  young 
gorilla  has  reached  about  80  per  cent,  of  his  bodily  stature,  and  the 
child's  brain  has  reached  about  80  per  cent,  of  its  full  size. 

Even  during  foetal  life,  as  well  as  afterwards,  the  relative  weight  of  the 
brain  keeps  on  declining.  It  is  about  18  per  cent,  of  the  body- weight  in  the 
third  month,  16  per  cent,  in  the  fourth,  14  per  cent,  in  the  fifth;  and  the 
ratio  falls  slowly  till  it  comes  to  about  12  per  cent,  at  birth,  say  10  per  cent, 
a  year  afterwards,  and  little  more  than  2  per  cent,  at  twenty*.  Many  statistics 
indicate  a  further  decrease  of  brain-weight,  actual  as  well  as  relative.  The 
fact  has  been  doubted  and  denied ;  but  Raymond  Pearl  has  shewn  evidence 
of  a  slow  decline  continuing  throughout  adult  life  f. 

The  latter  part  of  the  table  shews  the  decreasing  weights  of  the 
organs  compared  with  the  body  as  a  whole:  brain,  which  was 
12  per  cent,  of  the  body- weight  at  birth,  falling  to  2  per  cent,  at 
five-and-twenty ;  heart  from  0-76  to  0-46  per  cent.;  liver  from 
4-6  to  2-78  per  cent.  The  thyroid  gland  (as  we  know  it  in  the  rat) 
grows  for  a  few  weeks,  and  then  diminishes  during  all  the  rest  of 
the  creature's  Hfetime;  even  during  the  brief  period  of  its  own 
growth  it  is  growing  slower  than  the  body  as  a  whole. 

It  is  plain,  then,  that  there  is  no  simple  and  direct  relation,  holding 

*■  Cf.  J.  Ariens  Kappers,  Proc.  K.  Akad.  Wetensch.,  Amsterdam,  xxxix,  No.  7,  1936. 
t  R.  Pearl,  Variation  and  correlation  in  brain-weight,  Biometrika,  iv,  pp.  13-104, 
1905. 


Ill]  OF  PARTS  OR  ORGANS  187 

good  throughout  life,  between  the  size  of  the  body  and  its  organs; 
and  the  ratio  of  magnitude  tends  to  change  not  only  as  the  individual 
grows,  but  also  with  change  of  bodily  size  from  one  individual,  one 
race,  one  species  to  another.  In  giant  and  pigmy  breeds  of  rabbits, 
the  organs  have  by  no  means  the  same  ratio  to  the  body- weight ;  but 
if  we  choose  individuals  of  the  same  weight,  then  the  ratios  tend  to 
be  identical,  irrespective  of  breed*.  The  larger  breeds  of  dogs  are 
for  the  most  part  lighter  and  slenderer  than  the  small,  and  the  organs 
change  their  proportions  with  their  size.  The  spleen  keeps  pace 
with  the  weight  of  the  body ;  but  the  liver,  hke  the  brain,  becomes 
relatively  less.  It  falls  from  about  6  per  cent,  of  the  body-weight 
in  little  dogs  to  rather  over  2  per  cent,  in  a  great  hound  f. 

The  changing  ratio  with  increasing  magnitude  is  especially 
marked  in  the  case  of  the  brain,  which  constitutes  (as  we  have  just 
seen)  an  eighth  of  the  body- weight  at  birth,  and  but  one-fiftieth  at 
twenty-five.  This  faUing  ratio  finds  its  parallel  in  comparative 
anatomy,  in  the  general  law  that  the  larger  the  animal  the  smaller 
(relatively)  is  the  brain  J.  A  falhng  ratio  of  brain- weight  during  life 
is  seen  in  other  animals.  Max  Weber  §  tells  us  that  in  the  lion,  at 
five  weeks,  four  months,  eleven  months  and  lastly  when  full-grown, 
the  brain  represents  the  following  fractions  of  the  weight  of  the  body : 
viz.  i/18,  1/80,  1/184  and  1/546.  And  Kellicott  has  shewn  that  in  the 
dogfish,  while  certain  organs,  e.g.  pancreas  and  rectal  gland,  grow 
pari  passu  with  the  body,  the  brain  grows  in  a  diminishing  ratio, 
to  be  represented  (roughly)  by  a  logarithmic  curve ||. 

In  the  grown  man,  Raymond  Pearl  has  shewn  brain-weight  to 
increase  with  the  stature  of  the  indix'idual  and  to  decrease  with 
his  age,  both  in  a  straight-hne  ratio,  or  linear  regression,  as  the 

*  R.  C.  Robb,  Hereditary  size-limitation  in  the  rabbit,  Journ.  Exp.  Biol,  vi, 
1929. 

t  Cf.  H.  Vorsteher,  Einfiuss  d.  Gesamtgrosse  auf  die  Zusammensetzung  des 
Kbrpers;   Diss.,  Leipzig,  1923. 

X  Oliver  Goldsmith  argues  in  his  Animated  Nature  as  follows,  regarding  the  un- 
likelihood of  dwarfs  or  giants:  "Had  man  been  born  a  dwarf,  he  could  not  have 
been  a  reasonable  creature;  for  to  that  end,  he  must  have  a  jolt  head,  and  then  he 
would  not  have  body  and  blood  enough  to  supply  his  brain  with  spirits;  or  if  he 
had  a  small  head,  proportionable  to  his  body,  there  would  not  be  brain  enough  for 
conducting  life.     But  it  is  still  worse  with  giants,  etc." 

§   Die  Sdugethiere,  p.  117. 

II  Amer.  Journ.  of  Anatomy,  viii,  pp.  319-353,  1908. 


188  THE  RATE  OF  GROWTH  [ch. 

statisticians  call  it.     Thus  the  following  wholly  empirical  equations 
give  the  required  ratios  in  the  case  of  Swedish  males : 

Brain-weight  (gms.)  =  1487-8  —  1-94  x  age,  or 
=  915-06  +  2-86  X  stature. 

In  the  two  sexes,  and  in  different  races,  these  empirical  constants 
will  be  greatly  changed*;  and  Donaldson  has  further  shewn  that 
correlation  between  brain-weight  and  body-weight  is  much  closer 
in  the  rat  than  in  manf. 


Weight  of 

• 

entire 

Weight  of 

animal 

brain 

Ratios 

(gm.) 

(gm.) 

J, 

In 

r 

W 

w 

w:W  y^iv:-^W 

M>»»=Fr 

Marmoset 

335 

12-5 

1:26 

1:20 

w  =  2-30 

Spider  monkey 

1,845 

126 

15 

M 

1-56 

Felis  minuta 

1,234 

23-6 

52 

1-2 

2-25 

F.  domestica 

3,300 

31 

107 

2-4 

2-36 

Leopard 

27,700 

164 

168 

12 

200 

Lion 

119,500 

219 

546 

1-3 

217 

Dik-dik 

4,575 

37 

124 

2-7 

2-30 

Steinbok 

8,600 

49-5 

173 

2-9 

2-32 

Impala 

37,900 

148-5 

255 

2-75 

211 

Wildebeest 

212,200 

443 

479 

2-8 

201 

Zebra 

255,000 

541 

472 

2-7 

1-98 

»> 

297,000 

555 

536 

2-8 

200 

Rhinoceros 

765,000 

655 

1170 

3-6 

209 

Elephant 

3,048,000 

5,430 

560 

20 

1-74 

Whale  (Globiocephalus) 

1,000,000 

2,511 

400 

20 

1-77 

Mean  2-23  206 

Brandt,  a  very  philosophical  anatomist,  argued  some  seventy 
years  ago  that  the  brain,  being  essentially  a  hollow  structure,  a 
surface  rather  than  a  mass,  ought  to  be  equated  with  the  surface 
rather  than  the  mass  of  the  animal.  This  we  may  do  by  taking 
the  square-root  of  the  brain-weight  and  the  cube-root  of  the  body- 
weight;  and  while  the  ratios  so  obtained  do  not  point  to  equality, 
they  do  tend  to  constancy,  especially  if  we  hmit  our  comparison  to 
similar  or  related  animals.  Or  we  may  vary  the  method,  and  ask 
(as  Dubois  has  done)  to  what  power  the  brain-weight  must  be  raised 

*  Biometrika,  iv,  pp.  13-105,  1904. 

t  H.  H.  Donaldson,  A  comparison  of  the  white  rat  with  man,  etc.,  Boas  Memorial 
Volume,  New  York,  1906,  pp.  5-26. 


Ill]  OF  PARTS  OR  ORGANS  189 

to  equal  the  body-weight;  and  here  again  we  find  the  same  tendency 
towards  uniformity*. 

The  converse  to  the  unequal  growth  of  organs  is  found  in  their 
unequal  loss  of  weight  under  starvation.  Chossat  found,  in  a 
well-known  experiment,  that  a  starved  pigeon  had  lost  93  per  cent, 
of  its  fat,  about  70  per  cent,  of  hver  and  spleen,  40  per  cent,  of  its 
muscles,  and  only  2  per  cent,  of  brain  and  nervous  tissues  f.  The 
salmon  spends  many' weeks  in  the  river  before  spawning,  without 
taking  food.  The  muscles  waste  enormously,  but  the  reproductive 
bodies  continue  to  grow. 

As  the  internal  organs  of  the  body  grow  at  different  rates,  so  that 
their  ratios  one  to  another  alter  as  time  goes  on,  so  is  it  with  those 
hnear  dimensions  whose  inconstant  ratios  constitute  the  changing 
form  and  proportions  of  the  body.  In  one  of  Quetelet's  tables 
he  shews  the  span  of  the  outstretched  arms  from  year  to  year,  com- 
pared with  the  vertical  stature.  It  happens  that  height  and  span 
are  so  nearly  co-equal  in  man  that  direct  comparison  means  little ; 
but  the  ratio  of  span  to  height  (Fig.  48)  undergoes  a  significant  and 
remarkable  change.  The  man  grows  faster  in  stretch  of  arms  than 
he  does  in  height,  and  span  which  was  less  at  birth  than  stature  b}'' 
about  I  per  cent,  exceeds  it  by  about  4  per  cent,  at  the  age  of 
twenty.  Quetelet's  data  are  few  for  later  years,  but  it  is  clear 
enough  that  span  goes  on  increasing  in  proportion  to  stature.  How 
far  this  is  due  to  actual  growth  of  the  arms  and  how  far  to  increasing 
breadth  of  the  chest  is  another  story,  and  is  not  yet  ascertained. 

*  Cf.  A.  Brandt,  Sur  le  rapport  du  poids  du  cerveau  a  celui  du  corps  chez 
diflFerents  aniraaux,  Bull,  de  la  Soc.  Imp.  des  naturalistes  de  Moscou,  XL,  p.  525, 
1867;  J.  Baillanger,  De  I'etendu  de  la  surface  du  cerveau,  Ann.  Med.  Psychol. 
XVII,  p.  1,  1853;  Th.  van  Bischoff,  Das  Hirngewicht  des  Menschen,  Bonn,  1880 
(170  pp.),  cf.  Biol.  Centralhl.  i,  pp.  531-541,  1881;  E.  Dubois,  On  the  relation 
between  the  quantity  of  brain  and  the  size  of  the  body,  Proc.  K.  Akad.  Wetensch., 
Amsterdam,  xvi,  1913.  Also,  Th.  Ziehen,^  Maszverhaltnisse  des  Gehirns,  in 
Bardeleben's  Handh.  d.  Anatomie  des  Menschen;  P.  Warneke,  Gehirn  u.  Korper- 
gewichtsbestimmungen  bei  Saugern,  Journ.  f.  Psychol,  u.  Neurol,  xiii,  pp.  355-403, 
1909;  B.  Klatt,  Studien  zum  Domestikationsproblem,  Bibliotheca  genetica,  ii, 
1921;  etc.  The  case  of  the  heart  is  somewhat  analogous ;  see  Parrot,  Zooi.  Ja^r&. 
(System.),  vii,  1894;   Piatt,  in  Biol.  Centralhl.  xxxix,  p.  406,  1919. 

t  C.  Chossat,  Recherches  sur  I'inanition,  Mem.  Acad,  des  Sci.,  Paris,  1843, 
p.  438. 


190 


THE  RATE  OF  GROWTH 


[CH. 

The  growth-rates  of  head  and  body  differ  still  more;    for  the 
height  of  the  head  is  no  more  than  doubled,  but  stature  is  trebled, 


Height  of  the  head  in  man  at  various  ages  * 
(After  Quetelet,  p.  207,  abbreviated) 

Men  Women 


Stature 

Head 

Stature 

Head 

Age 

m. 

ra. 

Ratio 

m. 

m. 

Rat 

Birth 

0-50 

Oil 

4-5 

0-49 

Oil 

4-4 

1  year 

0-70 

015 

4-5 

0-69 

015 

4-5 

2  years 

0-79 

017 

4-6 

0-78 

0-17 

4-5 

3     „ 

0-86 

0-18 

4-7 

0-85 

0-18 

4-7 

5     „ 

0-99 

019 

51 

0-97 

019 

51 

10     „ 

1-27 

0-21 

6-2 

1-25 

0-20 

6-2 

20     „ 

1-51 

0-22 

70 

1-49 

0-21 

7  0 

25     ,. 

1-67 

0-23 

7-3 

1-57 

0-22 

71 

30     „ 

1-69 

0-23 

7-4 

1-58 

0-22 

71 

40     „ 

1-69 

0-23 

7-4 

1-58 

0-22 

71 

Fig.  48. 


Ratio  of  stature  in  man,  to  span  of  outstretched  arms. 
From  Quetelet's  data. 


between  infancy  and  manhood.  Diirer  studied  and  illustrated  this 
remarkable  phenomenon,  and  the  difference  which  accompanies  and 


*  A  smooth  curve,  very  similar  to  this,  is  given  by  Karl  Pearson  for  the  growth 
in  "auricular  height"  of  the  girl's  head,  in  Biometrika,  ni,  p.  141,  1904. 


Ill]  OF  PARTS  OR  ORGANS  191 

results  from  it  in  the  bodily  form  of  the  child  and  the  man  is  easy 
to  see. 

The  following  table  shews  the  relative  sizes  of  certain  parts  and 
organs  of  a  young  trout  during  its  most  rapid  development;  and 
so  illustrates  in  a  simple  way  the  varying  growth-rates  in  different 
parts  of  the  body*.  It  would  not  be  difficult,  from  a  picture  of  the 
little  trout  at  any  one  of  these  stages,  to  draw  its  approximate 
form  at  any  other  by  the  help  of  the  numerical  data  here  set 
forth.  In  like  manner  a  herring's  head  and  tail  grow  longer, 
the  parts  betw^een  grow  relatively  less,  and  the  fins  change  their 
places  a  httle;  the  same  changes  take  place  with  their  specific 
differences  in  related  fishes,  and  herring,  sprat  and  pilchard 
owe  their  specific  characters  to  their  rates  of  growth  or  modes  of 
increment  f. 

Trout  (Salmo  fario) ;  proportionate  growth  of  various  organs 
(From  Jenkinson's  data) 


Days 

Total 

1st 

Ventral 

2nd 

Tail 

Breadth 

old 

length 

Eye 

Head 

dorsal 

fin 

dorsal 

fin 

•  of  tail 

40 

100 

100 

100 

100 

100 

100 

100 

100 

63 

130 

129 

148 

149 

149 

108 

174 

156 

77 

155 

147 

189 

(204) 

(194) 

139 

258 

220 

92 

173 

179 

220 

(193) 

(182) 

155 

308 

272 

106 

195 

193 

243 

173 

165 

173 

337 

288 

Sachs  studied  the  same  phenomenon  in  plants,  after  a  method 
in  use  by  Stephen  Hales  a  hundred  and  fifty  years  before.  On  the 
growing  root  of  a  bean  ten  narrow  zones  were  marked  off,  starting 
from  the  apex,  each  zone  a  millimetre  long.  After  twenty-four 
hours'  growth  (at  a  given  temperature)  the  whole  ten  zones  had 
grown  from  10  to  33  mm.,  but  the  several  zones  had  grown  very 
unequally,  as  shewn  in  the  annexed  table  J  (p.  192): 

*  Cf.  J.  W.  Jenkinson,  Growth,  variability  "and  correlation  in  young  trout,  Bio- 
metrika,  vin,  pp.  444-466.  1912. 

I  Cf.  E.  Ford,  On  the  transition  from  larval  to  adolescent  herring,  Journ.  Mar. 
Biol.  Assoc.  XVI,  p.  723;  xviii,  p.  977,  1930-31.  8o  also  in  larval  eels,  tail  and 
body  grow  at  different  rates,  which  rates  differ  in  different  species;  cf.  Johannes 
Schmidt,  Meddel.  Kommlss.  Havsiindersok.  1916;  L.  Bertin,  Bull.  Zool.  France, 
1926,  p.  327. 

I  From  Sachs's  Textbook  of  Botany,  1882,  p.  820. 


192 


THE  RATE  OF  GROWTH 


[CH. 


Graded  growth  of  bean-root 


Increment 

Increment 

Zone 

mm. 

Zone 

mm. 

Apex 

1-5 

6th 

1-3 

2nd 

5-8 

7th 

0-5 

3rd 

8-2 

8th 

0-3 

4th 

3-5 

9th 

0-2 

5th 

1-6 

10th 

01 

"...  I  marked  in  the  same  manner  as  the  Vine,  young  Honeysuckle  shoots, 
etc. . . . ;  and  I  found  in  them  all  a  gradual  scale  of  unequal  extensions,  those  parts 
extending  most  which  were  tenderest,"  Vegetable  Staticks,  Exp.  cxxiii. 

The  lengths  attained  by  the  successive  zones  He  very  nearly 
on  a  smooth  curve  or  gradient;  for  a  certain  law,  or  principle 
of  continuity,  connects  and  governs  the  growth-rates  along  the 
growing  axis.     This  curve  has  its  family  likeness  to  those  differential 


10 


2  3         4  5         6  7         8 

Zones 
Fig.  49.     Rate  of  growth  of  bean-root,  in  successive  zones 
of  1  mm.  each,  beginning  at  the  tip. 

curves  which  we  have  already  studied,  in  which  rate  of  growth  was 
plotted  against  time,  as  here  it  is  plotted  against  successive  spatial 
intervals  of  a  growing  structure ;  and  its  general  features  are  those 
of  a  curve,  a  skew  curve,  of  error.  Had  the  several  growth-rates 
been  transverse  to  the  axis,  instead  of  being  longitudinal  and 
parallel  to  it,  they  would  have  given  us  a  leaf-shaped  structure, 
of  which  our  curve  would  represent  the  outline  on  either  side;  or 
again,  if  growth  had  been  symmetrical  about  the  axis,  it  might  have 
given  us  a  turnip-shaped  soHd  of  revolution.  There  is  always  an 
easy  passage  from  growth  to  form. 


Ill]  CONCERNING  GRADIENTS  193 

A  like  problem  occurs  when  we  deal  with  rates  of  growth  in 
successive  natural  internodes;  and  we  may  then  pass  from  the 
actual  growth  of  the  internodes  to  the  varying  number  of  leaves 
which  they  successively  produce.  Where  we  have  whorls  of  leaves 
at  each  node,  as  in  Equisetum  or  in  many  water-weeds,  then  the 
problem  is  simphfied;  and  one  such  case  has  been  studied  by 
Ra3rmond  Pearl*.  In  Ceratophyllum  the  mean  number  of  leaves 
increases  with  each  successive  whorl,  but  the  rate  of  increase 
diminishes  from  whorl  to  whorl  as  we  ascend.  On  the  main  stem 
the  rate  of  change  is  very  slow;  but  in  the  small  twigs,  or  tertiary 
branches,  it  becomes  rapid,  as  we  see  from  the  following  abbreviated 
table : 

Number  of  leaves  per  whorl  on  the  tertiary  branches  of 
Ceratophyllum 


Order  of  whorl     ... 

1 

2 

3 

4 

5 

6 

Mean  no.  of  leaves 
Smoothed  no. 

6-55 
6-5 

8-07 
8-0 

1 

9-00 
90 

9-20 
9-5 

9-75 
9-8 

10-00 
100 

Raymond  Pearl  gives  a  logarithmic  formula  to  fit  the  case;  but 
the  main  point  is  that  the  numbers  form  a  graded  series,  and  can 
be  plotted  as  a  simple  curve. 

In  short,  a  large  part  of  the  morphology  of  the  organism  depends 
on  the  fact  that  there  is  not  only  an  average,  or  aggregate,  rate  of 
growth  common  to  the  whole,  but  also  a  gradation  of  rate  from  one 
part  to  another,  tending  towards  a  specific  rate  characteristic  of  each 
part  or  organ.  The  least  change  in  the  ratio,  one  to  another,  of 
these  partial  or  locahsed  rates  of  growth  will  soon  be  manifested 
in  more  and  more  striking  differences  of  form;  and  this  is  as  much 
as  to  say  that  the  time-element,  which  is  imphcit  in  the  idea  oigrowthy 
can  never  (or  very  seldom)  be  wholly  neglected  in  our  consideration 
of  form  I . 

A  flowering  spray  of  Montbretia  or  lily-of-the-valley  exemplifies 
a  growth-gradient,  after  a  simple  fashion  of  its  own.     Along  the 

*  On  variation  and  differentiation  in  Ceratophylly,m,  Carnegie  Inst.  Publications, 
No.  58,  1907;  see  p.  87. 

t  Herein  lies  the  easy  answer  to  a  contention  raised  by  Bergson,  an(J  to  which 
he  ascribes  much  importance,  that  "a  mere  variation  of  size  is  one  thing,  and 
a  change  of  form  is  another."  Thus  he  considers  "a  change  in  the  form  of  leaves" 
to  constitute  "a  profound  morphological  difference"  {Creative  Evolution,  p.  71). 


194  THE  RATE  OF  GROWTH  [ch. 

stalk  the  growth-rate  falls  away ;  the  florets  are  of  descending  age, 
from  flower  tO'  bud:  their  graded  differences  of  age  lead  to  an 
exquisite  gradation. of  size  and  form;  the  time-interval  between 
one  and  another,  or  the  "space-time  relation"  between  them  all, 
gives  a  peculiar  quality — ^we  may  call  it  phase-beauty — to  the 
whole.  A  clump  of  reeds  or  rushes  shews  this  same  phase-beauty, 
and  so  do  the  waves  on  a  cornfield  or  on  the  sea.  A  jet  of  water 
is  not  much,  but  a  fountain  becomes  a  beautiful  thing,  and  the 
play  of  many  fountains  is  an  enchantment  at  Versailles. 

On  the  weight-length  coefficient,  or  ponderal  itidex 

So  much  for  the  visible  changes  of  form  which  accompany 
advancing  age,  and  are  brought  about  by  a  diversity  of  rates  of 
growth  at  successive  points  or  in  different  directions.  But  it  often 
happens  that  an  animal's  change  of  form  may  be  so  gradual  as  to 
pass  unnoticed,  and  even  careful  measurement  of  such  small  changes 
becomes  difficult  and  uncertain.  Sometimes  one  dimension  is  easily 
determined,  but  others  are  hard  to  measure  with  the  same  accuracy. 
The  length  of  a  fish  is  easily  measured ;  but  the  breadth  and  depth 
of  plaice  or  haddock  are  vaguer  and  more  uncertain.  We  may  then 
make  use  of  that  ratio  of  weight  to  length  which  we  spoke  of  in  the 
last  chapter:  viz.  that  W  oc  L^,  or  W  ^  kL^,  or  W/L^  =  k,  where 
k,  the  "ponderal  index,"  is  a  constant  to  be  determined  for  each 
particular  case*. 

We  speak  of  this  /;  as  a  "constant,"  with  a  mean  value  specific 
to  each  species  of  animal  and  dependent  on  the  bodily  proportions 
or  form  of  that  animal;  yet  inasmuch  as  the  animal  is  continually 
apt  to  change  its  bodily  proportions  during  life,  k  also  is  continually 
subject  to  change,  and  is  indeed  a  very  delicate  index  of  such 

*  This  relation,  and  how  important  it  is,  were  clearly  recognised  by  Herbert 
Spencer  in  his  Recent  Discussions  in  Science,  etc.,  1871.     The  formula  has  been 

X  y/w 

often,  and  often  independently,  employed:    first  perhaps  in  the  form  — y-  x  100, 

by  R.  Livi,  L'indice  ponderale,  o  rapporto  tra  la  statura  e  il  peso,  Atti  Soc.  Romamt 
Antropologica,  v,  1897.  Values  of  k  for  man  and  many  animals  are  given  by 
H.  Przibram,  in  Form  und  Formel,  1922.  On  its  use  as  an  index  to  the  condition 
or  habit  of  body  of  an  individual,  see  von  Rhode,  in  Abderhalden's  Arbeitsmethoden, 
IX,  4.  The  constant  k  might  be  called,  more  strictly,  ki,  leaving  kf,  and  k,i  for 
the  similar  constants  to  be  deri-ved  from  the  breadth  and  depth  of  the  fish. 


Ill 


THE  PONDERAL  INDEX 


195 


progressive  changes :  delicate — because  our  measurements  of  length 
are  very  accurate  on  the  whole,  and  weighing  is  a  still  more  dehcate 
method  of  comparison. 

Thus,  in  the  case  of  plaice,  when  we  deal  with  mean  values  for 
large  numbers  and  with  samples  so  far  "homogeneous"  that  they 
are  taken  at  one  place  and  time,  we  find  that  k  is  by  no  means 
constant,  but  varies,  and  varies  in  an  orderly  way,  with  increasing 
size  of  the  fish.  The  phenomenon  is  unexpectedly  complex,  much 
more  so  than  I  was  aware  of  when  I  first  wrote  this  book.     Fig.  50 


120  — 


.2 

o     110 


§    100 

^       U 


90 


'  1 

T — :—t     r  -j— T-  1     I 

1  ' 

_ 

December 

^ 

^ 



— 

— 

~-y 

March 

^ 

^'^"^^■"•..•^ 

— 

, ,  1 

1     1    1    1    1    1     1    1     1 

\  , 

1   1    1    1    1 

1   1 

22 


25 


40 


30  35 

Length  (cm.) 

Fig.  50.     Changes  in  the  weight-length  coefficient  of  plaice  with 
increasing  size;  from  March  and  December  samples. 

shews  the  weight-length  coefficient,  or  ponderal  index,  in  two 
large  samples,  one  taken  in  the  month  of  March,  the  other  in 
December.  In  the  latter  sample  k  increases  steadily  as  the  plaice 
grow  from  about  25  to  40  cm.  long;  weight,  that  is  to  say,  increases 
more  rapidly  than  the  cube  of  the  length,  and  it  follows  that  length 
itself  is  increasing  less  rapidly  than  some  other  linear  dimension. 
In  other  words,  the  plaice  grow  thicker,  or  bulkier,  with  length  and 
age.  The  other  sample,  taken  in  the  month  of  March,  is  curiously 
different;  for  now  k  rises  to  a  maximum  when  the  fish  are  some- 
where about  30  cm.  long,  and  then  decHnes  slowly  with  further 
increase  in  size  of  the  fish;  and  k  itself  is  less  in  March  than  in 
December,  the  discrepancy  being  slight  in  the  small  fish  and  great 
in  the  large.  ,  The  "point  of  inflection"  at  30  cm.  or  thereby  marks 


196  THE  RATE  "OF  GROWTH  [ch. 

an  epoch  in  the  fish's  life;  it  is  about  the  size  when  sexual  maturity 
begins,  or  at  least  near  enough  to  suggest  a  connection  between  the 
two  phenomena*. 

A  step  towards  further  investigation  would  be  to  determine  k  for  the  two 
sexes  separately,  and  to  see  whether  or  no  the  point  of  inflection  occurs,  as 
maturity  is  known  to  be  reached,  at  a  smaller  size  in  the  male.  This  d'Ancona 
has  done,  not  for  the  plaice  but  for  the  shad  {Alosa  finta).  He  finds  that  the 
males  are  the  first  to  reach  maturity,  first  to  shew  a  retardation  of  the  rate 
of  growth,  first  to  reach  a  maximal  value  of  th^  ponderal  index,  and  in  all 
probability  the  first  to  diet • 

Again  we  may  enquire  whether,  or  how,  k  varies  with  the  time 
of  year;  and  this  torrelation  leads  to  a  striking  result  J.  For  the 
ponderal  index  fluctuates  periodically  with  the  seasons,  falling 
steeply  to  a  minimum  in  March  or  April,  and  rising  slowly  to  an 
annual  maximum  in  December  (Fig.  51)  §.  The  main  and  obvious 
explanation  hes  in  the  process  of  spawning,  the  rapid  loss  of  weight 
thereby,  and  the  slow  subsequent  rebuilding*  of  the  reproductive 
tissues;  whence  it  follows  that,  without  ever  seeing  the  fish  spawn, 
and  without  ever  dissecting  one  to  see  the  state  of  its  reproductive 
system,  we  may  by  this  statistical  method  ascertain  its  spawning 
season,  and  determine  the  beginning  and  end  thereof  with  con- 
siderable accuracy.  But  all  the  while  a  similar  fluctuation,  of 
much  less  amphtude,  is  to  be  found  in  voung  plaice  before  the 
spawning  age;  whence  we  learn  that  the*  fluctuation  is  not  only 
due  to  shedding  and  replacement  of  spawn,  but  in  part  also  to 
seasonal  changes  in  appetite  and  general  condition. 

Returning  to  our  former  instance,  we  now  see  that  the  March 
and  December  samples  of  plaice,  which  shewed  such  discrepant 
variations  of  the  ponderal  index  with  increasing  size,  happen  to 

♦  The  carp  shews  still  more  striking  changes  than  does  the  plaice  in  the  weight - 
length  coefl&cient:  in  other  words,  still  greater  changes  in  bodily  shape  with 
advancing  age  and  increasing  size;  cf.  P.  H.  Stnithers,  The  Champlain  Watershed, 
Albany,  New  York,  1930. 

■j-  U.  d'Ancona,  II  problema  dell'  accrescimento  dei  pesci,  etc.,  Mem.  R.  Acad, 
dei  Lincei  (6),  n,  pp.  497-540,  1928. 

J  Cf.  Lammel,  Ueber  periodische  Variationen  in  Organismen,  Biol.  Centralbl. 
xxn,  pp.  368-376,  1903. 

§  When  we  restrict  ourselves,  for  simphcity'a  sake,  to  fish  of  one  particular 
size,  we  need  not  determine  the  values  of  k,  for  changes  in  weight  are  obvious 
enough;  but  when  we  have  small  numbers  and  various  sizes  to  deal  with,  the 
determination  of  k  helps  very  much. 


Ill 


THE  PONDERAL  INDEX 


197 


coincide  with  the  beginning  and  end  of  the  spawning  season;  the 
fish  were  full  of  spawn  in  December,  but  spent  and  lean  in  March. 
The  weight-length  ratio  was,  of  necessity,  higher  at  the  former 
season;  and  the  faUing-off  in  condition,  and  in  bulk,  which  the 
March  sample  indicates,  is  more  and  more  pronounced  in  the  larger 
and  therefore  more  heavily  spawn-laden  fish. 


JFMAMJJASONDJ 
Fig.  51.     Periodic  annual  change  in  the  weight-length  ratio  of  plaice. 

Periodi-c  relation  of  weight  to  length  in  plaice  of  55  cm.  long 


Average  weight 

WJL^ 

W/L^  (smoothed) 

decigrams 

Jan. 

204 

1-23 

116 

Feb. 

174 

104 

108 

March 

162 

0-97 

0-99 

April 

159 

0-95 

0-97 

May 

162 

0-98 

0-98 

June 

171 

103 

101 

July 

169 

101 

104 

August 

178 

1-07 

104 

Sept. 

173 

104 

Ml 

Oct. 

203 

1-22 

116 

Nov. 

203 

1-22 

1-21 

Dec. 

200 

1-20 

1-22 

Mean 


180 


1-08 


198  THE  RATE  OF  GROWTH  [ch. 

Plaice  caught  in  a  certain  area,  March  1907  and  December  1905. 
Variation  of  k,  the  weight-length  coefficient,  with  size 


March  sample 

December  sample 

A 

r 

Do. 

Do. 

cm. 

gm. 

WjL^      smoothed 

gm. 

W/L^ 

smoothed 

23 

113 

0-93 









24 

128 

0-93 

0-94 

—  ' 

— 



25 

152 

0-97 

0-96 







26 

178 

0-96 

0-98 

177 

101 



27 

193 

0-98 

0-99 

209 

106 

1-06 

28 

221 

101 

1-00 

241 

MO 

1-08 

29 

250 

102 

101 

264 

108 

109 

30 

271 

1-00 

101 

294 

109 

109 

31 

300 

101 

100 

325 

109 

110 

32 

328 

100 

100 

366 

112 

112 

33 

354 

0-99 

0-99 

410 

114 

113 

34 

384 

0-98 

0-98 

449 

M4 

115 

35 

419 

0-98 

0-98 

501 

117 

117 

36 

454 

0-97 

0-97 

556 

119 

117 

37 

492 

0-95 

0-96 

589 

116 

118 

38 

529 

0-96 

0-96 

652 

119 

119 

39 

564 

0-95 

0-95 

719 

1-21 

1-22 

40 

614 

096 

0-95 

809 

1-26 

— 

41 

647 

0-94 

0-94 

— 





42 

679 

0-92 

0-93 

— 

— 

— 

43 

732 

0-92 

0-93 

— 

•    — 

— 

H 

800 

0-94 

0-94 

— 

— 

— 

4l5 

875 

0-96 

— 

— 

— 

— 

These  weights  and  measurements  of  plaice  are  taken  from  the  Department  of 
Agriculture  and  Fisheries'  Plaice-Report,  i,  pp.  65,  107,  1908;   ii,  p.  92,  1909. 

Japanese  goldfish*  are  exposed  to  a  much  wider  range  of  tem- 
perature than  our  plaice  are  called  on  to  endure ;  they  hibernate  in 
winter  and  feed  greedily  in  the  heat  of  summer.  Their  weight  is 
low  in  winter  but  rises  in  early  spring,  it  falls  as  low  as  ever  at  the 
height  of  the  spawning  season  in  the  month  of  May;  so  for  one 
weight-length  fluctuation  which  the  plaice  has,  the  goldfish  has  a 
twofold  cycle  in  the  year.  The  index  reaches  its  second  and  higher 
maximum  in  August,  and  falls  thereafter  till  the  end  of  the  year. 
That  it  should  begin  to  fall  so  soon,  and  fall  so  quickly,  merely  means 
that  late  autumn  is  a  time  of  groipth ;  the  fish  are  not  losing  weight, 
but  growing  longer  f. 

*  Cf.  Kichiro  Sasaki,  Tohoku  Soi.  Reports  (4),  i,  pp.  239-260,  1926. 

t  Much  has  been  written  on  the  weight-length  index  in  fishes.  See  (int.  al.) 
A.  Meek,  The  growth  of  flatfish,  Northumberland  Sea  Fisheries  Ctee,  1905,  p.  58; 
W,  J.  Crozier,  Correlations  of  weight,  length,  etc,  in  the  weakfish,  Cynoscion 


hi]  the  PONDERAL  index  199 

It  is  the  rule  in  fishes  and  other  cold-blooded  vertebrates  that 
growth  is  asymptotic  and  size  indeterminate,  while  in  the  warm- 
blooded growth  comes,  sooner  or  later,  to  an  end.  But  the 
characteristic  form  is  established  earlier  in  the  former  case,  and 
changes  less,  save  for  the  minor  fluctuations  we  have  spoken  of. 
In  the  higher  animals,  such  as  ourselves,  the  whole  course  of  life 
is  attended  by  constant  alteration  and  modification  of  form;    and 


Fig.  52.     The  ponderal  index,  or  weight-length  coefficient,  in 
man.     From  Quetelet's  data. 

we  may  use  our  weight-length  formula,  or  ponderal  index,  to  illus- 
trate (for  instance)  the  changing  relation  between  height  and  weight 
in  boyhood,  of  which  we  spoke  before  (Fig.  52). 


regalis,  Bull.  U.S.  Bureau  of  Fisheries,  xxxni,  pp.  141-147,  1913;  Selig  Hecht, 
Form  and  growth  in  fishes,  Journ.  of  Morphology,  xxvii,  pp.  379-400,  1916; 
J.  Johnstone  (Plaice),  Trans.  Liverpool  Biolog.  Soc.  xxv,  pp.  186-224,  1911; 
J.  J.  Tesch  (Eel),  Journ.  du  Conseil,  iii,  1927;  Frances  N.  Clark  (Sardine),  Calif. 
Fish.  Bulletin,  No.  19,  1928  (with  full  bibliography).  For  a  discussion  on  statistical 
lines,  apart  from  any  assumptions  such  as  the  "law  of  the  cubes,"  see  G.  Buncker, 
Korrelation  zwischen  Lange  u.  Gewicht,  etc.,  Wissensch.  Meerestmtersuch.  Helgoland, 
XV,  pp.  1-26,  1923. 


THE  RATE  OF  GROWTH 


[CH. 


The  weight-length  coefficient,  or  ponderal  index,  k,  in  young  Belgians 
(From  Quetelefs  figures) 


(years) 

WJL^ 

Age  (years) 

WjL^ 

0 

2-55 

10 

1-25 

1 

2-92 

11 

118 

2 

2-34 

12 

1-23 

3 

2-08 

13 

1-29 

4 

1-87 

14 

1-23 

5 

1-72 

15 

1-23 

6 

1-56 

16 

1-28 

7 

1-48 

20 

1-30 

8 

1-39 

25 

1-36 

9 

1-29 

The  infant  is  plump  and  chubby/  and  the  ponderal  index  is  at 
its  highest  at  a  year  old.  As  the  boy  grows,  it  is  in  stature  that  he 
does  so  most  of  all;  his  ponderal  index  falls  continually,  till  the 
growing  years  are  over,  and  the  lad  "fills  out"  and  grows  stouter 
again.  During  prenatal  Ufe  the  index  varied  httle,  and  less  than 
we  might  suppose : 


Relation  between  length  and  weight  of  the  human  foetus 
(From  Scammon's  data) 


Length 

Weight 

cm. 

gm. 

7-7 

13 

12-3 

41 

17-3 

115 

223 

239 

27-2 

405 

32-3 

750 

37-2 

1163 

42-2 

1758 

46-9 

2389 

51-7 

3205 

2-9 
2-2 
2-2 
2-2 

io 

2-2 
2-3 
2-3 
2-3 
2-3 


As  a  further  illustration  of  the  rate  of  growth,  and  of  unequal 
growth  in  various  directions,  we  have  figures  for  the  ox,  extending 
over  the  first  three  years  of  the  animal's  life,  and  giving  (1)  the 
weight  of  the  animal,  month  by  month,  (2)  the  length  of  the  back, 
from  occiput  to  tail,  and  (3)  the  height  to  the  withers.  To  these 
I  have  added  (4)  the  ratio  of  length  to  height,  (5)  the  weight-length 
coefficient,  k,  and  (6)  a  similar  coefficient,  or  index-number,  k' ,  for 


Ill] 


THE  PONDERAL  INDEX 


201 


the  height,  of  the  animal.  All  these  ratios  change  as  time  goes  on. 
The  ratio  of  length  to  height  increases,  at  first  considerably,  for  the 
legs  seem  disproportionately  long  at  birth  in  the  ox,  as  in  other 

Relations  between  the  weight  and  certain  linear  dimensions  of  the  ox 


{Data  from  i 

Jo^nevin*y 

abbreviated) 

Length 

Age 

Weight 

of  back 

Height 

lonths 

kgm. 

m. 

m. 

L/H 

k  =  WjL^ 

k'  =  Wli 

0 

37 

0-78 

0-70 

Ml 

0-78 

1-08 

1 

55 

0-94 

0-77 

1-22 

0-66 

1-21 

2 

86 

1-09 

0-85 

1-28 

0-67 

1-41 

3 

121 

1-21 

0-94 

1-28 

0-69 

1-46 

4 

150 

1-31 

0-95 

1-38 

0-66 

1-75 

5 

179 

1-40 

104 

1-35 

0-65 

1-60 

6 

210 

1-48 

109 

1-36 

0-64 

1-64 

7 

247 

1-52 

112 

1-36 

0-70 

1-75 

8 

267 

1-58 

115 

1-38 

0-68 

1-79 

9 

283 

1-62 

116 

1-39 

0-66 

1-80 

10 

304 

1-65 

119 

1-39 

0-68 

1-79 

11 

328 

1-69 

1-22 

1-39 

0-67 

1-79 

12 

351 

1-74 

1-24 

1-40 

0-67 

1-85 

13 

375 

1-77 

1-25 

1-41 

0-68 

1-90 

14 

391 

1-79 

1-26 

1-41 

0-69 

1-94 

15 

406 

1-80 

1-27 

1-42 

0-69 

1-98 

16 

418 

1-81 

1-28 

1-42 

0-70 

2-09 

17 

424 

1-83 

1-29 

1-42 

0-69 

1-97 

18 

424 

1-86 

1-30 

1-43 

0-66 

1-94 

19 

428 

1-88 

1-31 

1-44 

0-65 

1-92 

20 

438 

1-88 

1-31 

1-44 

0-66 

1-94 

21 

448 

1-89 

1-32 

1-43 

0-66 

1-94 

22 

464 

1-90 

1-33 

1-43 

0-68 

1-96 

23 

481 

1-91 

1-35 

1-42 

0-69 

1-98 

24 

501 

1-91 

1-35 

1-42 

.0-71 

203 

25 

521 

1-92 

1-36 

1-41 

0-74 

2-08 

26 

534 

1-92 

1-36 

1-41 

0-75 

212 

27 

547 

1-93 

1-36 

1-41 

0-76 

2-16 

28 

555 

1-93 

1-36 

1-41 

0-77 

219 

29 

562 

1-93 

1-36 

1-41 

0-78 

2-22 

30 

586 

1-95 

1-38 

1-41 

0-79 

2-22 

31 

611 

1-97 

1-40 

1-40 

0-80 

2-21 

32 

626 

1-98 

1-42 

1-40 

0-80 

219 

33 

641 

200 

1-44 

1-39 

0-81 

216 

34 

656 

201 

1-45 

1-38 

0-81 

213 

ungulate  animals;  but  this  ratio  reaches  its  maximum  and  falls  off 
a  little  during  the  third  year :  so  indicating  that  the  beast  is  growing 
more   in   height   than  length,   at   a   time   when   growth    in    both 

*  Ch.  Cornevin,  ^^tudes  sur  la  croissanee,  Arch,  de  Physiol,  norm,  et  pathol.  (5),  iv, 
p.  477,  1892.  Cf.  also  R.  Gartner,  Ueber  das  W'achstum  d.  Tiere,  Landxvirtsch. 
Jahresher.  lvii,  p.  707,  1922. 


202 


THE  RATE  OF  GROWTH 


[CH. 


dimensions  is  nearly  over*.  The  ratio  W/H^  increases  steadily, 
and  at  three  years  old  is  double  what  it  was  at  birth.  It  is  the 
most  variable  of  the'  three  ratios;  and  it  so  illustrates  the  some- 
what obvious  but  not  unimportant  fact  that  k  varies  most  for  the 
dimension  which  varies  least,  or  grows  most  uniformly;  in  other 
words,  that  the  values  of  k,  as  determined  at  successive  epochs  for 
any  one  dimension,  are  a  measure  of  the  variability  of  the  other  two. 
The  same  ponderal  index  serves  as  an  index  of  "build,"  or 
bodily  proportion;  and  its  mean  values  have  been  determined  for 
various  ages  and  f6r  many  races  of  mankind.     Within  one  and  the 


ie>u 

1     • 

r- 

• 

y 

y- 

150 

y 

y 

X« 

140 

my 

X' 

y 

130 

^ 

1 

1       1 

1    , 

1 

1         1 

\ 

60 


70 


65 

Height  (inches) 

Fig.  53.     Ratio  of  height  to  weight  in  man.     From  Goringe's  data. 

same  race  it  varies  with  stature ;  for  tall  men,  and  boys  too,  are  apt 
to  be  slender  and  lean,  and  short  ones  to  be  thickset  and  strong. 
And  so  much  does  the  weight-length  ratio  change  with  build  or 
stature  that,  in  the  following  table  of  mean  heights  and  weights  of 
men  between  five  and  six  feet  high,  it  will  be  seen  that  weight, 
instead  of  varying  as  the  cube  of  the  height,  is  (within  the  hmits 
shewn)  in  nearly  simple  linear  relation  to  it  (Fig.  53)  f. 

*  As  a  matter  of  fact,  the  data  shew  that  the  animal  grows  under  7  per  cent, 
in  length,  but  over  11  per  cent,  in  height,  between  the  twentieth  and  the  thirtieth 
month  of  its  age. 

t  Had  the  weights  varied  as  the  cube  of  the  height,  the  tallest  men  should 
have  weighed  close  on  200  lb.,  instead  of  160  lb. 


Ill]  OF  WHALES  AND  ELEPHANTS  203 

Ratio  of  height  to  weight  in  man  * 


No.  of 

Height 

Weight 

instances 

in. 

lb. 

WIH 

Wjm 

59 

60-5 

125 

207 

5-62 

118 

61-5 

129 

213 

5-55- 

220 

62-5 

133 

213 

•     5-45 

285 

63-5 

136 

214 

5-30 

327 

64-5 

139 

215 

519 

386 

65-5 

143 

218 

5-09 

346 

66-5 

146 

2-20 

4-97 

289 

67-5 

153 

2-27 

4-96 

220 

68-5 

151 

2-20 

4-71 

116 

69-5 

156 

224 

4-64 

58 

70-5 

160 

2-27 

4-57 

The  same  index  may  be  used  as  a  measure  of  the  condition,  even 
of  the  quaUty,  of  an  animal;  three  Burmese  elephants  had  the 
following  heights,  weights,  and  reputations!: 


Height 

Weight 

WjH^ 

A 
B 
C 

7  ft.  lOi  in. 

8  1 
7       5 

7,511  lb. 

7,216 

4,756 

1-54 
1-36 
115 

A  famous  elephant 

A  good  elephant 

A  weak,  poor  elephant 

But  a  great  African  elephant,  10  ft.  10  in.  high,  weighed  14,640  Ib.J : 
whence  the  weight-height  coefficient  was  no  more  than  1-15.  That  is 
to  say,  the  African  elephant  is  considerably  taller  than  the  Indian, 
and  the  weight-height  ratio  is  correspondingly  less. 

Lastly,  by  means  of  the  same  index  we  may  judge,  to  a  first  rough 
approximation,  the  weight  of  a  large  animal  such  as  a  whale,  where 
weighing  is  out  of  the  question.  Sigurd  Rusting  has  given  us 
many  measurements,  and  many  foetal  weights,  from  the  Antarctic 
whale-fishery:  among  which,  choosing  at  random,  we  find  that  a 
certain  foetus  of  the  blue  whale,  or  Sibbald's  rorqual,  measured 
4  ft.  6  in.  long,  and  weighed  23  kilos,  or  say  46  lb.  A  whale  of  the 
same  kind,  45  ft.  long,  should  then  weigh  46  x  10^  lb.,  or  about 
23  tons;  and  one  of  90  ft.,  23  x  2^  tons,  or  over  180  tons.  Again 
in  seven  young  unborn  whales,  measuring  from  39  to  54  inches  and 
weighing  from  10  to  23  kilos,  the  mean  value  of  the  index  was  found 

*  Data  from  Sir  C.  Goringe,  The  English  Convict,  H.M.  Stationery  Office,  1913. 
See  also  J.  A,  Harris  and  others.  The,  Measurement  of  Man,  Minnesota,  1930,  p.  41. 
t  Data  from  A.  J.  Milroy,  On  the  management  of  elephants,  Shillong,  1921. 
%  D.  P.  Quireng,  in  Qrcrwth,  iii,  p.  9,  1939. 


204  THE  RATE  OF  GROWTH  [ch. 

to  be  15-2,  in  gramme-inches.     From  this  we  calculate  the  weight 
of  the  great  rorqual,  as  follows : 

1 5*2  X  SOO^ 

At  25  ft.,  or  300  inches,  W  = -— —  =  4,100,000  g. 

lUu 

=  4,100  kg. 

=  4  tons,  nearly. 

At    50  ft.,  TF  =  4  X  23  tons  =  32  tons. 

100  ft.        =  32  X  23  tons  =  256  tons. 

106  tons  (the  largest  known)  W  =  305  tons,  nearly. 

The  two  independent  estimates  are  in  close  agreement. 

Of  surface  and  volume 

While  the  weight-length  relation  is  of  especial  importance,  and 
is  wellnigh  fundamental  to  the  understanding  of  growth  and  form 
and  magnitude,  the  corresponding  relation  of  surface-area  to  weight 
or  volume  has  in  certain  cases  an  interest  of  its  own.  At  the  surface 
of  an  animal  heat  is  lost,  evaporation  takes  place,  and  oxygen  may 
be  taken  in,  all  in  due  proportion  as  near  as  may  be  to  the  bulk 
of  the  animal;  and  again  the  bird's  wing  is  a  surface,  the  area  of 
which  must  be  in  due  proportion  to  the  size  of  the  bird.  In  hollow 
organs,  such  as  heart  or  stomach,  area  is  the  important  thing  rather 
than  weight  or  mass;  and  we  have  seen  how  the  brain,  an  organ 
not  obviously  but  essentially  and  developmentally  hollow,  tends  to 
shew  its  due  proportions  when  reckoned  as  a  surface  in  comparison 
with  the  creature's  mass. 

Surface  cannot  keep  pace  with  increasing  volume  in  bodies  of 
similar  form;  wing-area  does  not  and  cannot  long  keep  pace  with 
the  bird's  increasing  bulk  and  weight,  and  this  is  enough  of  itself 
to  set  limits  to  the  size  of  the  flying  bird.  It  is  the  ratio  between 
square-root-of-surface  and  cube-root-of-volume  which  should,  in 
theory,  remain  constant;  but  as  a  matter  of  fact  this  ratio  varies 
(up  to  a  certain  extent)  with  the  circumstances,  and  in  the  case  of 
the  bird's  wing  with  varying  modes  and  capabilities  of  _flight.  The 
owl,  with  his  silent,  effortless  flight,  capable  of  short  swift  spurts 
of  attack,  has  the  largest  spread  of  wings  of  all ;  the  kite  outstrips 
the  other  hawks  in  spread  of  wing,  in  soaring,  and  perhaps  in  speed. 


Ill]  OF^SURFACE  AND  VOLUME  205 

Stork  and  seagull  have  a  great  expanse  of  wing;  but  other  skilled 
and  speedy  fliers  have  long  narrow  wings  rather  than  large  ones. 
The  peregrine  has  less  wing-area  than  the  goshawk  or  the  kestrel; 
the  swift  and  the  swallow  have  less  than  the  lark. 

Mean  ratio,  VS/^^^W,  between  wing-area  and  weight  of  birds 
{From  Mouillard's  data) 

Ratio 


Owls 

1  1 

species 

2-2 

Hawks 

7 

Gulls 

1 

Waders 

3 

Petrels 

2 

Plovers 

3 

Passeres 

4 

1-3 

Ducks 

2 

1-2 

To  measure  the  length  of  an  animal  is  easy,  to  weigh  it  is  easier  still,  but 
to  estimate  its  surface-area  is  another  thing.  Hence  we  know  but  little  of 
the  surface-weight  ratios  of  animals,  and  what  we  know  is  apt  to  be  uncertain 
and  discrepant.  Nevertheless,  such  data  as  we  possess  average  down  to  mean 
values  which  are  more  uniform  than  we  might  expect*. 

Mean  ratio,  VS/VW,  in  various  animals  [cm.  gm.  units) 


Ape 

11-8 

I             Sheep  (shorn) 

8 

Man 

11 

1             Snake 

12-5 

Dog 

10-11 

1             Frog 

10-6 

Cat,  horse 

10 

Birds 

10 

Rabbit 

9-75 

1             Tortoise 

10 

Cow,  pig,  rat 

9 

A  further  note  on  unequal  growth,  or  heterogony 

An  organism  is  so  complex  a  thing,  and  growth  so  complex  a 
phenomenon,  that  for  growth  to  be  so  uniform  and  constant  in  all 
the  parts  as  to  keep  the  whole  shape  unchanged  would  indeed 
be  an  unlikely  and  an  unusual  circumstance.  Rates  ''^ary,  propor- 
tions change,  and  the  whole  configuration  alters  accordingly.  In  so 
humble  a  creature  as  a  medusoid,.  manubrium  and  disc  grow  at 
different  rates,  and  certain  sectors  of  the  disc  faster  than  others, 
as  when  the  little  Ephyra-lfnya.  "develops"  into  the  great  Aurdia- 
jellyfish.     Many  fishes  grow  from  youth  to  age  with  no  visible, 

*  From  Fr.  G.  Benedict,  Oberflachenbestimmung  verschiedener  Tiergattungen, 
Ergebnisae  d.  Pkysiologiey  xxxvi,  pp.  300-346,  1934  (with  copious  bibliography). 


206  THE  RATE  OF  GROWTH  [ch. 

hardly  a  measurable,  change  of  form*;  but  the  shapes  and  looks 
of  man  and  woman  go  on  changing  long  after  the  growing  age  is 
over,  even  all  their  lives  long.  A  centipede  has  its  many  pairs  of 
legs  alike,  to  all  intents  and  purposes;  they  begin  alike  and  grow 
uniformly.  But  a  lobster  has  his  great  claws  a,nd  his  small,  his 
lesser  legs,  his  swimmerets  and  the  broad  flaps  of  his  tail ;  all  these 
begin  ahke,  and  diverse  rates  of  growth  make  up  the  difference 
between  them.  Moreover,  we  may  sometimes  watch  a  single  Umb 
growing  to  an  unusual  size,  perhaps  in  one  sex  and  not  in  the  other, 
perhaps  on  one  side  and  not  on  the  other  side  of  the  body:  such 
are  the  "horns,"  or  mandibles,  of  the  stag-beetle,  only  conspicuous 
in  the  male,  and  the  great  unsymmetrical  claws  of  the  lobster,  or 
of  that  extreme  case  the  httle  fiddler-crab  {Uca  pugnax).  For  such 
well-marked  cases  of  differential  growth-ratio  between  one  part  and 
another,  JuHan  Huxley  has  introduced  the  term  heterogonyf. 

Of  the  fiddler-crabs  some  four  hundred  males  were  weighed,  in 
twenty-five  graded  samples  all  nearly  of  a  size,  and  the  weights 
of  the  great  claw  and  of  the  rest  of  the  body  recorded  separately. 
To  begin  with  the  great  claw  was  about  8  per  cent.,  and  at  the  end 
about  38  per  cent.,  of  the  total  weight  of  the  unmutilated  body. 
In  the  female  the  claw  weighs  about  8  per  cent,  of  the  whole  from 
beginning  to  end;  and  this  contrast  marks  the  disproportionate, 
or  heterogonic,  rate  of  growth  in  the  male.  We  know  nothing 
about  the  actual  rate  of  growth  of  either  body  or  claw,  we  cannot 
plot  either  against  time;  but  we  know  the  relative  proportions,  or 
relative  rates  of  growth  of  the  two  parts  of  the  animal,  and  this  is 
all  that  matters  meanwhile.  In  Fig.  54,  we  have  set  off  the  successive 
weights  of  the  body  as  abscissae,  up  to  700  mgm.,  or  about  one-third 
of  its  weight  in  the  adult  animal;  and  the  ordinates  represent  the 
corresponding  weights  of  the  claw.  We  see  that  the  ratio  between 
the  two  magnitudes  follows  a  curve,  apparently  an  exponential 
curve;   it  does  in  fact  (as  Huxley  has  shewn)  follow  a  compound 

*  Cf,  S.  Hecht,  Form  and  growth  in  fishes,  Journ.  Morphology,  xxvii,  pp.  379- 
400,  1916;  F.  S.  and  D.  W.  Haramett,  Proportional  length-growth  of  garfish 
(Lepidosteus),  Growth,  in,  pp.  197-209,  1939. 

t  See  Problems  of  Relative  Growth,  1932,  and  many  papers  quoted  therein. 
The  term,  as  Huxley,  tells  us,  had  been  used  by  Pezard;  but  it  Jiad  been  used,  in 
another  sense,  by  Rolleston  long  before  to  mean  an  alternation  of  generations, 
or  production  of  offspring  dissimilar  to  the  parent. 


^73  )([M,j^  J 


III]  OF  UNEQUAL  GROWTH  207 

interest  law,  which  (calhng  y  and  x  the  weights  of  the  claw  and  of 
the  rest  of  the  body)  may  be  expressed  by  the  usual  formula  for 
compound  interest, 

y  =  bx^,  or  log  y  =  \ogb  +  k  log  x; 

and  the  coefficients  (6  and  Jc)  work  out  in  the  case  of  the  fiddler-crab, 
to  begin  with,  at 

2/  =  0-0073  xi-«2.      . 

300 


100        200        300        400        500       600        700 
Weight  of  body  (mgm.) 

Eig.  54.     Relative  weights  of  body  and  claw  in  the  fiddler- 
crab  { Uca  'pugnax). 

But  after  a  certain  age,  or  certain  size,  th^e  coefficients  no  longer 
hold,  and  new  coefficients  have  to  be  found.  Whether  or  no,  the 
formula  is  mathematical  rather  than  biological;  there  is  a  lack  of 
either  biological  or  physical  significance  in  a  growth-rate  which 
happens  to  stand,  during  part  of  an  animaFs  fife,  at  62  per  cent, 
compound  interest. 

Julian  Huxley  holds,  and  many  hoki  with  him,  that  the  exponential 
or  logarithmic  formula,  or  the  compound-interest  .law,  is  of  general 
application  to  cases  of  differential  growth-rates.  I  do  not  find  it  to 
be  so :  any  more  than  we  have  found  organ,  organism  or  population 
to  increase  by  compound  interest  or  geometrical  progression,  save 


208  THE  feATE  OF  GROWTH  [ch. 

under  exceptional  circumstances  and  in  transient  phase.  Undoubtedly 
many  of  Huxley's  instances  shew  increase  by  compound  interest, 
during  a  phase  of  rapid  and  unstinted  growth;  but  I  find  many 
others  following  a  simple-interest  rather  than  a  compound-interest 
law. 

Relative  weights  of  claw  and  body  in  fiddler-crabs  (Uca  pugnax). 
(Data  abbreviated  from  Huxley,  Problems  of  Relative  Growth, 
p.  12) 


rt.  of  body 

less  claw 

Wt.  of 

Ratio 

Wt.  of 

Wt.  of 

Ratio 

(mgm.) 

claw 

% 

body 

claw 

% 

58 

5 

8-6 

618 

243 

39-3 

80 

9 

11-2 

743 

319 

42-9 

109 

14 

12-8 

872 

418 

47-9 

156 

25 

160 

983 

461 

46-9 

200 

38 

190 

1080 

537 

49-7 

238 

53 

22-3 

1166 

594 

50-9 

270 

59 

21-9 

1212 

617 

50-9 

300 

78 

260 

1299 

670 

51-6 

355 

105 

29-7 

1363 

699 

51-3 

420 

135 

321 

1449 

773 

53-7 

470 

165 

351 

1808 

1009 

55-8 

536 

196 

36-6 

2233 

1380 

61-7 

In  the  common  stag-beetle  {Lv^anus  cervus)  we  have  the  following 
measurements  of  mandible  and  elytron  or  wing-case:  which  two 
organs  make  up  the  bulk  of,  and  may  for  our  purpose  be  held 
as  constituting,  the  "total  length"  of  the  beetle.  Here  a  simple 
equation  meets  the  case;  in  other  words,  the  length  of  elytron  or 
of  mandible  plotted  against  total  length  gives  what  is  to  all  intents 
and  purposes  a  straight  hne,  indicating  a  simple-interest  rather 
than  a  compound-interest  rate  of  increase. 

Measurements  of  iS  stag-beetles  (Lucanus  cervus)*  (mm.) 


Number  of  specimens 

1 

4 

5 

10 

5 

7 

11 

5 

Length,  total  (x) 

310 

38-7 

40-5 

42-6 

450 

46-9 

49-2 

53-6 

Length  of  elytron  (y) 

250 

30-9 

31-5 

32-6 

33-8 

351 

36-4 

39-2 

(     „      calculated)  (y') 

26-9 

30-8 

31-7 

32-8 

340 

350 

36-2 

38-5 

Length  of  mandible  (z) 

60 

7-8 

90 

100 

11-2 

11-9 

12-8 

14-4 

(     „     calculated)  (z') 

5-9 

7-7 

9-3 

101 

110 

11-7 

12-6 

14-2 

*  Data,  from  Julian  Huxley,  after  W.  Bateson  and  H.  H.  Brindley,  in  P.Z.S. 
1892,  pp.  585-594. 


Ill]   OF  UNEQUAL  GROWTH  OR  HETEROGONY  209 

From  the  observed  data  we  may  solve,  by  the  method  of  least 
squares,  the  simple  equations 

y  =  a  -{-  bx,     z=  c  +  dx, 

or  in  other  words,  find  the  equations  of  the  straight  lines  in  closest 
agreement  with  the  observed  data.     The  solutions  are  as  follows*: 

y  -  11-02  +  0-512^,   and  z^-  5-64  +  0-368x, 

the  two  coefficients  0-368  and  0-512  signifying  the  difference  between 
the  rates  *of  increase  of  the  two  organs.  The  number  of  samples  is 
not  very  large,  and  some  deviation  is  to  be  expected;  nevertheless, 
the  calculated  straight  lines  come  close  to  the  observed  values. 

40 


10 


20 


Fig. 


30  40  50 

Total  length  (mm.) 

55.     Relative  growth  of  body  and  mandible  in  reindeer-beetle 
{Cydommatus  tarandus). 


The  reindeer-beetle  {Cydommatus  tarandus),  belonging  to  the 
same  family,  shews  much  the  same  thing.  The  mandible  grows  in 
approximately  linear  ratio  to  the  body,  save  that  it  tends  to  be  at 
first  a  little  above,  and  later  on  a  little  below,  this  Unear  ratio 
(Fig.  55). 

Measurements  o/ Cydommatus  tarandus f  (mm.) 


Length  of  mandible  (y) 

3-9 

10-7 

^    141 

19-9 

24-0 

30-7 

34-5 

Total  length  (x) 

20-4 

331 

38-4 

47-3 

54-2 

66-1 

74-0 

Total  length  calculated: 
x  =  l7y  +  U-l 

20-3 

31-9 

37-7 

47-5 

54*5 

65-9 

72-4 

*  As  determined  for  me  by  Dr  A.  C.  Aitken,  F.R.S. 

t  Data,  much  abbreviated,  from  Huxley,  after  E.  Dudieh,  Archivf.  Naturgesch.  (A), 
1923. 


22-0 

48-3 

58-0 

73-5 

891 

102-0 

1120 

420 

65-3 

74-5 

85-5 

99-3 

112-6 

1200 

421 

65-2 

73-7 

87-4 

971 

112-5 

121-2 

210  THE  RATE  OF  GROWTH  [ch. 

The  facial  and  cranial  parts  of  a  dog's  skull  tend  to  grow  at 
different  rates  (Fig.  56) ;  and  changes  in  the  ratio  between  the  two 
go  a  long  way  to  explain  the  differences  in  shape  between  one  dog's 
skull  and  another's,  between  the  greyhound's  and  the  pug's.  But 
using  Huxley's  own  data  (after  Becher)  for  the  sheepdog,  I  find  the 
ratio  between  the  facial  and  cranial  portions  of  the  skull  to  be,  once 
again,  a  simple  linear  one. 

Measurements  of  skull  of  sheep-dog  (30  specimens)  *  {mm.) 

Mean  length  of  facial  region  (y) 

Mean  length  of  cranial  region- (x) 

Calculated  values  for  cranial 
region:  x  =  22-7 +0-88y 

And  now,  returning  to  the  fiddler-crab,  we  find  that  after  the 
crab  has  reached  a  certain  size  and  the  first  phase  of  rapid  growth 
is  over,  claw  and  body  grow  in  simple  linear  relation  to  one  another, 
and  the  heterogonic  or  compound-interest  formula  is  no  longer 
required: 

Fiddler-crab  (Uca  pugnax):  ratio  of  growth-rates ^  in  later  stages ^ 
of  claw  and  body  (mgm.) 

Weight  of  body  less      872      983       1080      1165      1212      1291       1363       1449 
claw  {x) 

Weight  of  large  418      461         537        594        617        670        699        778 

claw  {y) 

Do.,  calculated:  413       480        538        590        617        665        708         759 

y=0-6a;-110 

*  Data  from  A.  Becher,  in  Archivf.  Naturgesch.  (A),  1923;  see  Huxley,  Problems 
of  Relative  Growth,  p.  18,  and  Biol.  Centralbl.  loc.  cit.  Here,  and  in  the  previous 
case  of  Cydommatus,  the  equation  has  been  arrived  at  in  a  very  simple  way.  Take 
any  two  values,  x^,  x^,  and  the  corresponding  values,  yi,  y^.    Then  let 

x-Xj^  ^  y-yi 
^-^  Vz-Vx 
Of -65-3  v-48-3 


e.g. 

or 

from  which  a;  =  22-7+0-88y. 

We  may  with  advantage  repeat  this  process  with  other  values  of  x  and  y;  and 
take  the  mean  of  the  results  so  obtained. 


112-6-65-3      1020-48-3^ 

a; -653  _y-48-3 
47-3     "■     53-7    * 


TIT] 


OF  HETEROGONY 


211 


Once  again  we  find  close  agreement  between  the  observed  and 
calculated  values,  although  the  observations  are  somewhat  few  and 
the  equation  is  arrived  at  in  a  simple  way.  We  may  take  it  as 
proven  that  the  relation  between  the  two  growth-rates  is  essentially 
linear. 

A  compound-interest  law  of  growth  occurs,  as  Malthus  knew, 
in  cases,  and  at  times,  of  rapid  and  unrestricted  growth.  But 
unrestricted  growth  occurs  under  special  conditions  and  for  brief 


50  100 

Length  of  cranial  portion  (mm.) 


150 


Fig.  56.     Relative  growth  of  the  cranial  and  facial  portions  of  the 
skull  in  the  sheepdog.    Cf.  Huxley,  p.  18,  after  Becher, 


periods;  it  is  the  exception  rather  than  the  rule,  whether  in  a 
population  or  in  the  single  organism.  In  cases  of  diiferential 
growth  the  compound-interest  law  manifests  itself,  for  the  same 
reason,  when  one  of  the  two  growth-rates  is  rapid  and  "unre- 
stricted," and  when  the  discrepancy  between  the  two  growth-rates  is 
consequently  large,  for  instance  in  the  fiddler-crabs.  The  compound- 
interest  law  is  a  very  natural  mode  of  growth,  but  its  range  is 


212 


THE  RATE  OF  GROWTH 


[CH. 


limited.     A  linear  relation,  or  simple-interest  law,  seems  less  likely 
to  occur;   but  the  fact  is,  it  does  occur,  and  occurs  commonly. 

On  so-called  dimorphism 

In  a  well-known  paper,  Bateson  and  Brindley  shewed  that  among 
a  large  number  of  earwigs  collected  in  a  particular  locality,  the 
males  fell  into  two   groups,   characterised   by  large  or  by  small 


DD 


Fig.  57.     Tail-forceps  of  earwig.     From  Martin  Burr,  after  Willi  Kuhl. 
150 


^  100 


mm.  10 


Length  in  mm. 

Fig.  .58.     Variability  of  length  of  tail-forceps  in  a  sample  of  earwigs. 

After  Bateson  and  Brindley,  P.Z.S.  1892,  p.  588. 

tail-forceps  (Fig.  57),  with  few  instances  of  intermediate  magnitude*. 
This  distribution  into  two  groups,  according  to  magnitude,  is 
illustrated    in    the    accompanying    diagram    (Fig.    58);    and    the 

*  W.  Bateson  and  H.  H.  Brindley,  On  some  cases  of  variation  in  secondary 
sexual  characters  [Forficula,  Xylotrupa],  statistically  examined,  P.Z.S.  1892, 
pp.  585-594.  Cf.  D.  M.  Diakonow,  On  dimorphic  variability  of  Forficula,  Joum. 
Genet,  xv,  pp.  201-232,  1925;  and  Julian  Huxley,  The  bimodal  cephalic  horn  of 
Xylotrupa,  ibid,  xviii,  pp.  45-53,  1927. 


Ill 


OF  DIMORPHIC  GROWTH 


213 


phenomenon  was  described,  and  has  been  often  quoted,  as  one 
of  dimorphism  or  discontinuous  variation.  In  this  diagram  the 
time-element  does  not  appear;  but 'it  looks  as,  though  it  lay  close 
behind.  For  the  two  size-groups  into  which  the  tails  of  the  earwigs 
fall  look  curiously  hke  two  age-groups  such  as  we  have  already 
studied  in  a  fish,  where  the  ages  and  therefore  also  the  magnitudes  of  a 
random  sample  form  a  discontinuous  series  (Fig.  59).  And  if,  instead 
of  measuring  the  whole  length  of  our  fish,  we  had  confined  ourselves 
to  particular  parts,  .such  as'  head,  or  tail  or  fin,  we  should  have 
obtained  discontinuous  curves  of  distribution  for  the  magnitudes 


200 


150 


100 


Ocm.  \5  20  25  30 

Length  in  cm. 
Fig.  59.     Length  of  body  in  a  random  sample  of  plaice. 

of  these  organs,  just  as  for  the  whole  body  of  the  fish,  and  just  as 
for  the  tails  of  Bateson's  earwigs.  The  differences,  in  short,  with 
which  Bateson  was  dealing  were  a  question  of  magnitude,  and  it 
was  only  natural  to  refer  these  diverse  magnitudes  to  diversities  of 
growth;  that  is  to  say,  it  seemed  natural  to  suppose  that  in  this 
case  of  "dimorphism,"  the  tails  of  the  one  group  of  earwigs  (which 
Bateson  called  the  "high  males")  had  either  grown  faster,  or  had 
been  growing  for  a  longer  period  of  time,  than  those  of  the  "low 
males."  If  the  whole  random  sample  of  earwigs  were  of  one  and 
the  same  age,  the  dimorphism  would  appear  to  be  due  to  two 
alternative  values  for  the  mean  growth-rate,  individual  earwigs 
varying  around  one  mean  or  the  other.     If,  on  the  other  hand,  the 


214 


THE  RATE  OF  GROWTH 


[CH. 


two  groups  of  earwigs  were  of  diiferent  ages,  or  had  passed  through 
one  moult  more  or  less,  the  phenomenon  would  be  simple  indeed, 
and  there  would  be  no  more  to  be  said  about  it*.  Diakonow  made 
the  not  unimportant  observation  that  in  earwigs  living  in  un- 
favourable conditions  only  the  short-tailed  type  tended  to  appear. 
In  apparent  close  analogy  with  the  case  of  the  earwigs,  and  in 
apparent  corroboration  of  their  dimorphism  being  due  to  age, 
Fritz  Werner  measured  large  numbers  of  water-fleas,  all  apparently 
adult,  found  his  measurements  falling  into  groups  and  so  giving 
multimodal  curves.  The  several  cusps,  or  modes,  he  interpreted 
without  difficulty  as  indicating  diiferences  of  age,  or  the  number 
of  moults  which  the  creatures  had  passed  through f  (Fig.  BO). 


9 
/i.  60 


10    II    12   13   14  15    16    17   18   19  20  21    22  23  24 

80  100  120  140  160/1 

Length  in  ^ 

Fig.  60.     Measurements  of  the  dorsal  edge  in  a  population  of 
Chydorus  sphaericus,  a  water-flea. 


From  Fritz  Werner. 


An  apparently  analogous  but  more  difficult  case  is  that  of  a 
certain  little  beetle,  Onthofhagus  taurus,  which  bears  two  "horns" 
on  its  head,  of  variable  size  or  prominence.  Linnaeus  saw  in  it 
a  single  species,  Fabricius  saw  two;  and  the  question  long  remained 
an  open  one  among  the  eniomologists.  We  now  know  that  there 
are  two  "modes,"  two  predominant  sizes  in  a  continuous  range  of 

*  The  number  of  moults  is  known  to  be  variable  in  many  species  of  Orthoptera, 
and  even  occasionally  in  higher  insects;  and  how  the  number  of  moults  may  be 
influenced  by  hunger,  damp  or  cold  is  discussed  by  P.  P.  Calvert,  Proc.  Amer. 
Pkilos.  Soc.  Lxviii,  p.  246,  1929.  On  the  number  of  moults  in  earwigs,  see  E.  B. 
Worthington,  Entomologist,  1926,  and  W.  K.  Weyrauch,  Biol.  Centralbl.  1929, 
pp.  543-5^8. 

■f  Fritz  Werner,  Variationsanalytische  Untersuchungen  an  Chydoren,  Ztschr.  f. 
Morphologie  u.  Oekologie  d.  Tiere,  n,  pp.  58-188,  1924. 


Ill] 


OF  DIMORPHIC  GROWTH 


215 


variation*.  In  the  "complete  metamorphosis"  of  a  beetle  there 
is  no  room  for  a  moult  more  or  less,  and  the  reason  for  the  two 
modal  sizes  remains  hidden  (Fig.  61). 

But  new  hght  has  been  thrown  on  the  case  of  the  earwigs,  which 
may  help  to  explain  other  obscure  diversities  of  shape  and  size 
within  the  class  of  insects.  At  metamorphosis,  and  even  in  a  simple 
moult,  the  external  organs  of  an  insect  may  often  be  seen  to  unfold, 
as  do,  for  instance,  the  wings  of  a  butterfly;  they  then  quickly 
harden,  in  a  form  and  of  a  size  with  which  ordinary  gradual  growth 

I 


200 


Head  of  0.'tauru8;two 
forms  of  maJe 


^N 


A. 


0  0*5  1-0 

Length  of  horn  (mm.) 
Fig.  61.    Two  forms  of  the  male,  in  the  beetle  Ontkophagvs  taurus. 

has  had  nothing  directly  to  do.  This  is  a  very  peculiar  phenomenon, 
and  marks  a  singular  departure  from  the  usual  interdependence  of 
growth  and  form.  When  the  nymph,  or  larval  earwig,  is  about  to 
shed  its  skin  for  the  last  time,  the  tail-forceps,  still  soft  and  tender, 
are  folded  together  and  wrapped  in  a  sheath;  they  need  to  be 
distended,  or  inflated,  by  a  combined  pressure  of  the  body-fluid 
(or  haemolymph)  and  an  intake  of  respiratory  air.     If  all  goes  well, 

*  Rene  Paulian,  Bull.  Soc.  Zool.  Fr.  1933;    also  Le  polymorpMsme  des  mMea  de 
Cddoptires,  Paris,  1935,  p.  8. 


216  THE  RATE  OF  GROWTH  [ch. 

the  forceps  expand  to  their  full  size;  if  the  .reature  be  weak  or 
underfed,  inflation  is  incomplete  and  tho  tail-forceps  remain  small. 
In  either  case  it  is  an  affair  of  a  few  critical  moments  during  the 
final  ecdysis;  in  ten  minutes  or  less,  the  chitin  has  hardened,  and 
shape  and  size  change  no-  more.  Willi  Kuhl,  who  has  given  us  this 
interesting  explanation,  suggests  that  the  dimorphism  observed  by 
Bateson  and  by  Diakonow  is  not  an  essential  part  of  the  pheno- 
menon; he  has  found  it  in  one  instance,  but  in  other  and  much  larger 
samples  he  has  found  all  gradations,  but  only  a  single,-  well-marked 
unimodal  peak*. 

The  effect  of  temperature'f 

The  rates  of  growth  which  we  have  hitherto  dealt  with  are  mostly 
based  on  special  investigations,  conducted  under  particular  local 
conditions;  for  instance,  Quetelet's  data,  so  far  as  we  have  used 
them  to  illustrate  the  rate  of  growth  in  man,  are  drawn  from  his 
study  of  the  Belgian  people.  But  apart  from  that  "fortuitous" 
individual  variation  which  we  have  already  considered,  it  is  obvious 
that  the  normal  rate  of  growth  will  be  found  to  vary,  in  man  and 
in  other  animals,  just  as  the  average  stature  varies,  in  different 
locahties  and  in  different  "races."  This  phenomenon  is  a  very 
complex  one,  and  is  doubtless  a  resultant  of  many  undefined  con- 
tributory causes;  but  we  at  least  gain  something  in  regard  to  it 
when  we  discover  that  rate  of  growth  is  directly  affected  by 
temperature,  and  doubtless  by  other  physical  conditions.  Reaumur 
was  the  first  to  shew,  and  the  observation  was  repeated  by  Bonnet  {, 
that  the  rate  of  growth  or  development  of  the  chick  was  dependent 
on  temperature,  being  retarded  at  temperatures  below  and  somewhat 

*  Willi  Kuhl,  Die  Variabilitat  der  abdominalen  Korperanhange  bei  Forficula, 
Ztsch.  Morph.  u.  Oek.  d.  Tiere,  xii,  p.  299,  1924.  Cf.  Malcolm  Burr,  Discovery, 
1939,  pp.  340^345. 

t  The  temperature  limitations  of  life,  and  to  some  extent  of  growth,  are  sum- 
marised for  a  large  number  of  species  by  Davenport,  Exper.  Morphology,  cc.  viii, 
xviii,  and  by  Hans  Przibram,  Exp.  Zoologie,  iv,  c.  v.^ 

X  Reaumur,  L'art  de  faire  eclorre  et  elever  en  toute  saison  des  oiseaux  domestiques, 
soil  par  le  moyen  de  la  chaleur  du  fumee,  soil  par  le  moyen  de  celle  du  feu  ordinaire, 
Paris,  1749.  He  had  also  studied,  a  few  years  before,  the  effects  of  heat  and  cold 
on  growth-rate  and  duration  of  life  in  caterpillars  and  chrysalids:  Memoires,  ii, 
p.  1,  de  la  dnree  de  la  vie  des  crisalides  (1736).  See  also  his  Observations  du 
Thermometre,  etc.,  Mem.  Acad.,  Paris,  1735,  pp.  345-376. 


Ill]  THE  EFFECT'  OF  TEMPERATURE  217 

accelerated  at  temperatures  above  the  normal  temperature  of 
incubation,  that  is  to  say  the  temperature  of  the  sitting- hen.  In 
the  case  of  plants  the  fact  that  growth  is  greatly  affected  by  tem- 
perature is  a  matter  of  famiUar  knowledge;  the  subject  was  first 
carefully  studied  by  Alphonse  De  Candolle,  and  his  results  and  those 
of  his  followers  are  discussed  in  the  textbooks  of  botany*. 

That  temperature  is  only  one  of  the  climatic  factors  determining  growth  and 
yield  is  well  known  to  agriculturists;  and  a  method  of  "multiple  correlation" 
has  been  used  to  analyse  the  several  influences  of  temperature  and  of  rainfall 
at  different  seasons  on  the  future  yield  of  our  own  crops  f.  The  same  joint 
influence  can  be  recognised  in  the  bamboo;  for  it  is  said  (by  Lock)  that  the 
growth-rate  of  the  bamboo  in  Ceylon  is  proportional  to  the  humidity  of  the 
atmosphere,  and  again  (by  Shibata)  that  it  is  proportional  to  the  temperature 
in  Japan.  But  BlackmanJ  suggests  that  in  Ceylon  temperature  conditions 
are  all  that  can  be  desired,  but  moisture  is  apt  to  be  deficient,  while  in  Japan 
there  is  rain  in  abundance  but  the  average  temperature  is  somewhat  low: 
so  that  in  the  one  country  it  is  the  one  factor,  and  in  the  other  country  the 
other,  whose  variation  is  both  conspicuous  and  significant.  After  all,  it  is 
probably  rate  of  evaporation,  the  joint  result  of  temperature  and  humidity, 
which  is  the  crux  of  the  matter§.  "Climate"  is  a  subtle  thing,  and  includes 
a  sort  of  micro-meteorology.  A  sheltered  corner  has  a  climate  of  its  own;  one 
side  of  the  garden -wall  has  a  different  climate  to  the  other;  and  deep  in  the 
undergrowth  of  a  wood  celandine  and  anemone  enjoy  a  climate  many  degrees 
warmer  than  what  is  registered  on  the  screen ||. 

Among  the  mould-fungi  each  several  species  has  its  own  optimum  tempera- 
ture for  germination  and  growth.  At  this  optimum  temperature  growth  is 
further  accelerated  by  increase  of  humidity ;  and  the  further  we  depart  from 
the  optimum  temperature,  the  narrower  becomes  the  range  of  humidity  within 
which  growth  can  proceed^.  Entomologists  know,  in  like  manner,  how  over- 
abundance of  an  insect-pest  comes,  or  is  apt  to  come,  with  a  double  optimum 
of  temperature  and  humidity. 

*  Cf.  {int.  at.)  H.  de  Vries,  Materiaux  pour  la  connaissance  de  I'influence  de  la 
temperature  sur  les  plantes,  Arch.  Neerlandaises,  v,  pp.  385-401,  1870;  C.  Linsser, 
Periodische  Erscheinungen  des  Pflanzenlebens,  Mem.  Acad,  des  Sc,  8t  Petersbourg 
(7),  XI,  XII,  1867-69;  Koppen,  Warme  und  Pflanzenwachstum,  Bull.  Soc.  Imp. 
Nat.,  Moscou,  xliii,  pp.  41-110,  1871;  H.  Hoffmann,  Thermisehe  Vegetations- 
constanten,  Ztschr.  Oesterr.  Ges.  f.  Meteorologie,  xvii,  pp.  121-131,  1881;  Pheno- 
logische  Studien,  Meteorolog.  Ztschr.  iii,  pp.  H3-120,  1886. 

t  See  [int.  al.)  R.  H.  Hooker,  Journ.  Roy.  Statist.  Soc.  1907,  p.  70;  Journ.  Roy. 
Meteor.  Soc.  1922,  p.  46. 

I  F.  F.  Blackman,  Ann.  Bot.  xix,  p.  281,  1905. 

§   Szava-Kovatz,  in  Petermann's  Mitteilungen,  1927,  p.  7. 

II  Cf.  E.  J.  Salisbury,  On  the  oecological  aspects  of  Meteorology,  Q.J.R.  Meteorol. 
Soc.  July  1939. 

^  R.  G.  Tomkins,  Proc.  R.S.  (B),  cv,  pp.  375^01,  1929. 


218 


THE  RATE  OF  GROWTH 


[CH. 


The  annexed  diagram  (Fig.  62),  showing  growth  in  length  of  the 
roots  of  some  common  plants  at  various  temperatures,  is  a  sufl&cient 
illustration  of  the  phenomenon.  We  see  that  there  is  always  a 
certain  temperature  at  which  the  rate  is  a  maximum ;  while  on  either 
side  of  the  optimum  the  rate  falls  off,  after  the  fashion  of  the  normal 
curve  of  error.  We  see  further,  from  the  data  given  by  Sachs  and 
others,  that  the  optimum  is  very  much  the  same  for  all  the  common 
plants  of  our  own  chmate.    For  these  it  is  somewhere  about  26°  C. 


80 


70 


M^ie  16  20  22  24  26  28  30  32  34  36  38  40° 

Temp. 

Fig.  62.    Relation  of  rate  of  growth  to  temperature  in  certain 

plants.     From  Sachs's  data. 


(say  77°  F.),  or  about  the  temperature  of  a  warm  summer's  day; 
while  it  is  considerably  higher,  naturally,  in  such  plants  as  the  melon 
ot*  the  maize,  which  are  at  home  in  warmer  countries  than  our  own. 
The  bacteria  have,  in  like  manner,  their  various  optima,  and  some- 
times a  high  one.  The  tuberculosis-bacillus,  as  Koch  shewed,  only 
begins  to  grow  at  about  28°  C,  and  multipUes  most  rapidly  at 
37-38°,  the  body- temperature  of  its  host. 

The  setting  and  ripening  of  fruit  is  a  phase  of  growth  stiU  more 
dependent  on  temperature;  hence  it  is  a  "dehcate  test  of  climate," 
and  a  proof  of  its  constancy,  that  the  date-palm  grows  but  bears 


Ill]  THE  EFFECT  OF  TEMPERATURE  219 

no  fruit  in  Judaea,  and  the  vine  bears  freely  at  Eshcol,  but  not  in 
the  hotter  country  to  the  south*.  Shellfish  have  their  own  appro- 
priate spawning-temperatures;  it  needs  a  warm  summer  for  the 
oyster  to  shed  her  spat,  and  Hippopus  and  Tridacna,  the  great  clams 
of  the  coral-reefs,  only  do  so  when  the  water  has  reached  the  high 
temperature  of  30°  C.  For  brown  trout,  6°  C.  is  found  to  be  a 
critical  temperature,  a  minimum  short  of  which  they  do  not  grow 
at  all;  it  follows  that  in  a  Highland  burn  their  growth  is  at  a 
standstill  for  fully  half  the  yearf. 

That  a  rise  of  temperature  accelerates  growth  is  but  part  of  the 
story,  and  is  not  always  true.  Several  insects,  experimentally 
reared,  have  been  found  to  diminish  in  size  as  the  temperature 
increased  J;  and  certain  flies  have  been  found  to  be  larger  in  their 
winter  than  their  summer  broods.  The  common  copepod,  Calmius 
finmarchicus,  has  spring,  summer  and  autumn  broods,  which  (at 
Plymouth §)  are  large,  middle-sized  and  small;  but  the  large  spring 
brood  are  hatched  and  reared  in  the  cold  "winter"  water,  and  the 
small  autumn-winter  brood  in  the  warmest  water  of  the  year.  In 
the  cold  waters  of  Barents  Sea  Calanus  grows  larger  still;  of  an 
allied  genus,  a  large  species  lives  in  the  Antarctic,  a  small  one  in 
the  tropics,  a  middle-sized  is  common  in  the  temperate  oceans. 
The  large  size  of  many  Arctic  animals,  coelenterates  and  crustaceans, 
is  well  known;  and  so  is  that  of  many  tropical  forms,  Hke  Fungia 
among  the  corals,  or  the  great  Tritons  and  Tridacnas  among 
molluscs.  Another  common  phenomenon  is  the  increasing  number 
of  males  in  late  summer  and  autumn,  as  in  the  Rotifers  and  in  the 
above-mentioned  Calani.  All  these  things  seem  somehow  related 
to  temperature;  but  other  physical  conditions  enter  iilto  the  case, 
for  instance  the  amount  of  dissolved  oxygen  in  the  cold  waters,  and 
the  physical  chemistry  of  carbonate  of  Hme  in  the  warm||. 

The  vast  profusion  of  life,  both  great  and  small,  in  Arctic  seas,  the  multitude 
of  individuals  and  the  unusual  size  to  which  many  species  grow,  has  been 
often  ascribed  to  a  superabundance  of  dissolved  oxygen,  but  oxygen  alone 
would  not  go   far.     The   nutrient  salts,   nitrates   and  phosphates,   are   the 

*  Cf.  J.  W.  Gregory,  in  Geogr.  Journ.  1914,  and  Journ.  E.  Geogr.  Soc.  Oct.  1930. 

t  Cf.  C.  A.  Wingfield,  op.  cit.  supra,  p.  176. 

X  B.  P.  Uvarow,  Trans.  Ent.  Soc.  Lond.  lxxix,  p.  38,  1931. 

§   W.  H.  Golightly  and  LI.  Lloyd,  in  Nature,  July  22,  1939. 

li   Cf.  B.  G.  Bogorow  and  others,  in  the  Journ.  M.B.A.  xix,  1933-34. 


220  THE  RATE  OF  GROWTH  [ch. 

limiting  factor  in  the  growth  of  that  micro-vegetation  with  which  the  whole 
cycle  of  life  begins.  The  tropical  oceans  are  often  very  bare  of  these  salts; 
in  our  own  latitudes  there  is  none  too  much,  and  the  spring-growth  tends  to 
use  up  the  supply.  But  we  have  learned  from  the  Discovery  Expedition 
that  these  salts  are  so  abundant  in  the  Antarctic  that  plant-growth  is  never 
checked  for  stint  of  them.  Along  the  Chilean  coast  and  in  S.W.  Africa, 
cold  Antarctic  water  wells  up  from  below  the  warm  equatorial  current.  It 
is  ill-suited  for  the  growth  of  corals,  which  build  their  reefs  in  the  warmer 
waters  of  the  eastern  side ;  but  it  teems  with  nourishment,  breeds  a  plankton- 
fauna  of  the  richest  kind,  which  feeds  fishes  preyed  on  by  innumerable  birds, 
the  guano  of  which  is  sent  all  over  the  world.  Now  and  then  persistent  winds 
thrust  the  cold  current  aside ;  a  new  warm  current,  el  Nino  of  the  Chileans, 
upsets  the  old  equilibrium ;  the  fishes  die,  the  water  stinks,  the  birds  starve. 
The  same  thing  happens  also  at  Walfisch  Bay,  where  on  such  rare  occasions 
dead  fish  lie  piled  up  high  along  the  shore. 

It  is  curiously  characteristic  of  certain  physiological  reactions, 
growth  among  them,  to  be  affected  not  merely  by  the  temperature 
of  the  moment,  but  also  by  that  to  which  the  organism  has  been 
previously  and  temporarily  exposed.  In  other  words,  acclimatisation 
to  a  certain  temperature  may  continue  for  some  time  afterwards  to 
affect  all  the  temperature  relations  of  the  body*.  That  temporary 
c<!>ld  may,  under  certain  circumstances,  cause  a  subsequent  accelera- 
tion of  growth  is  made  use  of  in  the  remarkable  process  known  as 
vernalisation.  An  ingenious  man,  observing  that  a  winter  wheat  failed 
to  flower  when  sown  in  spring,  argued  that  exposure  to  the  cold  of 
winter  was  necessary  for  its  subsequent  rapid  growth ;  and  this  he 
verified  by  "  chiUing"  his  seedlings  for  a  month  to  near  freezing-point, 
after  which  they  grew  quickly,  and  flowered  at  the  same  time  as  the 
spring  wheat.  The  economic  advantages  are  great  of  so  shortening 
the  growing  period  of  a  crop  as  to  protect  it  from  autumn  frosts  in  a 
cold  chmate  or  summer  drought  in  a  hot  one ;  much  has  been  done, 
especially  by  Lysenko  in  Russia,  with  this  end  in  viewf. 

The  most  diverse  physiological  processes  may  be  afl'ected  by 
temperature.  A  great  astronomer  at  Mount, Wilson,  in  California, 
used  some  idle  hours  to  watch  the  "trail-running"  ants,  which  run 
all  night  and  all  day.  Their  speed  increases  so  regularly  with  the 
temperature  that  the  time  taken  to  run  30  cm.  suffices  to  tell  the 

*  Cf.    Kenneth   Mellanby,    On    temperature   coefficients    and    acclimatisation, 
Nature,  3  August  1940. 
t  Of.  {int.  al.)  V.  H.  Blackman,  in  Nature,  June  13,  1936. 


Ill]  THE  TEMPERATURE  COEFFICIENT  221 

temperature  to  1°  C. !  Of  two  allied  species,  one  ran  nearly  half  as 
fast  again  as  the  other,  at  the  same  temperature*. 

While  at  low  temperatures  growth  is  arrested  and  at  temperatures 
unduly  high  hfe  itself  becomes  impossible,  we  have  now  seen  that 
within  the  range  of  more  or  less  congenial  temperatures  growth 
proceeds  the  faster  the  higher  the  temperature.  The  same  is  true 
of  the  ordinary  reactions  of  chemistry,  and  here  Van't  Hoff  and 
Arrheniusf  have  shewn  that  a  definite  increase  in  the  velocity  of 
the  reaction  follows  a  definite  increase  of  temperature,  according  to 
an  exponential  law:  such  that,  for  an  interval  of  n  degrees  the 
velocity  varies  as  x",  x  being  called  the  "temperature  coefficient" 
for  the  reaction  in  question  J.  The  law  holds  good  throughout  a 
considerable  range,  but  is  departed  from  when  we  pass  beyond 
certain  normal  limits ;  moreover,  the  value  of  the  coefficient  is  found 
to  keep  to  a  certain  order  of  magnitude — somewhere  about  2  for 
a  temperature-interval  of  10°  C. — which  means  to  say  that,  the 
velocity  of  the  reaction  is  just  about  doubled,  more  or  less,  for  a 
rise  of  10°  C. 

This  law,  which  has  become  a  fundamental  principle  of  chemical 
mechanics,  is  applicable  (with  certain  qualifications)  to  the  pheno- 
mena of  vital  chemistry,  as  Van't  Hoff  himself  was  the  first  to  declare ; 
and  it  follows  that,  on  much  the  same  fines,  one  may  speak  of  a 
"temperature  coefficient"  of  growth.  At  the  same  time  we  must 
remember  that  there  is  a  very  important  difference  (though  we  need 
not  call  it  a  fundamental  one)  between  the  purely  physical  and  the 

*  Harlow  Shapley,  On  the  thermokinetics  of  Dolichoderine  ants,  Proc.  Nat. 
Acad.  Sci.  x,  pp.  436-439,  1924. 

t  Van't  HofF  and  Cohen,  Studien  zur  chemischen  Dynamik,  1896;  Sv.  Arrhenius, 
Ztschr.  f.  phys.  Chemie,  iv,  p.  226. 

X  For  various  instances  of  a  temperature  coefficient  in  physiological  processes, 
see  (e.g.)  Cohen,  Physical  Chemistry  f or ...  Biologists  (Enghsh  edition),  1903; 
Kanitz  and  Herzog  in  Zeitschr.  f.  Elektrochemie,  xi,  1905;  F.  F.  Blackman,  Ann. 
Bot.  XIX,  p.  281,  1905;  K.  Peter,  Arch.f.  Entw.  Mech.  xx,  p.  130,  1905;  Arrhenius, 
Ergebn.  d.  Physiol,  vii,  p.  480,  1908,  and  Quantitative  Laws  in  Biological  Chemistry, 
1915;  Krogh  in  Zeitschr.  f.allgem.  Physiologie,xyi, -pp.  163,178,1914;  James  Gray, 
Proc.  E.S.  (B),  xcv,  pp.  6-15,  1923;  W.  J.  Crozier,  many  papers  in  Journ.  Gen. 
Physiol.  1924;  J.  Belehradek,  in  Biol.  Reviews,  v,  pp.  1-29,  1930.  On  the  general 
subject,  see  E.  Janisch,  Temperaturabhangigkeit  biologischer  Vorgange  und  ihrer 
kurvenmassige  Analyse,  Pfluger's  Archiv,  ccix,  p.  414,  1925;  G.  and  P.  Hertwig, 
Regulation  von  Wachstum . .  .  durch  Umweltsfaktoren,  in  Hdb.  d.  normal,  u.  pathol. 
Physiologie,  xvi,  1930. 


222  THE  RATE  OF  GROWTH  [ch. 

physiological  phenomenon,  in  that  in  the  former  we  study  (or  seek 
and  profess  to  study)  one  thing  at  a  time,  while  in  the  living  body 
we  have  constantly  to  do  with  factors  which  interact  and  interfere; 
increase  in  the  one  case  (or  change  of  any  kind)  tends  to  be  con- 
tinuous, in  the  other  case  it  tends  to  be  brought,  or  to  bring 
itself,  to  arrest.  This  is  the  simple  meaning  of  that  Law  of 
Optimum,  laid  down  by  Errera  and  by  Sachs  as  a  general  principle 
of  physiology ;  namely  that  every  physiological  process  which  varies 
(Hke  growth  itself)  with  the  amount  or  intensity  of  some  external 
influence,  does  so  under  such  conditions  that  progressive  increase  is 
followed  by  progressive  decrease;  in  other  words,  the  function  has 
its  optimum  condition,  and  its  curve  shews  a  definite  maximu^n. 
In  the  case  of  temperature,  as  Jost  puts  it,  it  has  on  the  one  hand 
its  accelerating  effect,  which  tends  to  follow  Van't  Hoff's  law.  But 
it  has  also  another  and  a  cumulative  effect  upon  the  organism: 
"Sie  schadigt  oder  sie  ermiidet  ihn,  und  je  hoher  sie  steigt  desto 
rascher  macht  sie  die  Schadigung  geltend  und  desto  schneller  scbxeitet 
sie  voran*."  It  is  this  double  effect  of  temperature  on  the  organism 
which  gives,  or  helps  to  give  us  our  "optim'im"  curves,  which  (like 
all  other  curves  of  frequency  or  error)  are  the  expression,  not  of  a 
single  solitary  phenomenon,' but  of  a  more  or  less  complex  resultant. 
Moreover,  as  Blackman  and  others  have  pointed  out,  our  "optimum" 
temperature  is  ill-defined  until  we  take  account  also  of  the  duration 
of  our  experiment;  for  a  high  temperature  may  lead  to  a  short  but 
exhausting  spell  of  rapid  growth,  while  the  slower  rate  manifested 
at  a  lower  temperature  may  be  the  best  in  the  end.  The  mile  and 
the  hundred  yards  are  won  by  different  runners;  and  maximum 
rate  of  worldng,^  and  maximum  amount  of  work  done,  are  two  very 
different  things  f. 

In  the  case  of  maize,  a  certain  series  of  experiments  shewed  that 
the  growth  in  length  of  the  roots  varied  with  the  temperature  as 
follows  J: 

*  On  such  limiting  factors,  or  counter-reactions,  see  Putter,  Ztschr.  f.  aUgem. 
Physiologic,  xvi,  pp.  574-627,  1914. 

t  Cf.  L.  Errera,  UOptimum,  1896  (Recueil  d'oeuvres,  Physiologie  genirale,  pp.  338- 
368,  1910) ;  Sachs,  Physiologie  d.  Pflanzen,  1882,  p.  233;  PfeflFer,  Pflanzenphysiologie, 
n,  p.  78,  194;  and  cf,  Jost,  Ueber  die  Reactionsgeschwindigkeit  ira  Organismus, 
Biol.  CentraWl.  xxvi,  pp.  225-244,  1906. 

t  After  Koppen,  Bull.  Soc.  Nat.  Moscou,  XLin,  pp.  41-101,  1871. 


Ill]  THE  TEMPERATURE  COEFFICIENT  223 


Temperature 

Growth  in  48  hours 

°C. 

mm. 

18-0 

11 

23-5 

10-8 

26-6 

29-6 

23-5 

26-5 

30-2 

64-6 

33-5 

69-5 

36-5 

20-7 

us 

write  our 

formula  in 

the  form 

V(>^) 

=  x^,    or  In 

laF..._^- 

\o0V  =  nA 

Then  choosing  two  values  out  of  the  above  experimental  series 
(say  the  second  and  the  second-last),  we  have  t  =  23-5,  n  =  10, 
and  F,  V  =  10-8  and  69-5  respectively. 

.         ,.     -             log  69-5  -  log  10-8      , 
Accordmgly,  — ^ — — =  log  x, 

0-8414  -  0-034 
or ^^  =  0-0808, 

and  therefore  the  temperature-coefficient 

=  antilog  0-0808  =  1-204  (for  an  interval  of  1°  C). 

This  first  approximation  might  be  much  improved  by  taking  account 
of  all  the  experimental  values,  two  only  of  which  we  have  yet  made 
use  of;  but  even  as  it  is,  we  see  by  Fig.  63  that  it  is  in  very  fair 
accordance  with  the  actual  results  of  observation,  within  those 
particular  limits  of  temperature  to  which  the  experiment  is  confined. 
For  an  experiment  on  Lupinus  albus,  quoted  by  Asa  Gray* 
I  have  worked  out  the  corresponding  coefficient,  but  a  httle  more 
carefully.  Its  value  I  find  to  be  1-16,  or  very  nearly  identical  with 
that  we  have  just  found  for  the  maize;  and  the  correspondence 
between  the  calculated  curve  and  the  actual  observations  is  now 
a  close  one. 

Miss  I.  Leitch  has  made  careful  observations  of  the  rate  of  growth  of  rootlets 
of  the  Pea;  and  I  have  attempted  a  further  analysis  of  her  principal  resultsf . 

*  Asa  Gray,  Botany,  p.  387. 

t  I.  Leitch,  Some  experiments  on  the  influence  of  temperature  on  the  rate 
of  growth  in  Pisum  sativum,  Ann.  Bot.  xxx,  pp.  25-46,  1916,  especially  Table  III, 
p.  45.  Cf.  Priestley  and  Pearsall,  Growth  studies,  Ann.  Bot.  xxxvi,  pp.  224-249, 
1922. 


224 


THE  RATE  OF  GROWTH 


[CH. 


In  Fig.  64  are  shewn  the  mean  rates  of  growth  (based  on  about  a  hundred 
experiments)  at  some  thirty-four  different  temperatures  between  0-8°  and 
29-3°,  each  experiment  lasting  rather  less  than  twenty-four  hours.  Working 
out  the  mean  temperature  coefficient  for  a  great  many  combinations  of  these 
values,  I  obtain  a  value  of  1-092  per  C.°,  or  2-41  for  an  interval  of  10°,  and 
a  mean  value  for  the  whole  series  shewing  a  rate  of  growth  of  just  about 
1  mm.  per  hour  at  a  temperature  of  20°.  My  curve  in  Fig.  64  is  drawn  from 
these  determinations;  and  it  will  be  seen  that,  while  it  is  by  no  means  exact 
at  the  lower  temperatures,  and  will  fail  us  altogether  at  very  high  tem- 
peratures, yet  it  serves  as  a  satisfactory  guide  to  the  relations  between  rate 
and  temperature  within  the  ordinary  limits  of  healthy  growth.     Miss  Leitch 


18       20        22        24        26         28        30        32       34°C 
Fig.  63.     Relation  of  rate  of  growth  to  temperature  in  maize.     Observed 
values  (after  Koppen),  and  calculated  curve. 

holds  that  the  curve  is  not  a  Van't  Hoff  curve ;  and  this,  in  strict  accuracy, 
we  need  not  dispute.  But  the  phenomenon  seems  to  me  to  be  one  into  which 
the  Van't  Hoff  ratio  enters  largely,  though  doubtless  combine'd  with  other 
factors  which  we  cannot  determine  or  eliminate. 

While  the  above  results  conform  fairly  well  to  the  law  of  the 
temperature-coefficient,  it  is  evident  that  the  imbibition  of  water 
plays  so  large  a  part  in  the  process  of  elongation  of  the  root  or 
stem  that  the  phenomenon  is  as  much  or  more  a  physical  than  a 
chemical  one:  and  on  this  account,  as  Blackman  has  remarked,  the 
data  commonly  given  for  the  rate  of  growth  in  plants  are  apt  to 
be    irregular,    and    sometimes    misleading*.     We    have    abundant 

*  F.  F.  Blackman,  Presidential  Address  in  Botany,  Brit.  Assoc.  Dublin,  1908. 


Ill] 


OF  TEMPERATURE  COEFFICIENTS 


225 


illustrations,  however,  among  animals,  in  which  we  may  study  the 
temperature-coefficient  under  circumstances  where,  though  the 
phenomenon  is  always  complicated,  true  metabohc  growth  or 
chemical  combination  plays  a  larger  role.  Thus  Mile.  Maltaux  and 
Professor  Massart*  have  studied  the  rate  of  division  in  a  certain 
flagellate,  Chilomonas  paramoecium,  and  found  the  process  to  take 
20 


/• 

— 

/ 

.7 

■~ 

7 

/ 

/ 

/ 

_ 

/ 

V 

- 

. 

/ 

/ 

— 

/ 

'/. 

/ 

/ 

1 — 

• 

/ 

/ 

^ 

— 

^i 

^^ 

• 

^ 

* 

- 

• 

^ 

y 

-^ 

• 
* 

"" 

per 
hour 

20 


1-8 

1-6 

1-4 

1-2 

1-0 

•8 

•6 

•4 

•2 


0  4°  8°        1 2°       1 6°       20°       24°  ^      28°      32°  ^ 

Fig,  64.     Relation  of  rate  of  growth  to  temperature  in  rootlets  of 
pea.     From  Miss  I.  Leitch's  data. 

29  minutes  at  15°  C,  12  at  25°,  and  only  5  minutes  at  35°  C.  These 
velocities  are  in  the  ratio  of  1 :  24:  5-76,  which  ratio  corresponds 
precisely  to  a  temperature-coefficient  of  24  for  each  rise  of  10°,  or 
about  1-092  for  each  degree  centigrade,  precisely  the  'Same  as  we 
have  found  for  the  growth  of  the  pea. 

By  means  of  this  principle  we  may  sometimes  throw  hght  on 
apparently  compUcated  experiments.     For  instance.  Fig.  65  is  an 

*  Rec.  de  VInst.  Bot.  de  Brzizelles,  vi,  1906. 


226 


THE  RATE  OF  GROWTH 


[CH. 


illustration,  which  has  been  often  copied,  of  0.  Hertwig's  work  on 
the  effect  of  temperature  on  the  rate  of  development  of  the  tadpole*. 


'1          ~fZi 

Z                   ^4^ 

1                   1       1 

/                    /         1 

~/      ~X  X- 

T_                 J-       L 

«' 

/                       /         1 

1                       I       1 

1            ill 

01      «^       .... 

-          t    1    14^ 

"^ 

-     4  -T  ttU- 

.    I  2^    riii 

.      t^'^     rtlf 

'^ 

v^      IJ  LE 

-  z^^     Tti-l^t 

M 

R  1  J^iItX- 

»^ 

VJV    ptj 

^  ^   t     Qj   3 

-l 

f-  /-    ziy  i 

f~ 

/^^'^  z  it-,/ 

"                                                 JJ- 

-i^'^-^Z   Jl   J 

\                                           JV    . 

t-  z  ^^zt'^ 

\  .,o^j>       y^  L/. 

%^^yjLt 

£^^^   V 

^        / 

^^^-""^^^^^^     .-^ 

>^  ""^^ 

'^'^^  ^c^^  -^^  ^^^ 

^^^^^^^^^     --' 

-■^ 

^  —  "  ^ — ::h^  "^  "^ 

j                                         TemneXcLl 

uie     Ceatign.a.d.e 

«*•  «•  2^"  «/• 


/*•  //"  It,'  I-.'  IV  li"  If  II'  lof  9"   a"    ^   ef   s' 


Fig.  65.  Diagram  shewing  time  taken  (in  days),  at  various  temperatures  {°  C), 
to  reach  certain  stages  of  development  in  the  frog:  viz.  I,  gastrula;  II, 
medullary  plate;  III,  closure  of  medullary  folds;  IV,  tail- bud;  V,  tail  and 
gills;  VI,  tail-fin;  VII,  operculum  beginning;  VIII,  do.  closing;  IX,  first 
appearance  of  hind-legs.     From  Jenkinson,  after  0.  Hertwig,  1898. 

*  0.  Hertwig,  Einfluss  der  Temperatur  auf  die  Entwicklung  von  Rana  fusca 
und  R.  esculenta,  Arch.  f.  mikrosk.  Anat.  li,  p.  319,  1898.  Cf.  also  K.  Bialaszewicz, 
Beitrage  z.  Kenntniss  d.  Wachsthumsvorgange  bei  Amphibienembryonen,  Ball. 
Acad.  Sci.  de  Cracovie,  p.  783,  1908;  Abstr.  in  Arch.  f.  Entwicklungsmech.  xxviii, 
p.  160,  1909:  from  which  Ernst  Cohen  determined  the  value  of  Q^q  (Vortrdge  iib. 
physikal.  C hemic  f.  Arzte,  1901;  English  edit.  1903). 


Ill] 


OF  TEMPERATURE  COEFFICIENTS 


227 


From  inspection  of  this  diagram,  we  see  that  the  time  taken  to 
attain  certain  stages  of  development  (denoted  by  the  numbers 
III-VII)  was  as  follows,  at  20°  and  at  10°  C,  respectively. 


At  20°  C. 

At  10°  C. 

Stage  III 

2-0 

6-5  days 

„      IV 

2-7 

8-1     „ 

„      V 

30 

10-7     „ 

;,  VI 

4-0 

13-5     „ 

„      VII 

50 

16-8     ,. 

Total 


16-7 


55-6 


25'C.        20°  15°  10°  5' 

Fig.  66.     Calculated  values,  corresponding  to  preceding  figure. 

That  is  to  say,  the  time  taken  to  produce  a  given  result  at  10° 
was  (on  the  average)  somewhere  about  55-6/16-7,  or  3-33,  times  as 
long  as  was  required  at  20°  C. 

We  may  then  put  our  equation  in  the  simple  form, 

x^^  =  3-33. 
Or,  10  log  X  =  log  3-33  =  0-52244. 

Therefore  log  x  =  0-05224, 

and  X  =  1-128. 


228  THE  RATE  OF  GROWTH  [ch. 

That  is  to  say,  between  the  intervals  of  10°  and  20°  C,  if  it  take 
m  days,  at  a  certain  given  temperature,  for  a  certain  stage  of 
development  to  be  attained,  it  will  take  m  x  1-128"  days,  when  the 
temperature  is  n  degrees  less,  for  the  same  stage  to  be  arrived  at. 

Fig.  66  is  calculated  throughout  from  this  value;  and  it  will  be 
found  extremely  concordant  with  the  original  diagram,  as  regards 
all  the  stages  of  development  and  the  whole  range  of  temperatures 
shewn;  in  spite  of  the  fact  that  the  coefficient  on  which  it  is  based 
was  derived  by  an  easy  method  from  a  very  few  points  on  the 
original  curves.  In  hke  manner,  the  following  table  shews  the 
"incubation  period"  for  trout-eggs,  or  interval  between  fertihsation 
and  hatching,  at  different  temperatures  * : 

Incubation-period  of  trout-eggs 


Temperature 

Days'  interval 

°C. 

before  hatching 

2-8 

165 

3-6 

135 

3-9 

121 

4-5 

109 

50 

103 

5-7 

96 

6-3 

89 

6-6 

81 

7-3 

73 

8-0 

65 

90 

56 

10-0 

47 

111 

38 

12-2 

32 

Choosing  at  random  a  pair  of  observations,  viz.  at  3-6°  and  10°, 
and  proceeding  as  before,  we  have 

10°  -  3-6°  =  64°. 

Then  (64)  =  '-g, 

or  6-4  X  log  X  =  log  135  —  log  47 

=  2-1303  -  1-6721  =  0-4582 
and  log  X  =  0-4582  -  6-4       =  0-0716, 

X  =  1-179. 

*  Data  from  James  Gray,  The  growth  of  fish,  Journ.  Exper.  Biology,  vi,  p.  126, 
1928. 


Ill]  OF  TEMPERATURE  COEFFICIENTS  229 

Using  three  other  pairs  of  observations,  we  have  the  following 
concordant  results : 

At  12-2°  and  2-8^,  x  -  M91 

10-0°         3-6°  M79 

9-0°         5-7°;         M78 

8-0°         5-0°  M65 

Mean  M8 

A  very  curious  point  is  that  (as  Gray  tells  us)  the  young  fish  which 
have  hatched  slowly  at  a  low  temperature  are  bigger  than  those 
whose  growth  has  been  hastened  by  warmth. 

Again,  plaice-eggs  were  found  to  hatch  and  grow  to  a  certain 
length  (4-6  mm.),  as  follows*: 

Temperature  (°  C.)  Days 

41  230 

6-1  181 

8-0  13-3 

10-1  10-3 

120  8-3 

From  these  we  obtain,  as  before,  the  following  constants: 

At  12°     and  8°,    x  =  M3 

12°  4-1°         M4 

10-1°         6-1°         M5 

8-0°         4-1°         M5 

Mean  M4 

The  value  of  x  is  much  the  same  for  the  one  fish  as  for  the  other. 

Karl  Peter  t,  experimenting  on  echinoderm  eggs,  and  making  use 
also  of  Richard  Hert wig's  experiments  on  young  tadpoles,  gives  the 
temperature-coefficients  for  intervals  of  10°  C.  (commonly  written 
Qio)  as  follows,  to  which  I  have  added  the  corresponding  values 
forg,: 

Sphaerechinus    Qiq  =2-15    Qi^  1-08 
Echinus  ?'13  1-08 

Rana  2-86  Ml 

*  Data  from  A.  C.  Johansen  and  A.  Krogh,  Influence  of  temperature,  etc., 
Publ.  de  Circonstance,  No.  68,  1914.  The  function  is  here  said  to  be  a  linear  one — 
which  would  have  been  an  anomalous  and  unlikely  thing". 

t  Der  Grad  der  Beschleunigung  tierischer  Entwicklung  durch  erhohte  Tem- 
peratur,  Arch.  f.  Entw.  Mech.  xx,  p.  130,  1905.  More  recently  Bialaszewicz  has 
determined  the  coefficient  for  the  rate  of  segmentation  in  Rana  as  being  2-4  per  10°  C. 


230  THE  RATE  OF  GROWTH  fcH. 

These  values  are  not  only  concordant,  but  are  of  the  same  order 
of  magnitude  as  the  temperature-coefficient  in  ordinary  chemical 
reactions.  Peter  has  also  discovered  the  interesting  fact  that  the 
temperature-coefficient  alters  with  age,  usually  but  not  always 
decreasing  as  time  goes  on*: 

Sphaerechinus    Segmentation    Q^q  =2-29    Q^  =  1-09 
Later  stages  2-03  1-07 

Echinus  Segmentation  2-30  1-09 

Later  stages  2-08  1-08 

Rana  Segmentation  2-23  1-08 

Later  stages  3-34  1-13 

Furthermore,  the  temperature-coefficient  varies  with  the  tem- 
perature itself,  falhng  as  the  temperature  rises — a  rule  which  Van't 
Hoff  shewed  to  hold  in  ordinary  chemical  operations.  Thus  in  Rana 
the  temperature-coefficient  (Qiq)  at  low  temperatures  may  be  as 
high  as  5-6;  which  is  just  another  way  of  saying  that  at  low 
temperatures  development  is  exceptionally  retarded. 

As  the  several  stages  of  development  are  accelerated  by  warmth, 
so  is  the  duration  of  each  and  all,  and  of  life  itself,  proportionately 
curtailed.  The  span  of  life  itself  may  have  its  temperature- 
coefficient — in  so  far  as  Life  is  a  chemical  process,  and  Death  a 
chemical  result.  In  hot  climates  puberty  comes  early,  and  old  age 
(at  least  in  women)  follows  soon;  fishes  grow  faster  and  spawn 
earlier  in  the  Mediterranean  than  in  the  North  Sea.  Jacques  Loeb  f 
found  (in  complete  agreement  with  the  general  case)  that  the  larval 
stages  of  a  fly  are  abbreviated  by  rise  of  temperature;  that  the 
mean  duration  of  life  at  various  temperatures  can  be  expressed  by 
a  temperature-coefficient  of  the  usual  order  of  magnitude ;  that  this 
coefficient  tends,  as  usual,  to  fall  as  the  temperature  rises;  and 
lastly — what  is  not  a  little  curious — ^that  the  coefficient  is  very  much 
the  same,  in  fact  all  but  identical,  for  the  larva,  pupa  and  imago  of 
the  fly. 

*  The  diflferences  are,  after  all,  of  small  order  of  magnitude,  as  is  all  the  better 
seen  when  we  reduce  the  ten-degree  to  one-degree  coefficients. 

t  J-  Loeb  and  Northrop,  On  the  influence  of  food  and  temperature  upon  the 
duration  of  life,  Journ.  Biol  Chemistry,  xxxii,  pp.  103-121,  1917. 


Ill 


OF  TEMPERATURE  COEFFICIENTS 


231 


Temperature-coefficients  (Q^q)  of  Drosophila 


Larva 

Pupa 

Imago 

15-20°  C. 

115 

117 

118 

20-25°  C. 

106 

1-08 

107 

And  Japanese  students,  studying  a  little  fresh- water  crustacean,  have 
carried  the  experiment  much  beyond  the  range  of  Van't  Hoff 's  law, 
and  have  found  length  of  hfe  to  rise  rapidly  to  a  maximum  at  about 
13-14°  C,  and  to  fall  slowly,  in  a  skew  curve,  thereafter*  (Fig.  67). 


20  30  40 

Temperature,  °C. 
Fig.  67.     Length  of  life,  at  various  temperatures,  in  a  water-flea. 

If  we  now  summarise  the  various  temperature-coefficients  (Q^) 
which  we  have  happened  to  consider,  we  are  struck  by  their 
remarkably  close  agreement: 

Yeast  Qi  =  M3 


Lupin 

M6 

Maize 

1-20 

Pea 

1-09 

Echinoids 

1-08 

Drosophila  (mean)" 

M2 

Frog,  segmentation 

1-08 

„     tadpole 

M3 

Mean 

M2 

*  A.  Terao  and  T.  Tabaka,  Duration  of  life  in  a  water-flea,  Moina  sp.;   Joum. 
Imp.  Fisheries  Inst.,  Tokyo,  xxv,  No.  3,  March  1930. 


232  THE  RATE  OF  GROWTH  [ch. 

The  constancy  of  these  results  might  tempt  us  to  look  on  the 
phenomenon  as  a  simple  one,  though  we  well  know  it  to  be  highly 
complex.  But  we  had  better  rest  content  to  see,  as  Arrhenius  saw 
in  the  beginning,  a  general  resemblance  rather  than  an  identity 
between  the  temperature-coefficients  in  physico-chemical  and 
biological  processes*. 

It  was  seen  from  the  first  that  to  extend  Van't  Hoff's  law  from  physical 
chemistry  to  physiology  was  a  bold  assumption,  to  all  appearance  largely 
justified,  but  always  subject  to  severe  and  cautious  limitations.  If  it  seemed 
to  simplify  certain  organic  phenomena,  further  study  soon  shewed  how  far 
from  simple  these  phenomena  were.  Living  matter  is  always  heterogeneous, 
and  from  one  phase  to  another  its  reactions  change;  the  temperature- 
coefficient  varies  likewise,  and  indicates  at  the  best  a  summation,  or  integration, 
of  phenomena.  Nevertheless,  attempts  have  been  made  to  go  a  little  further 
towards  a  physical  explanation  of  the  physiological  coefficient.  Van't  Hoff 
suggested  a  viscosity-correction  for  the  temperature -coefficient  even  of  an 
ordinary  chemical  reaction;  the  viscosity  of  protoplasm  varies  in  a  marked 
degree,  inversely  with  the  temperature,  and  the  viscosity-factor  goes,  perhaps, 
a  long  way  to  account  for  the  aberrations  of  the  temperature-coefficient.  It 
has  even  been  suggested  (by  Belehradekf)  that  the  temperature- coefficients 
of  the  biologist  are  merely  those  of  protoplasmic  viscosity.  For  instance,  the 
temperature-coefficients  of  mitotic  cell-division  have  been  shewn  to  alter 
from  one  phase  to  another  of  the  mitotic  process,  being  much  greater  at  the 
start  than  at  the  end| ;  and  so,  precisely,  has  it  been  shewn  that  protoplasmic 
viscosity  is  high  at  the  beginning  and  low  at  the  end  of  the  mitotic  process  §. 

On  seasonal  growth 

There  is  abundant  evidence  in  certain  fishes,  such  as  plaice  and 
haddock,  that  the  ascending  curve  of  growth  is  subject  to  seasonal 
fluctuations  or  interruptions,  the  rate  during  the  winter  months 
bejng  always  slower  than  in  the  months  of  summer.  Thus  the 
Newfoundland  cod  have  their  maximum  growth-rate  in  June,  and 
in  January-February  they  cease  to  grow;  it  is  as  though  we  super- 
imposed a  periodic  annual  sine-curve  upon  the  continuous  curve  of 
growth.  Furthermore,  as  growth  itself  grows  less  and  less  from 
year  to  year,  so  will  the  difference  between  the  summer  and  the 

♦  Cf.  L.  V.  Heilbronn,  Science,  lxii,  p.  268,  1925. 
t  J.  Belehradek,  in  Biol.  Reviews,  v,  pp.  30-58,  1930. 

X  Cf.  E.  Faure-Fremiet,  La  cinetique  du  developpement,  1925;  also  B.  Ephrussi, 
C.R.  cLxxxii,  p.  810,  1926. 

§   See  {int.  al.)  L.  V.  Heilbronn,  The  Colloid  Chemistry  of  Protoplasm,  1928. 


Ill] 


OF  SEASONAL  GROWTH 


233 


winter  rates  grow  less  and  less.  The  fluctuation  in  rate  represents 
a  vibration  which  is  gradually  dying  out;  the  amplitude  of  the 
sine-curve  diminishes  till  it  disappears;  in  short  our  phenomenon 
is  simply  expressed  by  what  is  known  as  a  "damped  sine-curve*." 

Growth  in  height  of  German  military  cadets,  in  half-yearly  periods 

Increment  (em.) 


Height  (cm. 

) 

f 

A 

■> 

Number 
observed 

, 

^ 

Winter 
^-year 

Summer 
^-year 

Age 

October 

April 

October 

Year 

12 

11-12 

139-4 

141-0 

143-3 

1-6 

2-3 

3-9 

80 

12-13 

143-0 

144-5 

1474 

1-5 

2-9 

4-4 

146 

13-14 

147-5 

149-5 

152-5 

2-0 

3-0 

5-0 

162 

14-1.-) 

152-2 

155-0 

158-5 

2-8 

3-5 

6-3 

162 

15-16 

158-5 

160-8 

163-8 

2-3 

30 

5-3 

150 

16-17 

163-5 

165-4 

167-7 

1-9 

2-3 

4-2 

82 

17-18 

167-7 

168-9 

170-4 

1-2 

1-5 

2-7 

22 

18-19 

169-8 

170-6 

171-5 

0-8 

0-9 

1-7 

6 

19-20 

170-7 

171-1 

171-5 
Mean 

0-4 
1-6 

0-4 
2-2 

0-8 

cm.  4 


years 

Fig.  68.    Half-yearly  increments  of  growth,  in  cadets  of  various  ages. 
From  Daffner's  data. 

The  same  thing  occurs  in  man,  though  neither  in  his  case  nor  in 
that  of  the  fish  have  we  sufficient  data  for  its  complete  illustration. 
We  can  demonstrate  the  fact,  however,  by  help  of  certain  measure- 
ments of  the  height  of  German  cadets,  measured  at  half-yearly 
intervals  t-  In  the  accompanying  diagram  (Fig.  68)  the  half-yearly 
increments  are  set  forth  from  the  above  table,  and  it  will  be  seen 


*  The  scales,  on  the  other  hand,  make  most  of  their  growth  during  the  int/er- 
mediate  seasons:  and  with  this  peculiarity,  that  a  few  broad  zones  are  added  to 
the  scale  in  spring,  and  a  larger  number  of  narrow  circuli  in  autumn :  see  Contrib. 
to  Canadian  Biology,  iv,  pp.  289-305,  1929;  Ben  Dawes,  Growth... in  plaice, 
Journ.  M.B.A.  xvii,  pp.  103-174,  1930. 

t  From  Daffner,  Da^  Wachstum  des  Menschen^  p.  329,  1902. 


234 


THE  RATE  OF  GROWTH 


CH. 


that  they  form  two  even  and  entirely  separate  series.  Danish  school- 
boys show  just  the  same  periodicity  of  growth  in  stature. 

The  seasonal  effect  on  visible  growth-rate  is  much  alike  in  fishes 
and  in  man,  in  spite  of  the  fact  that  the  bodily  temperature  of  the  one 
varies  with  the  milieu  externe  and  that  of  the  other  keeps  constant 
to  within  a  fraction  of  a  degree. 

While  temperature  is  the  dominant  cause,  it  is  not  the  only  cause 
of  seasonal  fluctuations  of  growth;  for  alternate  scarcity  and 
abundance  of  food  is  often,  as  in  herbivorous  animals,  the  ostensible 


18 


10 


a 
o 

S 


/   •^A 

-1 1 1 J 1 1 X 

/          /         M 

/             \ 

/        /               '^ 

\                                       I          \ 

/        / 

V                        /        \ 

/         / 
/          / 

\\                /       \ 

y    / 

\\                         /    Steers\ 

/ 

\ 

/ 

\  \                  /                  \ 

v.^ 

y 

\     \                /                      \ 

1 

1         1         1        1 

\              /                HeiVfers 

\ 
\ 
1          1          I.I          t         1         .IN 

JUN  AUG  OCT  DEC  FEB  APR  dUN  AUG  OCT  DEC  FEB  APR 

15  20  25  30  35 

Age  in  months 

Fig.  69.     Seasonal  growth  of  S.  African  cattle:  Sussex  half-breeds. 
After  Schiitte. 


reason.  Before  turnips  came  into  cultivation  in  the  eighteenth 
century  our  own  cattle  starved  for  half  the  year  and  grew  fat  the 
other,  and  in  many  countries  the  same  thing  happens  still.  In 
South  Africa  the  rainy  season  lasts  from  November  to  February; 
by  January  the  grass  is  plentiful,  by  June  or  July  the  veldt  is 
parched  until  rain  comes  again.  Cattle  fatten  from  January  to 
March  or  April;  from  July  to  October  they  put  on  Httle  weight, 
or  lose  weight  rather  than  put  it  on*. 

*  Cf.  D.  J.  Schiitte,  in  Onderstepoort  Journal,  Oct.  1935. 


J 


Ill]  OF  THE  GROWTH  OF  TREES  2^5 

The  growth  of  trees 

Some  sixty  years  ago  Sir  Robert  Christison,  a  learned  and  versatile 
Edinburgh  professor,  was  the  first  to  study  the  "exact  measurement" 
of  the  girth  of  trees*;  and  his  way  of  putting  a  girdle  round  the 
tree,  and  fitting  a  recording  device  to  the  girdle,  is  copied  in  the 
"  dendrographs  "  t  used  in  forestry  today.  The  Edinburgh  beeches 
begin  to  enlarge  their  trunks  in  late  May  or  June,  when  in  full  leaf, 
and  cease  growing  some  three  months  later;'  the  buds  sprout  and 
the  leaves  begin  their  work  before  the  cambium  wakens  to  activity. 
The  beech-trees  in  Maryland  do  likewise,  save  that  the  dates  are 
a  little  earlier  in  the  year;  and  walnut-trees  on  high  ground  in 
Arizona  shew  a  like  short  season  of  growth,  differing  somewhat  in 
date  or  "phase,"  just  as  it  did  in  Edinburgh,  from  one  year  to 
another. 

Deciduous  trees  stop  growing  after  the  fall  of  the  leaf,  but  ever- 
greens grow  all  the  year  round,  more  or  less.  This  broad  fact  is 
illustrated  in  the  following  table,  which  happens  to  relate  to  the 

Mean  monthly  increase  in  girth  of  trees  at  San  Jorge,  Uruguay :  from 
C.  E.  HalVs  data.  Values  given  in  ^percentages  of  total  annual 
increment  J 


Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov.    Dec. 

Evergreens 

91 

8-8 

8-6 

8-9 

7-7 

5-4 

4-3 

6-0 

9-1 

Ill 

10-8  10-2 

Deciduous  trees 

20-3 

14-6 

9-0 

2-3 

0-8 

0-3 

0-7 

13 

3-5 

9-9 

16-7  210 

southern  hemisphere,  and  to  the  climate  of  Uruguay.  The  measure- 
ments taken  were  those  of  the  girth  of  the  tree,  in  mm.,  at  three 
feet  from  the  ground.  The  evergreens  included  Pinus,  Eucalyptus 

*  Sir  R.  Christison,  On  the  exact  measurement  of  trees.  Trans.  Edinb.  Botan.  Soc. 
XIV,  pp.  164-172,  1882.  Cf.  also  Duhamel  du  Monceau,  Des  semis,  et  plantation 
des  arbres,  Paris,  1750.  On  the  general  subject  see  {int.  al.)  Pfeffer's  Physiology 
of  Plants,  II,  Oxford,  1906;  A.  Maliock,  Growth  of  trees,  Proc.  R.S.  (B),  xc, 
pp.  186-191,  1919.  Maliock  used  an  exceedingly  delicate  optical  method,  in 
which  interference-bands,  produced  by  two  contiguous  glass  plates,  shew  a  visible 
displacement  on  the  slightest  angular  movement  of  the  plates,  even  of  the  order 
of  a  millionth  of  an  inch. 

t  W.  S.  Glock,  A.  E.  Douglass  and  G.  A.  Pearson,  Principles ...  of  tree-ring 
analysis,  Carnegie  Inst.  Washington,  No.  486,  1937;  D.  T.  MacDougal,  Tree  Growth, 
Leiden,  1938,  240  pp. 

%  Trans.  Edinb.  Botan.  Soc.  xviii,  p.  456,  1891. 


236 


THE  RATE  OF  GROWTH 


[CH. 


and  Acacia;  the  deciduous  trees  included  Quercus,  Populus,  Robinia 
and  Melia.  The  result  (Fig.  70)  is  much  as  we  might  expect. 
The  deciduous  trees  cease  to  grow  in  winter-time,  and  during 
all  the  months  when  the  trees  are  bare;  during  the  warm  season 
the  monthly  values  are  regularly  graded,  approximately  in  a  sine- 
curve,  with  a  clear  maximum  (in  the  southern  hemisphere)  about 
the  month  of  December.     In  the  evergreens  the  amplitude  of  the 


Fig.  70.     Periodic  annual  fluctuation  in  rate  of  growth  of  trees  in 
the  southern  hemisphere.     From  C.  E.  Hall's  data. 


annual  wave  is  much  less ;  there  is  a  notable  amount  of  growth  all 
the  year  round,  and  while  there  is  a  marked  diminution  in  rate 
during  the  coldest  months,  there  is  a  tendency  towards  equality 
over  a  considerable  part  of  the  warmer  season.  In  short,  the 
evergreens,  at  least  in  this  case,  do  not  grow  the  faster  as  the 
temperature  continues  to  rise;  and  it  seems  probable  that  some  of 
them,  especially  the  pines,  are  definitely  retarded  in  their  growth, 
either  by  a  temperature  above  their  optimum  or  by  a  deficiency  of 
moisture,  during  the  hottest  season  of  the  year. 

Fig.  71  shews  how  a  cypress  never  ceased  to  grow,  but  had  alternate 


Ill 


OF  THE  GROWTH  OF  TREES 


237 


spells  of  quicker  and  slower  growth,  according  to  conditions  of 
which  we  are  not  informed.  Another  figure  (Fig.  72)  illustrates  the 
growth  in  three  successive  seasons  of  the  Calif ornian  redwood,  a  near 
ally  of  the  most  gigantic  of  trees.  Evergreen  though  the  redwood 
is,  its  growth  has  periods  of  abeyance;  there  is  a  second  minimum 
about  midsummer,  and  the  chief  maximum  of  the  year  may  be  that 
before  or  after  this. 


Fig.  71.     Growth  of  cypress  (C.  macrocarpa),  shewing  seasonal  periodicity. 
From  MacDougal's  data:   smoothed  curve. 


1931  1932  1933 

Fig.  72.     Fortnightly  increase  of  girth  in  Californian  redwood  (Sequoia 
sempervirens),  shewing  seasonal  periodicity.   After  MacDougal. 

In  warm  countries  tree-growth  is  apt  to  shew  a  double  maximum, 
for  the  cold  of  winter  and  the  drought  of  summer  are  equally 
antagonistic  tQ  it.  Trees  grow  slower — and  grow  fewer — the  farther 
north  we  go,  till  only  a  few  birches  and  willows  remain, -stunted  and 
old;  it  is  nearly  a  hundred  years  ago  since  Auguste  Bravais* 
shewed  a  steadily  decreasing  growth-rate  in  the  forests  between 
50°  and  70°  N. 

*  Recherches  sur  la  croissance  du  pin  silvestre  dans  le  nord  de  I'Europe,  Mdm. 
couronnees  de  VAcad.  R.  de  Belgique,  xv,  64  pp.,  1840-41. 


238 


THE  RATE  OF  GROWTH 


[CH. 


The  delicate  measuring  apparatus  now  used  shews  sundry  minor 
but  beautiful  phenomena.  A  daily  periodicity  of  growth  is  a 
common  thing*  (Fig.  73).  In  the  tree-cactuses  the  trunk  expands 
by  day  and  shrinks  again  after  nightfall ;  for  the  stomata  close  in  sun- 
light, and  transpiration  is  checked  until  the  sun  goes  down.  But 
it  is  more  usual  for  the  trunk  to  shrink  from  sunrise  until  evening 
and  to  swell  from  sunset  until  dawn ;   for  by  dayhght  the  leaves  lose 


70°  F 


Black  Poplar 

feet  above  ground, 59  inches 


July  24 


26 


Fig.  73.     Growth  of  black  poplar,  shewing  daily  periodicity. 
After  A.  Mallock. 

water  faster,  and  in  the  dark  they  lose  it  slower,  than  the  roots 
replace  it.  The  rapid  midday  loss  of  water  even  at  the  top  of  a 
tail  Sequoia  is  quickly  followed  by  a  measurable  constriction  of  the 
trunk  fifty  or  even  a  hundred  yards  below  f. 


*  The  diurnal  periodicity  is  beautifully  shewn  in  the  case  of  the  hop  by  Johannes 
Schmidt,  C.R.  du  Laboratoire  Carlsberg,  x,  pp.  235-248,  Copenhagen,  1913. 

•f  This  rapid  movement  is  accounted  for  by  Dixon  and  Joly's  "cohesion-theory" 
of  the  ascent  of  sap.  The  leaves  shew  innumerable  minute  menisci,  or  cup-shaped 
water-surfaces,  in  their  intercellular  air-spaces.  As  water  evaporates  from  these 
the  little  cups  deepen,  capillarity  increases  its  pull,  and  suffices  to  put  in  motion 
the  strands  or  columns  of  water  which  run  continuously  through  the  vessels  of 
wood,  and  withstand  rupture  even  under  a  pull  of  100-200  atmospheres.  See 
{int.  al.)  H.  H.  Dixon  and  J.  Joly,  On  the  ascent  of  sap,  Phil.  Trans.  (B),  clxxxvi, 
p.  563,  1895;   also  Dixon's  Transpiration  and  the  Ascent  of  Sap,  8vo,  London,  1914. 


Ill]         OF  THE  SECULAR  GROWTH  OF  TREES        239 

In  the  case  of  trees,  the  seasonal  periodicity  of  growth  and  the 
direct  influence  of  weather  are  both  so  well  marked  that  we  are 
entitled  to  make  use  of  the  phenomenon  in  a  converse  way,  and  to 
draw  deductions  (as  Leonardo  da  Vinci  did*)  as  to  climate  during 
past  years  from  the  varying  rates  of  growth  which  the  tree  has 
recorded  for  us  by  the  thickness  of  its  annual  rings.  Mr  A.  E. 
Douglass,  of  the  University  .of  California,  has  made  a  careful  study 
of  this  question,  and  I  received  from  him  (through  Professor  H.  H. 
Turner)  some  measurements  of  the  average  width  of  the  annual 
rings  in  Californian  redwood,  five  hundred  yfears  old,  in  which  trees 
the  rings  are  very  clearly  shewn.  For  the  first  hundred  years  the 
mean  of  two  trees  was  used,  for  the  next  four  hundred  years  the 
mean  of  five;  and  the  means  of  these  (and  sometimes  of  larger 
numbers)  were  found  to  be  very  concordant.  A  correction  was 
appHed  by  drawing  a  nearly  straight  fine  through  the  curve  for  the 
whole  period,  which  line  was  assumed  to  represent  the  slowly 
diminishing  mean  width  of  annual  ring  accompanying  the  increasing 
size,  or  age,  of  the  tree;  and  the  actual  growth  as  measured  was 
equated  with  this  diminishing  mean.  The  figures  used  give,  then, 
the  ratio  of  the  actual  growth  in  each  year  to  the  mean  growth 
of  the  tree  at  that  epoch. 

It  was  at  once  manifest  that  the  growth-rate  so  determined 
shewed  a  tendency  to  fluctuate  in  a  long  period  of  between  100  and 
200  years.  I  then  smoothed  the  yearly  values  in  groups  of  100 
(by  Gauss's  method  of  "moving  averages"),,  so  that  each  number 
thus  found  represented  the  mean  annual  increase  during  a  century: 
that  is  to  say,  the  value  ascribed  to  the  year  1500  represented  the 
average  annual  growth  during  the  whole  period  between  1450  and 
1550,  and  so  on.  These  values,  so  simply  obtained,  give  us  a  curve 
of  beautiful  and  surprising  smoothness,  from  which  we  draw  the 
direct  conclusion  that  the  climate  of  Arizona,  during  the  last  five 
hundred  years,  has  fluctuated  with  a  regular  periodicity  of  almost 
precisely  150  years.  I  have  drawn,  more  recently,  and  also  from 
Mr  Douglass's  data,  a  similar  curve  for  a  group  of  pine  trees  in 
Calaveras  County  |.     These  trees  are  about  300  years  old,  and  the 

*  Cf.  J.  Playfair  McMurrich,  Leonardo  da  Vinci,  1930,  p.  247. 
t  When  this  was  first  written  I  had  not  seen  Mr  Douglass's  paper  On  a  method 
of  estimating  rainfall  by  the  growth  of  trees.  Bull.  Amer.  Geograph.  Soc.  XLVI, 


240 


THE  RATE  OF  GROWTH 


[CH. 


data  are  reduced,  as  before,  to  moving  averages  of  100  years,  but 
without  further  correction.  The  agreement  between  the  growth- 
rate  of  these  pines  and  that  of  the  great  Sequoias  during  the  same 
period  is  very  remarkable  (Fig.  74). 

We  should  be  left  in  doubt,  so  far  as  these  observations  go, 
whether  the  essential  factor  be  a  fluctuation  of  temperature  or  an 
alternation  of  drought  and  humidity;  but  the  character  of  the 
Arizona  chmate,  and  the  known  facts  of  recent  years,  encourage 
the  behef  that  the  latter  is  the  more  direct  and  more  important 
factor.  In  a  New  England  forest  many  trees  of  many  kinds  were 
studied  after  a  hurricane;  they  shewed  on  the  whole  no  correlation^ 


1440 


CQ 


Fig.  74.  Long-period  fluctuation  in  growth  of  Arizona  redwood  (Sequoia),  from 
A.D.  1390  to  1910;  and  of  yellow  pine  from  Calaveras  County,  from  a.d.  1620 
to  1920.     (Smoothed  in  100-year  periods.) 

between  growth-rate  and  temperature,  with  the  remarkable  exception 
(in  the  conifers)  of  a  clear  correlation  with  the  temperature  of 
March  and  April,  a  month  or  two  before  the  season's  growth  began. 
In  a  cold  spring  the  melting  snows  and  early  rains  ran  off  into  the 
rivers,  in  a  warm  and  early  one  they  sank  into  the  soil  * ;  in  other 
words,  humidity  was  still  the  controlling  factor.  An  ancient  oak' 
tree  in  Tunis  is  said  to  have  recorded  fifty  years  of  abundant  rain. 


pp.  321-335,  1914;  nor,  of  course,  his  great  work  on  Climatic  cycles  and  tree- 
growth,  Carnegie  Inst.  Publications,  1919,  1928,  1936.  Mr  Douglass  does  not 
fail  to  notice  the  long  period  here  described,  but  he  is  more  interested  in  the 
sunspot-cycle  and  other  shorter  cycles  known  to  meteorologists.  See  also  (int.  al.) 
E.  Huntingdon,  The  fluctuating  climate  of  North  America,  Geograph.  Journ. 
Oct.  1912;  and  Otto  Pettersson,  Climatic  variation  in  historic  and  prehistoric 
time,  Svens^M,  Hydrografisk-Biolog.  Skrifter,  v,  1914. 

♦  C.  J.  Lyon,  Amer.  Assoc.  Rep.  1939;   Nature,  Apr.  13,  1940,  p.  595. 


Ill]  OF  THE  INFLUENCE  OF  LIGHT  241 

with  short  intervals  of  drought,  during  the  eighteenth  century ;  then, 
after  1790,  longer  droughts  and  shorter  spells  of  rainy  seasons*. 

It  has  been  often  remarked  that  our  common  European  trees, 
such  as  the  elm  or  the  cherry,  have  larger  leaves  the  farther  north 
we  go ;  but  the  phenomenon  is  due  to  the  longer  hours  of  dayhght 
throughout  the  summer,  rather  than  to  intensity  of  illumination  or 
diJBFerence  of  temperature.  On  the  other  hand,  long  dayhght,  by 
prolonging  vegetative  growth,  retards  flowering  and  fruiting;  and 
late  varieties  of  soya  bean  may  be  forced  into  early  ripeness  by 
artificially  shortening  their  dayhght  at  midsummer  f. 

The  effect  of  ultra-violet  hght,  or  any  other  portion  of  the 
spectrum,  is  part,  and  perhaps  the  chief  part,  of  the  same  problem. 
That  ultra-violet  hght  accelerates  growth  has  been  shewn  both  in 
plants  and  animals f.  In  tomatoes,  growth  is  favoured  by  just  such 
ultra-violet  hght  as  comes  very  near  the  end  of  the  solar  spectrum  §, 
and  as  happens,  also,  to  be  especially  absorbed  by  ordinary  green- 
house glass II .  At  the  other  end  of  the  spectrum,  in  red  or  orange 
light,  the  leaves  become  smaller,  their  petioles  longer,  the  nodes 
more  numerous,  the  very  cells  longer  and  more  attenuated.  It  is 
a  physiological  problem,  and  as  such  it  shews  how  plant-hfe  is 
adapted,  on  the  whole,  to  just  such  rays  as  the  sun  sends;  but  it 
also  shews  the  morphologist  how  the  secondary  effects  of  chmate 
may  so  influence  growth  as  to  modify  both  size  and  form^.  An 
analogous  case  is  the  influence  of  hght,  rather  than  temperature, 
in  modifying  the  coloration  of  organisms,  such  as  certain  butterflies. 

*  Le  chene  Zeem  d'Ain  Draham,  Bull,  du  Directeur  General,  Tunisie,  1927. 

t  That  the  plant  grows  by  turns  in  darkness  and  in  light,  and  has  its  characteristic 
growth-phases  in  each,  longer  >or  shorter  according  to  species  and  variety  and 
normal  habitat,  is  a  subject  now  studied  under  the  name  of  "photoperiodism," 
and  become  of  great  practical  importance  for  the  northerly  extension  of  cereal 
crops  in  Canada  and  Russia.  Cf.  R.  G.  Whyte  and  M.  A.  Oljhovikov,  Nature, 
Feb.  18,  1939. 

I  Cf.  Kuro  Suzuki  and  T.  Hatano,  in  Proc.  Imp.  Acad,  of  Japan,  in,  pp.  94-96, 1927. 
§   Withrow  and  Benedict,  in  Bull,  of  Basic  Scient.  Research,  iii,  pp.  161-174,  1931. 

II  Cf.  E.  C.  Teodoresco,  Croissance  des  plantes  aux  lumieres  de  diverses  longueurs 
d'onde,  ^WTi.  Sc.  Nat,  Bat.  (8),  pp.  141-336,  1929;  N.  Pfeiffer,  Botan.  Gaz.  lxxxv, 
p.  127,  1929;  etc. 

II  See  D.  T.  MacDougal,  Influence  of  light  and  darkness,  etc.,  Mem.  N.  Y.  Botan. 
Garden,  1903,  392  pp.;  Growth  in  trees,  Carnegie  Inst.  1921,  1924,  etc.;  J.  Wiesner, 
Lichtgenuss  der  Pflunzen,  vn,  322  pp.,  1907;  Earl  S.  Johnston,  Smithson.  Misc. 
Contrib.  18  pp.,  1938;  etc.  On  the  curious  effect  of  short  spells  of  light  and  dark- 
ness, see  H.  Dickson,  Proc.  E.S.  (B),  cxv,  pp.  115-123,  1938. 


242  THE  RATE  OF  GROWTH  [ch. 

Now  if  temperature  or  light  affect  the  rate  of  growth  in  strict 
uniformity,  aHke  in  all  parts  and  in  all  directions,  it  will  only  lead 
to  local  races  or  varieties  differing  in  size,  as  the  Siberian  goldfinch 
or  bullfinch  differs  from  our  own.  But  if  there  be  ever  so  Httle  of  a 
discriminating  tendency  such  as  to  enhance  the  growth  of  one  tissue 
or  one  organ  more  than  another*,  then  it  must  soon  lead  to  racial, 
or  even  "specific,"  difference  of  form. 

It  is  hardly  to  be  doubted  that  chmate  has  some  such  dis- 
criminating influence.  The  large  leaves  of  our  northern  trees  are 
an  instance  of  it;  and  we  have  a  better  instance  of  it  still  in  Alpine 
plants,  whose  general  habit  is  dwarfed  though  their  floral  organs 
suffer  little  or  no  reduction  f.  Sunhght  of  itself  would  seem  to  be 
a  hindrance  rather  than  a  stimulant  to  growth;  and  the  familiar 
fact  of  a  plant  turning  towards  the  sun  means  increased  growth  on 
the  shady  side,  or  partial  inhibition  on  the  other. 

More  curious  and  still  more  obscure  is  the  moon's  influence  on 
growth,  as  on  the  growth  and  ripening  of  the  eggs  of  oysters,  sea- 
urchins  and  crabs.  Behef  in  such  lunar  influence  is  as  old  as  Egypt; 
it  is  confirmed  and  justified,  in  certain  cases,  nowadays,  but  the 
way  in  which  the  influence  is  exerted  is  quite  unknown  J. 

Osmotic  factors  in  growth 

The  curves  of  growth  which  we  have  been  studying  have  a 
twofold  interest,  morphological  and  physiological.  To  the  morpho- 
logist,  who  has  learned  to  recognise  form  as  a  "function  of  growth," 
the  most  important  facts  are  these:  (1)  that  rate  of  growth  is  an 
orderly  phenomenon,  with  general  features  common  to  various 
organisms,  each  having  its  own  characteristic  rates,  or  specific 
constants;  (2)  that  rate  of  growth  varies  with  temperature,  and  so 
with  season  and  with  climate,  and  also  with  various  other  physical 
factors,  external  and  internal  to  the  organism ;  (3)  that  it  varies  in 
different  parts  of  the  body,  and  along  various  directions  or  axes: 

*  Or  as  we  might  say  nowadays,  have  a  different  "threshold  value"  in  one 
organ  to  another. 

t  Cf.  for  instance,  NageH's  classical  account  of  the  effect  of  change  of  habitat 
on  alpine  and  other  plants,  Sitzungsber.  Baier.  Akad.  Wiss.  1865,  pp.  228-284. 

X  Cf.  Munro  Fox,  Lunar  periodicity  in  reproduction,  Proc.  R.S.  (B),  xcv, 
pp.  523-550,  1935;   also  Silvio  Ranzi,  Pubblic.  Slaz.  Zool.  Napoli,  xi,  1931. 


Ill]  OF  OSMOTIC  FACTORS  IN  GROWTH  243 

such  variations  being  harmoniously  "graded,"  or  related  to  one 
another  by  a  "principle  of  continuity,"  so  giving  rise  to  the 
characteristic  form  and  dimensions  of  the  organism  and  to  the 
changes  of  form  which  it  exhibits  in  the  course  of  its  development. 
To  the  physiologist  the  phenomenon  of  growth  suggests  many  other 
considerations,  and  especially  the  relation  of  growth  itself  to  chemical 
and  physical  forces  and  energies. 

To  be  content  to  shew  that  a  certain  rate  of  growth  occurs  in 
a  certain  organism  under  certain  conditions,  or  to  speak  of  the 
phenomenon  as  a  "reaction"  of  the  living  organism  to  its  environ- 
ment or  to  certain  stimuh,  would  be  but  an  example  of  that  "lack 
of  particularity"  with  which  we  are  apt  to  be  all  too  easily  satisfied. 
But  in  the  case  of  growth  we  pass  some  little  way  beyond  these 
limitations:  to  this  extent,  that  an  affinity  with  certain  types  of 
chemical  and  physical  reaction  has  been  recognised  by  a  great  number 
of  physiologists*. 

A  large  part  of  the  phenomenon  of  growth,  in  animals  and  still 
more  conspicuously  in  plants,  is  associated  with  "turgor,"  that  is 
to  say,  is  dependent  on  osmotic  conditions.  In  other  words,  the 
rate  of  growth  depends  (as  we  have  already  seen)  as  much  or  more 
on  the  amount  of  water  taken  up  into  the  living  cells  f,  as  on  the 
actual  amount  of  chemical  metabohsm  performed  by  them;  and 
sometimes,  as  in  certain  insect-larvae,  we  can  even  distinguish 
between  tissues  which  grow  by  increase  of  cell-size,  the  result 'of 
imbibition,  and  others  which  grow  by  multiplication  of  their 
constituent  cells  |.    Of  the  chemical  phenomena  which  result  in  the 

*  Cf.  F.  F.  Blackman,  Presidential  Address  in  Botany,  Brit.  Assoc,  Dublin, 
1908.  The  idea  was  first  enunciated  by  Baudrimont  and  St  Ange,  Recherches  sur 
le  developpement  du  foetus,  Mem.  Acad.  Sci.  xi,  p.  469,  1851. 

t  Cf.  J.  Loeb,  Untersuckungen  zur  physiologischen  Morphologie  der  Tiere,  1892; 
also  Experiments  on  cleavage,  Journ.  Morphology,  vii,  p.  253,  1892;  Ueber  die 
Dynamik  des  tierischen  Wachstums,  Arch.  f.  Entw.  Mech.  xv,  p.  669,  1902-3; 
Davenport,  On  the  role  of  water  in  growth,  Boston  Soc.  N.H.  1897;  Ida  H.  Hyde 
in  Amer.  Journ.  Physiology,  xii,  p.  241,  1905;  Bottazzi,  Osmotischer  Druck  und 
elektrische  Leitungsfahigkeit  der  Fliissigkeiten  der  Organismen,  in  Asher-Spiro's 
Ergebnisse  der  Physiologic,  vn,  pp.  160-402,  1908;  H.  A.  Murray  in  Journ.  Gener. 
Physiology,  ix,  p.  1,  1925;  J.  Gray,  The  role  of  water  in  the  evolution  of  the 
terrestrial  vertebrates,  Journ.  Exper.  Biology,  vi,  pp.  26-31,  1928;  and  A.  N.  J.  Heyn, 
Physiology  of  cell-elongation,  Botan.  Review,  vi,  pp.  515-574,  1940. 

X  Cf.  C.  A.  Berger,  Carnegie  Inst,  of  Washington,  Contributions  to  Embryology, 
xxvu,  1938. 


244 


THE  RATE  OF  GROWTH 


[CH. 


actual  increase  of  protoplasm  we  shall  speak  presently,  but  the  role 
of  water  in  growth  deserves  a  passing  word,  even  in  our  morpho- 
logical enquiry. 

The  lower  plants  only  Uve  and  grow  in  abundant  moisture;  lew 
fungi  continue  growing  when  the  humidity  falls  below  85  per  cent, 
of  saturation,  and  the  mould-fungi,  such  as  Penicillium,  need  more 
moisture  still  (Fig.  75).     Their  hmit  is  reached  a  little  below  90%. 


Humidity  ^ 

I.  75.     Growth  of  PeniciUium  in  relation  to  humidity. 

Growth  of  PeniciUium  (at  25°  C.)  * 


Humidity 

Growth  per  hour 

(%  of  saturation) 

(mm.) 

1000 

7-7 

97-0 

5-0 

94-2 

1-0 

926 

0-5 

90-8 

0-3 

Among  th^  coelenterate  animals  growth  and  ultimate  size  depend 
on  Httle  more  than  absorption  of  water  and  consequent  turgescence, 
the  process,  she  wing  itself  in  simple  ways.  A  sea-anemone  may  live 
to  ^n  immense  agef,  but  its  age  and  size  have  Httle  to  do  with  one 


*  From  R.  G.  Tomkins,  Studies  of  the  growth  of  moulds,  Proc.  B.S.  (B),  cv, 
pp.  375-^01,  1929. 

t  Like  Sir  John  Graham  Dalyell's  famous  "Granny,"  and  Miss  Nelson's  family 
of  CereiLS  (not  Sagartia)  of  which  one  still  lives  at  over  80  years  old.  Cf.  J.  H. 
Ashworth  and  Nelson  Annandale,  in  Trans.  R.  Physical  Soc.  Edin.  xxv,  pp.  1-14 
1904. 


Ill]  OF  TURGESCENCE  245 

another.  It  has  an  upper  limit  of  size  vaguely  characteristic  of  the 
species,  and  if  fed  well  and  often  it  may  reach  it  in  a  year ;  on  stinted 
diet  it  grows  slowly  or  may  dwindle  down;  it  may  be  kept  at 
wellnigh  what  size  one  pleases.  Certain  full-grown  anemones  were 
left  untended  in  war-time,  unfed  and  in  water  which  evaporated 
down  to  half  its  bulk;  they  shrank  down  to  little  beads,  and  grew 
up  again  when  fed  and  cared  for. 

Loeb  shewed,  in  certain  zoophytes,  that  not  only  must  the  cells 
be  turgescent  in  order  to  grow,  but  that  this  turgescence  is  possible 
only  so  long  as  the  salt-water  in  which  the  cells  lie  does  not  overstep 
a  certain  limit  of  concentration:  a  limit  reached,  in  the  case  of 
Tubularia,  when  the  salinity  amounts  to  about  3-4  per  cent.  Sea- 
water  contains  some  3-0  to  3-5  per  cent,  of  salts  in  the  open  sea, 
but  the  salinity  falls  much  below  this  normal,  to  about  2-2  per  cent., 
before  Tubularia  exhibits  its  full  turgescence  and  maximal  growth; 
a  further  dilution  is  deleterious  to  the  animal.  It  is  likely  enough 
that  osmotic  conditions  control,  after  this  fashion,  the  distribution 
and  local  abundance  of  many  zoophytes.  Loeb  has  also  shewn* 
that  in  certain  fish-eggs  (e.g.  of  Fundulus)  an  increasing  concentration, 
leading  to  a  lessening  water-content  of  the  egg,  retards  the  rate  of 
segmentation  and  at  last  arrests  it,  though  nuclear  division  goes  on 
for  some  time  longer. 

The  eggs  of  many  insects  absorb  water  in  large  quantities,  even 
doubling  their  weight  thereby,  and  fail  to  develop  if  drought  prevents 
their  doing  so ;  and  sometimes  the  egg  has  a  thin- walled  stalk,  or  else 
a  "hydropyle,"  or  other  structure  by  which  the  water  is  taken  inf. 

In  the  frog,  according  to  Bialaszewicz  J,  the  growth  of  the  embryo 
while  within  the  vitelline  membrane  depends  wholly  on  absorption 
of  water.  The  rate  varies  with  the  temperature,  but  the  amount 
of  water  absorbed  is  constant,  whether  growth  be  fast  or  slow. 
Moreover,  the  successive  changes  of  form  correspond  to  definite 
quantities  of  water  absorbed,  much  of  which  water  is  intracellular. 
The  solid  residue,  as  Davenport  Has  also  shewn,  may  even  diminish 


*  P finger's  Archiv,  lv,  1893. 

t  Cf.  V.  B.  Wigglesworth,  hised  Physiology,  1939,  p.  2. 

%  Beitrage  zur  Kenntniss  d.  Wachstumsvorgange  bei  Amphibienerabryonen,  Bull. 
Acad.  Sci.  de  Cracovie,  1908,  p.  783;  also  A.  Drzwina  and  C.  Bohn,  De  Taction. .  .dea 
solutions  salines  sur  les  larves  des  batraciens,  ibuL  1906. 


246  THE  RATE  OF  GROWTH  [ch. 

notably,  while  all  the  while  the  embryo  continues  to  grow  in  bulk 
and  weight.  But  later  on,  and  especially  in  the  higher  animals,  the 
water-content  diminishes  as  growth  proceeds  and  age  advances; 
and  loss  of  water  is  followed,  or  accompanied,  by  retardation  and 
cessation  of  growth.  A  crab  loses  water  as  each  phase  of  growth 
draws  to  an  end  and  the  corresponding  moult  approaches;  but  it 
absorbs  water  in  large  quantities  as  soon  as  the  new  period  of 
growth  begins*.  Moreover,  that  water  is  lost  as  growth  goes  on  has 
been  shewn  by  Davenport  for  the  frog,  by  Potts  for  the  chick,  and 
particularly  by  Fehhng  in  the  case  of  man.  Fehhng's  results  may 
be  condensed  as  follows : 

Age  in  weeks  (man)         6  17  22  24  26  30  35  39 

Percentage  of  water      97-5       91-8       92-0      89-9       86-4       83-7       82-9       74-2 

I 

The  following  illustrate  Davenport's  results  for  the  frog: 


Age  in  weeks  (frog) 

1 

2 

5 

7 

9 

14 

41 

84 

Percentage  of  water 

56-3 

58-5 

76-7 

89-3 

931 

950 

90-2 

87-5 

The  following  table  epitomises  the  drying-off  of  ripening  maize  f; 
it  shews  how  ripening  and  withering  are  closely  akin,  and  are  but 
two  phases  of  senescence  (Fig.  76): 


Days  (from  August  6) 

0 

22 

35 

49 

56 

63 

Percentage  of  water 

87 

81 

77 

68 

65 

58 

The  bird's  egg  provides  all  the  food  and  all  the  water  which  the 
growing  embryo  needs,  and  to  carry  a  provision  of  water  is  the 
special  purpose  of  the  white  of  the  egg ;  the  water  contained  in  the 
albumen  at  the  beginning  of  incubation  is  just  about  what  the 
chick  contains  at  the  end.  The  yolk  is  not  surrounded  by  water, 
which  would  diffuse  too  quickly  into  it,  nor  by  a  crystalloid  solution, 
whose  osmotic  value  would  soon  increase;  but  by  a  watery  albu- 
minous colloid,  whose  osmotic  pressure  changes  slowly  as  its  charge 
of  water  is  gradually  withdrawn  J. 

*  Cf.  A.  Krogh,  Osmotic  regulation  in  aquatic  animals,  Cambridge,  1939. 
t  Henry  and  Morrison,  1917;  quoted  by  Otto  Glaser,  on  Growth,  time  and  form, 
Biolog.  Reviews,  xiii,  pp.  2-58,  1938. 

X  Cf.  James  Gray,  in  Journ.  Exper.  Biology,  iv,  pp.  214-225,  1926. 


Ill 


OF  GAIN  OR  LOSS  OF  WATER 


247 


Distribution  of  water  in  a  hen's  egg 


Gm. 

of  water  contained  in 

Day  of 
incubation 

^ 

Loss  by 
evaporation 

Gain  by 
combustion 

Albumen 

Embryo 

Yolk 

0 

29-9 

0-0 

8-5 

00 

00 

6 

27-2 

0-4 

8-45 

2-4 

001 

12 

20-4 

4-6 

7-8 

5-6 

0-27 

18 

9-2 

181 

2-3 

8-8 

1-20 

20 

2-2 

27-4 

10 

9-8 

2-00 

The  actual  amount  of  water,  compared  with  the  dry  solids  in  the 
egg,  has  been  determined  as  follows: 


Day  of  incubation  (chick) 

5 

8 

11 

14 

17 

19 

Percentage  of  water 

94-7 

93-8 

92-3 

87-7 

82-8 

82-3 

10 


20 


30         40 
Days 
Fig.  76.     Percentage  of  water  in  ripening  maize.   From  Otto  Glaser. 

We  know  very  httle  of  the  part  which  all  this  water  plays:  how 
much  is  mere  "reaction-medium,"  how  much  is  fixed  in  hydrated 
colloids,  how  much,  in  short,  is  bound  or  unbound.  But  we  see  that 
somehow  or  other  water  is  lost,  and  lost  in  considerable  amount, 
as  the  embryo  draws  towards  completion  and  ceases  for  the  time 
being  to  grow. 

All  vertebrate  animals  contain  much  the  same  amount  of  water 
in  their  living  bodies,  say  85  per  cent,  or  thereby,  however  unequally 


248  THE  RATE  OF  GROWTH  [ch. 

distributed  in  the  tissues  that  water  may  be*.  Land  animals  have 
evolved  from  water  animals  with  Httle  change  in  this  respect, 
though  the  constant  proportion  of  water  is  variously  achieved. 
A.  newt  loses  moisture  by  evaporation  with  the  utmost  freedom,  and 
regains  it  by  no  less  rapid  absorption  through  the  skin;  while  a 
lizard  in  his  scaly  coat  is  less  hable  to  the  one  and  less  capable  of 
the  other,  and  must  drink  to  replace  what  water  it  may  lose. 

We  are  on  the  verge  of  a  difficult  subject  when  we  speak  of  the 
role  of  water  in  the  living  tissues,  in  the  growth  of  the  organism, 
and  in  the  manifold  activities  of  the  cell ;  and.  we  soon  learn,  among 
other  more  or  less  unexpected  things,  that  osmotic  equihbrium  is 
neither  universal  nor  yet  common  in  the  living  organism.  The  yolk 
maintains  a  higher  osmotic  pressure  than  the  white  of  the  egg — so 
long  as  the  egg  is  living;  and  the  watery  body  of  a  jellyfish,  though 
aot  far  off  osmotic  equilibrium,  has  a  somewhat  less  salinity  than 
the  sea-water.  In  other  words,  its  surface  acts  to  some  extent  as 
a  semipermeable  membrane,  and  the  fluid  which  causes  turgescence 
of  the  tissues  is  less  dense  than  the  sea- water  outside!. 

In  most  marine  invertebrates,  however,  the  body-fluids  con- 
stituting the  milieu  interne  are  isotonic  with  the  milieu  externe,  and 
vary  in  these  animals  pari  passu  with  the  large  variations  to  which 
sea- water  itself  is  subject.  On  the  other  hand,  the  dwellers  in 
fresh-water,  whether  invertebrates  or  fishes,  have,  naturally,  a  more 
concentrated  medium  within  than  without.  As  to  fishes,  diiferent 
kinds  shew  remarkable  differences.  Sharks  and  dogfish  have  an 
osmotic  pressure  in  their  blood  and  their  body  fluids  little  diiferent 

*  The  vitreous  humour  is  nearly  all  water,  the  enamel  has  next  to 'none,  the 
grey  matter  has  some  86  per  cent.,  the  bones,  say  22  per  cent. ;  lung  and  kidney 
take  up  mor^  than  they  can  hold,  and  so  become  excretory  or  regulatory  organs. 
Eggs,  whether  of  dogfish,  salmon,  frogs, '  snakes  or  birds,  are  composed,  roughly 
speaking,  of  half  water  and  half  solid  matter. 

t  Cf.  {int.  al.)  G.  Teissier,  Sur  la  teneur  en  eau. .  .de  Chrysaora,  Bull.  Soc.  Biol, 
de  France,  1926,  p.  266.  And  especially  A.  V.  Hill,  R.  A.  Gortner  and  others.  On 
the  state  of  water  in  colloidal  and  living  systems.  Trans.  Faraday  Soc.  xxvi, 
pp.  678-704,  1930.  For  recent  literature  see  (e.g.)  Homer  Smith,  in  Q.  Rev.  Biol. 
vn,  p.  1,  1932;  E.  K.  Marshall,  Physiol.  Rev.  xiv,  p.  133,  1934;  Lovatt  Evans, 
Recent  Advances  in  Physiology,  4th  ed.,  1930;  M.  Duval,  Recherches. .  .sur  le  milieu 
interieur  des  animaux  aquatiques,  Thhe,  Paris,  1925;  Paul  Portier,  Physiologic  des 
animaux  marins.  Chap,  iii,  Paris,  1938;  G.  P.  Wells  and  I.  C.  Ledingham,  Effects 
of  a  hypotonic  environment,  Journ.  Exp.  Biol,  xvii,  pp.  337-352,  1940. 


Ill]  OF  OSMOTIC  REGULATION  249 

from  that  of  the  sea-water  outside:  but  with  certain  chemical 
diiferences,  for  instance  that  the  chlorides  within  are  much 
diminished,  and  the  molecular  concentration  is  eked  out  by  large 
accumulations  of  urea  in  the  blood.  The  marine  teleosts,  on  the 
other  hand,  have  a  much  lower  osmotic  pressure  within  than  that 
of  the  sea-water  outside,  and  only  a  httle  higher  than  that  of  their 
fresh-water  allies.  Some,  hke  the  conger-eel,  maintain  an  all  but 
constant  internal  concentration,  very  different  from  that  outside; 
and  this  fish,  like  others,  is  constantly  absorbing  water  from  the  sea ; 
it  must  be  exuding  or  excreting  salt  continually*.  Other  teleosts 
differ  greatly  in  their  powers  of  regulation  and  of  tolerance,  the 
common  stickleback  (which  we  may  come  across  in  a  pool  or  in  the 
middle  of  the  North  Sea)  being  exceptionally  tolerant  or  "eury- 
halinef."  Physiology  becomes  "comparative"  when  it  deals  with 
differences  such  as  these,  and  Claude  Bernard  foresaw  the  existence 
of  just  such  differences:  "Chez  tous  les  etres  vivants  le  milieu 
interieur,  qui  est  un  produit  de  I'organisme,  conserve  les  rapports 
necessaires  d'echange  avec  le  milieu  exterieur;  mais  a  mesure  que 
I'organisme  devient  plus  parfait  le  milieu  organique  se  specifie,  et 
s'isole  en  quel  que  sorte  de  plus  en  plus  au  milieu  ambiant  J."  Claude 
Bernard  was  building,  if  I  mistake  not,  on  Bichat's  earUer  concept, 
famous  in  its  day,  of  life  as  "une  alternation  habituelle  d'action  de 
la  part  des  corps  exterieurs,*  et  de  reaction  de  la  part  du  corps 
vivant":  out  of  which  grew  his  still  more  famous  aphorism,  "La 
vie  est  I'ensemble  des  fonctions  qui  resistent  a  la  mort§." 

One  crab,  like  one  fish,  differs  widely  from  another  in  its  power 

*  Probably  by  help  of  Henle's  tubules  in  the  kidney,  which  structures  the  dogfish 
does  not  possess.  But  the  gills  have  their  part  to  play  as  water-regulators,  as 
also,  for  instance,  in  the  crab. 

t  The  grey  mullets  go  down  to  the  sea  to  spawn,  but  may  live  and  grow  in 
brackish  or  nearly  fresh- water.  The  several  species  differ  much  in  their  adaptability, 
and  Brunelli  sets  forth,  as  follows,  the  range  of  salinity  which  each  can  tolerate : 

M.  auratus  24-35  per  mille 

saliens  16-40 

chelo  10-40 
capita  5-40 

cephalus  4^0 

X  Introduction  d  Vetude  de  la  medecine  experimentale,  1855,  p.  110.  For  a  dis- 
cussion of  this  famous  concept  see  J.  Barcroft,  "La  fixite  du  milieu  interieur  est 
la  condition  de  la  vie  libre,"  Biol.  Reviews,  viii,  pp.  24-87,  1932. 

§  Sur  la  vie  et  la  mart,  p.  1. 


250  THE  RATE  OF  GROWTH  [ch. 

of  self-regulation;  and  these  physiological  differences  help  to  explain, 
in  both  cases,  the  limitation  of  this  species  or  that  to  more  or  less 
brackish,  or  more  or  less  saline,  waters.  In  deep-sea  crabs  {Hyas, 
for  instance)  the  osnjotic  pressure  of  the  blood  keeps  nearly  to  that 
of  the  milieu  exteme,  and  falls  quickly  and  dangerously  with  any 
dilution  of  the  latter;  but  the  httle  shore-crab  (Cardnus  moenas) 
can  hve  for  many  days  in  sea-water  diluted  down  to  one-quarter  of 
its  usual  sahnity.  Meanwhile  its  own  fluids  dilute  slowly,  but  not 
near  so  far;  in  other  words,  this  crab  combines  great  powers  of 
osmotic  regulation  with ,  a  large  capacity  for  tolerating  osmotic 
gradients  which  are  beyond  its  power  to  regulate.  How  the  unequal 
balance  is  maintained  is  yet  but  httle  understood.  But  we  do  know 
that  certain  organs  or  tissues,  especially  the  gills  and  the  antennary 
gland,  absorb,  retain  or  ehminate  certain  elements,  or  certain  ions, 
faster  than  others,  and  faster  than  mere  diffusion  accounts  for;  in 
other  words,  "ionic*'  regulation  goes  hand  in  hand  with  "osmotic" 
regulation,  as  a  distinct  and  even  more  fundamental  phenomenon*. 
This  at  least  seems  generally  true — and  only  natural — that  quickened 
respiration  and  increased  oxygen-consumption  accompany  all  such 
one-sided  conditions:  in  other  words,  the  "steady  state"  is  only 
maintained  by  the  doing  of  work  and  the  expenditure  of  energyf. 

To  the  dependence  of  growth  on  the  uptake  of  water,  and  to  the 
phenomena  of  osmotic  balance  and  its  regulation,  HoberJ  and  also 
Loeb  were  inclined  to  refer  the  modifications  of  form  which  certain 
phyllopod  Crustacea  undergo  when  the  highly  sahne  waters  which 
they  inhabit  are  further  concentrated,  or  are  abnormally  diluted. 
Their  growth  is  retarded  by  increased  concentration,  so  that 
individuals  from  the  more  saline  waters  appear  stunted  and  dwarfish ; 
and  they  become  altered  or  transformed  in  other  ways,  suggestive 
of  "degeneration,"  or  a  failure  to  attain  full  and  perfect  develop- 

*  See  especially  D.  A.  Webb,  Ionic  regulation  in  Cardnus  moenas,  Proc.  R.S.  (B), 
cxxix,  pp.  107-136,  1940. 

t  In  general  the  fresh- water  Crustacea  have  a  larger  oxygen -consumption  than 
the  marine.  Stenohaline  and  euryhaline  are  terms  applied  nowadays  to  species 
which  are.  confined  to  a  narrow  range  of  salinity,  or  are  tolerant  of  a  wide  one. 
An  extreme  case  of  toleration,  or  adaptability,  is  that  of  the  Chinese  woolly-handed 
crab,  Eriockeir,  which  has  not  only  acclimatised  itself  in  the  North  Sea  but  has 
ascended  the  Elbe  as  far  as  Dresden. 

X  R.  Hober,  Bedeutung  der  Theorie  der  Losungen  fiir  Physiologic  und  Medizin, 
Biol.  Centralbl.  xix,  p.  272,  1899. 


Ill]  OF  OSMOTIC  REGULATION  251 

merit*.  Important  physiological  changes  ensue.  The  consumption 
of  oxygen  increases  greatly  in  the  stronger  brines,  as  more  and  more 
active  "  osmo-regulation  "  is  required.  The  rate  of  multiplication  is 
increased,  and  parthenogenetic  reproduction  is  encouraged.  In  the 
less  sahne  waters  male  individuals,  usually  rare,  become  plentiful, 
and  here  the  females  bring  forth  their  young  alive ;  males  disappear 
altogether  in  the  more  concentrated  brines,  and  then  the  females 
lay  eggs,  which,  however,  only  begin  to  develop  when  the  sahnity 
is  somewhat  reduced. 

The  best-known  case  is  the  little  brine-shrimp,  Artemia  salina, 
found  in  one  form  or  another  all  the  world  over,  and  first  discovered 
nearly  two  hundred  years  ago  in  the  salt-pans  at  Lymington. 
Among  many  allied  forms,  one,  A.  milhausenii,  inhabits  the  natron- 
lakes  of  Egypt  and  Arabia,  where,  under  the  name  of  "loul,"  or 
"Fezzan-worm,"  it  is  eaten  by  the  Arabsf.  This  fact  is  interesting, 
because  it  indicates  (and  investigation  has  apparently  confirmed) 
that  the  tissues  of  the  creature  are  not  impregnated  with  salt,  as 
is  the  medium  in  which  it  hves.  In  short  Artemia,  hke  teleostean 
fishes  in  the  sea,  hves  constantly  in  a  "hypertonic  medium";  the 
fluids  of  the  body,  the  milieu  interne,  are  no  more  salt  than  are  those 
of  any  ordinary  crustacean  or  other  animal,  but  contain  only  some 
0-8  per  cent,  of  NaCl  J ,  while  the  milieu  externe  may  contain  from 
3  to  30  per  cent,  of  this  and  other  salts;  the  skin,  or  body- wall,  of. 
the  creature  acts  as  a  "semi-permeable  membrane,"  through  which 
the  dissolved  salts  are  not  permitted  to  diffuse,  though  water  passes 
freely.  When  brought  into  a  lower  concentration  the  animal  may 
grow  large  and  turgescent,  until  a  statical  equilibrium,  or  steady 
state,  is  at  length  attained. 

Among  the  structural  changes  which  result  from  increased  con- 

*  Schmankewitsch,  Zeitschr.  f.  wiss.  Zool.  xxix,  p.  429,  1877.  Schraankewitsch 
has  made  equally  interesting  observations  on  change  of  size  and  form  in  other 
organisms,  after  some  generations  in  a  milieu  of  altered  density ;  e.g.  in  the  flagellate 
infusorian  Ascinonema  acinus  Biitschli. 

t  These  "Fezzan- worms,"  when  first  described,  were  supposed  to  be  "insects' 
eggs";  cf.  Humboldt^  Personal  Narrative,  vi,  i,  8,  note;  Kirby  and  Spence,  Letter  x. 

X  See  D.  J.  Kuenen,  Notes,  systematic  and  physiological,  on  Artemia,  Arch. 
N4erland.  Zool.  iii,  pp.  365-449,  1939;  cf.  also  Abonyi,  Z.f.  w.  Z.  cxiv,  p.  134,  1915. 
Cf.  Mme.  Medwedewa,  Ueber  den  osmotischen  Druck  der  Haemolymph  v.  Artemia; 
in  Ztsch.  f.  vergl.  Physiolog.  v,  pp.  547-554,  1922. 


252 


THE  RATE  OF  GROWTH 


[CH. 


centration  of  the  brine  (partly  during  the  hfe-time  of  the  individual, 
but  more  markedly  during  the  short  season  which  suffices  for  the 
development  of  three  or  four,  or  perhaps  more,  successive  genera- 
tions), it  is  found  that  the  tail  comes  to  bear  fewer  and  fewer 
bristles,  and  the  tail-fins  themselves  tend  at  last  to  disappear: 
these  changes  corresponding  to  what  have  been  described  as  the 
specific  characters  of  A.  milhausenii,  and  of  a  still  more  extreme 
form,  A.  koppeniana]  while  on  the  other  hand,  progressive  dilution 
of  the  water  tends '  to  precisely  opposite  conditions,  resulting  in 
forms  which  have  also  been  described  as  separate  species,  and  even 


u 


^wwwWWWH 
I 


I 


Artewia  s.str. 


Callaonella 


Fig.  77.  Brine-shrimps  (Artemia),  from  more  or  less  saline  water.  Upper  figures 
shew  tail-segment  and  tail-fins;  lower  figures,  relative  length  of  cephalothorax 
and  abdomen.     After  Abonyi. 

referred  to  a  separate  genus,  Callaonella,  closely  akin  to  Branchipus 
(Fig.  77).  Pari  passu  with  these  changes,  there  is  a  marked  change 
in  the  relative  lengths  of  the  fore  and  hind  portions  of  the  body, 
that  is  to  say,  of  the  cephalothorax  and  abdomen:  the  latter 
growing  relatively  longer,  the  Salter  the  water.  In  other  words, 
not  only  is  the  rate  of  growth  of  the  whole  animal  lessened  by  the 
sahne  concentration,  but  the  specific  rates  of  growth  in  the  parts 
of  its  body  are  relatively  changed.  This  latter  phenomenon  lends 
itself  to  numerical  statement,  and  Abonyi  has  shewn  that  we  may 
construct  a  very  regular  curve,  by  plotting  the  proportionate  length 
of  the  creature's  abdomen  against  the  salinity,  or  density,  of  the 
water;  and  the  several  species  of  Artemia,  with  all  their  other 
correlated  specific  characters,  are  then  found  to  occupy  successive, 
more  or  less  well-defined,  and  more  or  less  extended,  regions  of  the 


Ill] 


OF  THE  BRINE-SHRIMPS 


253 


curve  (Fig.  78).  In  short,  the  density  of  the  water  is  so  clearly 
"specific,"  that  we  might  briefly  define  Artemia  jelskii,  for  instance, 
as  the  Artemia  of  density  1000-1010  (NaCl),  or  all  but  fresh  water, 
and  the  typical  A.  salina  (or  principalis)  as  the  Artemia  of  density 
1018-1025,  and  so  on*. 

Koppeniana 
160  """^ 


140 


120 


100 


1000 


1020 


1080 


1100 


1040  1060 

Density  of  water 
Fig.  78,     Percentage  ratio  of  length  of  abdomen  to  cephalothorax 
in  brine -shrimps,  at  various  salinities.     After  Abonyi. 

These  Artemiae  are  capable  of  living  in  waters  not  only  of  great 
density,  but  of  very  varied  chemical  composition,  and  it  is  hard  to 
say  how  far  they  are  safeguarded  by  semi-permeabihty  or  by  specific 
properties  and  reactions  of  the  living  colloids  "j".     The  natron-lakes, 

*  Different  authorities  have  recognised  from  one  to  twenty  species  of  Artemia. 
Daday  de  Dees  {Ann.  sci.  nat.  1910)  reduces  the  salt-water  forms  to  one  species 
with  four  varieties,  but  keeps  A.  jelskii  in  a  separate  sub-genus.  Kuenen  suggests 
two  species,  A.  salina  and  gracilis,  one  for  the  European  and  one  for  the  American 
forms.  According  to  Schmankewitsch  every  systematic  character  can  be  shewn 
to  vary  with  the  external  medium.  Cf.  Professor  Labbe  on  change  of  characters, 
specific  and  even  generic,  of  Copepods  according  to  the  ^H  of  saline  waters  at 
Le  Croisic,  Nature,  March  10,  1928. 

t  We  may  compare  Wo.  Ostwald's  old  experiments  on  Daphnia,  which  died  in 
a  pure  solution  of  NaCl  isotonic  with  normal  sea-water.  Their  death  was  not  to 
be  explained  on  osmotic  grounds;  but  was  seemingly  due  to  the  fact  that  the 
organic  gels  do  not  retain  their  normal  water- content  save  in  the  presence  of  such 
concentrations  of  MgClj  (and  other  salts)  as  are  present  in  sea-water. 


254  THE  RATE  OF  GROWTH  [ch. 

for  instance,  contain  large  quantities  of  magnesium  sulphate;  and 
the  Artemiae  continue  to  live  equally  well  in  artificial  solutions 
where  this  salt,  or  where  calcium  chloride,  has  largely  replaced  the 
common  salt  of  the  more  usual  habitat.  Moreover,  such  waters  as 
those  of  the  natron-lakes  are  subject  to  great  changes  of  chemical 
composition  as  evaporation  and  concentration  proceed,  owing  to  the 
different  solubilities  of  the  constituent  salts;  but  it  appears  that 
the  forms  which  the  Artemiae  assume,  and  the  changes  which  they 
undergo,  are  identical,  or  indistinguishable,  whichever  of  the  above 
salts  happen  to  exist  or  to  predominate  in  their  saline  habitat.  At 
the  same  time  we  still  lack,  so  far  as  I  know,  the  simple  but  crucial 
experiments  which  shall  tell  us  whether,  in  solutions  of  different 
chemical  composition,  it  is  at  equal  densities,  or  at  isotonic  concen- 
trations (that  is  to  say,  under  conditions  where  the  osmotic  pressure, 
and  consequently  the  rate  of  diffusion,  is  identical),  that  the  same 
changes  of  form  and  structure  are  produced  and  corresponding 
phases  of  equihbrium  attained. 

Sea-water  has  been  described  as  an  instance  of  the  "fitness  of  the 
environment*"  for  the  maintenance  of  protoplasm  in  an  appropriate 
milieu;  but  our  Artemias  suffice  to  shew  how  nature,  when  hard 
put  to  it,  makes  shift  with  an  environment  which  is  wholly  abnormal 
and  anything  but  "fit." 

While  Hober  and  others  f  have  referred  all  these  phenomena  to 
osmosis,  Abonyi  is  inclined  to  believe  that  the  viscosity,  or 
mechanical  resistance,  of  the  fluid  also  reacts  upon  the  organism; 
and  other  possible  modes  of  operation  have  been  suggested.  But 
we  may  take  it  for  certain  that  the  phenomenon  as  a  whole  is  not 
a  simple  one.  We  should  have  to  look  far  in  organic  nature  for 
what  the  physicist  would  call  simple  osmosis  % ;  and  assuredly  there 
is  always  at  work,  besides  the  passive  phenomena  of  intermolecular 


*  L.  H.  Henderson,  The  Fitness  of  the  Environment,  1913. 

t  Cf.  Schmankewitsch,  Z.  f.  w.  Zool.  xxv,  ISTi);  xxix,  1877,.  etc.;  transl.  in 
appendix  to  Packard's  Monogr.  of  N.  American  Phyllopoda,  1^83,  pp.  466-514; 
Daday  de  Dees,  Ann.  Sci.  Nat.  (Zool),  (9),  xi,  1910;  Samter  und  Heymons,  Abh. 
d.  K.  pr.  Akad.  Wiss.  1902;  Bateson,  Mat.  for  the  Study  of  Variation,  1894,  pp. 
96-101;  Anikin,  Mitlh.  Kais.  Univ.  Tomsk,  xiv:  Zool.  Centralhl.  vi,  pp.  756-760, 
1908;  Abonyi,  Z.f.  w.  Zool.  cxiv,  pp.  96-168,  1915  (with  copious  bibliography),  etc. 

%  Cf.  C.  F.  A.  Pantin,  Body  fluids  in  animals,  Biol.  Reviews,  \i,  p.  4,  1931; 
J.  Duclaux,  Chimie  apjMquee  a  la  biologic,  1937,  ii,  chap.  4. 


Ill]  OF  CATALYTIC  ACTION  255 

diffusion,  some  other  activity  to  play  the  part  of  a  regulatory 
mechanism*. 

On  growth  and  catalytic  action 

In  ordinary  chemical  reactions  we  have  to  deal  (1)  with  a  specific 
velocity  proper  to  the  particular  reaction,  (2)  with  variations  due 
to  temperature  and  other  physical  conditions,  (3)  with  variations 
due  to  the  quantities  present  of  the  reacting  substances,  according 
to  Van't  Hoff's  "Law  of  Mass  Action,"  and  (4)  in  certain  cases  with 
variations  due  to  the  presence  of  "catalysing  agents,"  as  BerzeHus 
called  them  a  hundred  years  agof.  In  the  simpler  reactions,  the 
law  of  mass  involves  a  steady  slo wing-down  of  the  process  as  the 
reaction  proceeds  and  as  the  initial  amount  of  substance  diminishes: 
a  phenomenon,  however,  which  is  more  or  less  evaded  in  the  organism, 
part  of  whose  energies  are  devoted  to  the  continual  bringing-up  of 
supphes. 

Catalytic  action  occurs  when  some  substance,  often  in  very 
minute  quantity,  is  present,  and  by  its  presence  produces  or 
accelerates  a  reaction  by  opening  "a  way  round,"  without  the 
catalysing  agent  itself  being  diminished  or  used  up  J.  It  diminishes 
the  resistance  somehow — little  as  we  know  what  resistance  means 

*  According  to  the  empirical  canon  of  physiology,  that,  as  Leon  Fredericq 
expresses  it  (Arch,  rfc  Zool.  1885),  '*L'etre  vivant  est  agence  de  telle  maniere  que 
chaque  influence  pertyrbatrice  provoque  d'elle-meme  la  mise  en  activite  de  Tappareil 
compensateur  qui  doit  neutraliser  et  reparer  le  dommage."  Herbert  Spencer  had 
conceived  a  similar  principle,  and  thought  he  recognised  in  it  the  vis  medicutrix. 
Nahirae.  It  is  the  physiological  analogue  of  the  "principle  of  Le  Chatelier  "  (1888), 
with  this  important  difference  that  the  latter  is  a  rigorous  and  quantitative  law, 
ba8e<i  on  a  definite  and  stable  equilibrium.  The  close  relation  between  the  two  is 
maintained  by  Le  Dantec  {La  titabilite  de  la  Vie,  1910,  p,  24),  and  criticised  by 
Lotka  {Physical  Biology,  p,  283  seq.). 

t  In  a  paper  in  the  Berliner  Jahrbuch  for  1836,  This  paper  was  translated  in 
the  Edinburgh  New  Philosophical  Journal  in  the  following  year;  and  a  curioas 
little  paper  On  the  coagulation  of  albumen,  and  catalysis,  by  Dr  Samuel  Brown, 
followed  in  the  Edinburgh  Academic  Annual  for  1840, 

X  Such  phenomena  come  precisely  under  the  head  of  what  Bacon  called 
Instances  of  Magic:  "By  which  I  mean  those  wherein  the  material  or  efficient 
cause  is  scanty  and  small  as  compared  with  the  work  or  effect  produced;  so  that 
even  when  they  are  common,  they  seem  like  miracles,  some  at  first  sight,  others 
even  after  attentive  consideration.  These  magical  effects  are  brought  about  in 
three  ways. .  .[of  which  one  is]  by  excitation  or  invitaticm  in  another  body,  as  in 
the  magnet  which  excites  numberless  needles  without  losing  any  of  its  virtue,  or 
in  yeast  and  such-like."     Nov.  Org,,  cap.  li. 


256  THE  RATE  OF  GROWTH  [ch. 

in  a  chemical  reaction.  But  the  velocity-curve  is  not  altered  in 
form ;  for  the  amount  of  energy  in  the  system  is  not  affected  by  the 
presence  of  the  catalyst,  the  law  of  mass  exerts  its  eifect,  and  the 
rate  of  action  gradually  slows  down.  In  certain  cases  we  have 
the  remarkable  phenomenon  that  a  body  capable  of  acting  as  a 
catalyser  is  necessarily  formed  as  a  product,  or  by-product,  of  the 
main  reaction,  and  in  such  a  case  as  this  the  reaction- velocity  will 
tend  to  be  steadily  accelerated.  Instead  of  dwindhng  away,  such 
a  reaction  continues  with  an  ever-increasing  velocity:  always 
subject  to  the  reservation  that  limiting  conditions  will  in  time  make 
themselves  felt,  such  as  a  failure  of  some  necessary  ingredient  (the 
"law  of  the  minimum"),  or  the  production  of  some  substance  which 
shall  antagonise  and  finally  destroy  the  original  reaction.  Such  an 
action  as  this  we  have  learned,  from  Ostwald,  to  describe  as  "auto- 
catalysis."  Now  we  know  that  certain  products  of  protoplasmic 
metabohsm — we  call  them  enzymes — are  very  powerful  catalysers, 
a  fact  clearly  understood  by  Claude  Bernard  long  ago*;  and  we 
are  therefore  entitled,  to  that  extent,  to  speak  of  an  autocatalytic 
action  on  the  part  of  protoplasm  itself. 

Going  a  httle  farther  in  the  footsteps  of  Claude  Bernard,  Chodat 
of  Geneva  suggested  (as  we  are  told  by  his  pupil  Monnier)  that 
growth  itself  might  be  looked  on  as  a  catalytic,  or  autocatalytic 
reaction:  "On  peut  bien,  ainsi  que  M.  Chodat  I'a  propose,  considerer 
I'accroissement  comme  une  reaction  chimique  complexe,  dans 
laquelle  le  catalysateur  est  la  cellule  vivante,  et  les  corps  en  presence 
sont  I'eau,  les  sels  et  Facide  carboniquet-" 

A  similar  suggestion  was  made  by  Loeb,  in  connection  with  the 

*  "Les  diastases  contiennent,  en  definitive,  le  secret  de  la  vie.  Or,  les  actions 
diastatiques  nous  apparaissent  comme  des  phenomenes  catalytiques,  en  d'autres 
termes,  des  accelerations  de  vitesse  de  reaction."  Cf.  M.  F.  Porchet,  Rewie 
Scientifique,  18th  Feb.  1911.  For  a  last  word  on  this  subject,  see  W.  Frankenberger, 
Katalytische  Umsetzungen  in  homogenen  u.  enzymatischen  Systemen,  Leipzig,  1937. 

t  Cf.  R.  Chodat,  Principes  de  Botanique  (2nd  ed.),  1907,  p.  133;  A.  Monnier,  La 
loi  d'accroissement  des  vegetaux,  Publ.  de  VInst.  de  Bot.  de  VUniv.  de  Geneve  (7), 
m,  1905.  Cf.  W.  Ostwald,  Vorlesungen  iiber  Naturphilosophie,  1902,  p.  342; 
Wo.  Ostwald,  Zeitliche  Eigenschaften  der  Entwicklungsvorgange,  in  Roux's 
Vortrdge,  Heft  5,  1908;  Robertson,  Normal  growth  of  an  individual,  and  its 
biochemical  significance,  Arch.f.  Entw.  Mech.  xxv,  pp.  581-614;  xxvi,  pp.  108-118, 
1908;  S.  Hatai,  Growth-curves  from  a  dynamical  standpoint,  Anat.  Record,  v, 
p.  373,  1911;  A.  J.  Lotka,  Ztschr.  f.  physikal.  Chemie,  Lxxn,  p.  511,  1910;  lxxx, 
p.  159,  1912;   etc. 


Ill]  OF  AUTOCATALYSIS  257 

synthesis  of  nuclear  protoplasm,  or  nuclein;  for  he  remarked  that, 
as  in  an  autocatalysed  chemical  reaction  the  rate  of  synthesis 
increases  during  the  initial  stage  of  cell-division  in  proportion  to  the 
amount  of  nuclear  matter  already  there.  In  other  words,  one  of 
the  products  of  the  reaction,  i.e.  one  of  the  constituents  of  the 
nucleus,  accelerates  the  production  of  nuclear  from  cytoplasmic 
material.  To  take  one  more  instance,  Blackman  said,  in  the  address 
already  quoted,  that  "the  botanists  (or  the  zoologists)  speak  of 
growth,  attribute  it  to  a  specific  power  of  protoplasm  for  assimila- 
tion, and  leave  it  alone  as  a  fundamental  phenomenon;  but  they 
are  much  concerned  as  to  the  distribution  of  new  growth  in  innu- 
merable specifically  distinct  forms.  While  the  chemist,  on  the 
other  hand,  recognises  it  as  a  famihar  phenomenon,  and  refers  it  to 
the  same  category  as  his  other  known  examples  of  autocatalysis." 

Later  on,  Brailsford  Robertson  upheld  the  autocatalytic  theory 
with  skill  and  learning*;  and  knowing  well  that  growth  was  no 
simple  solitary  chemical  reaction,  he  thought  that  behind  it  lay  some 
one  master-reaction,  essentially  autocatalytic,  by  which  protoplasmic 
synthesis  was  effected  or  controlled.  He  adduced  at  least  one 
curious  case,  in  the  growth  and  multiphcation  of  the  Infusoria, 
which  can  hardly  be  described  otherwise  than  as  catalytic.  Two 
minute  individuals  (of  Enchelys  or  Colpodiuyn)  kept  in  the  same  drop 
of  water,  so  enhance  each  other's  rate  of  asexual  reproduction  that 
it  may  be  many  times  as  great  when  two  are  together  as  when  one 
is  alone;  the  phenomenon  has  been  called  allelocatalysis.  When  a 
single  infusorian  is  isolated,  it  multiplies  the  quicker  the  smaller  the 
drop  it  is  in — a  further  proof  or  indication  that  something  is  being 
given  oif,  in  this  instance  by  the  living  cells,  which  hastens  growth 
and  reproduction.  But  even  the  ordinary  multiplication  of  a 
bacterium,  which  doubles  its  numbers  every  few  minutes  till  (were 
it  not  for  hmiting  factors)  those  numbers  would  be  all  but  incal- 
culable in  a  day,  looks  like  and  has  been  cited  as  a  simple  but  most 
striking  instance  of  the  potentiahties  of  protoplasmic  catalysis. 

It  is  not  necessary  for  us  to  pursue  this  subject  much  further. 

*  T.  B.  Robertson,  The  Chemical  Basis  of  Growth  and  Senescence,  1923;  and 
earlier  papers.  Cf.  his  Multiplication  of  isolated  infusoria,  Biochem.  Journ.  xv, 
pp.  598-611,  1921;  cf.  Journ.  Physiol,  lvi,  pp.  404-412,  1921;  R.  A.  Peters, 
Substances  needed  for  the  growth  of. .  .Colpodiy,m,  Journ.  Physiol,  lv,  p.  1,  1921. 


258  THE  RATE  OE  GROWTH  [ch. 

It  is  sufficiently  obvious  that  the  normal  S-shaped  curve  of  growth 
of  an  organism  resembles  in  its  general  features  the  velocity-curve 
of  chemical  autocatalysis,  and  many  writers  have  enlarged  on  the 
resemblance;  but  the  S-shaped  curve  of  growth  of  a  population 
resembles  it  just  as  well.  When  the  same  curve  depicts  the  growth 
of  an  individual,  and  of  a  population,  and  the  velocity  of  a  chemical 
reaction,  it  is  enough  to  shew  that  the  analogy  between  these  is  a 
mathematical  and  not  a  physico-chemical  one.  The  sigmoid  curve 
of  growth,  common  to  them  all,  is  sufficiently  explained  as  an 
interference  effect,  due  to  opposing  factors  such  as  we  may  use  a 
differential  equation  to  express :  a  phase  of  acceleration  is  followed 
by  a  phase  of  retardation,  and  the  causes  of  both  are  in  each  case 
complex,  uncertain  or  unknown. .  Nor  are  points  of  difference  lacking 
between  the  chemical  and  the  biological  phenomena.  As  the 
chemical  reaction  draws  to  a  close,  it  is  by  the  gradual  attainment 
of  chemical  equihbrium;  but  when  organic  growth  comes  to  an  end, 
it  is  (in  all  but  the  lowest  organisms)  by  reason  of  a  very  different 
kind  of  equilibrium,  due  in  the  main  to  the  gradual  differentiation 
of  the  organism  into  parts,  among  whose  pecuHar  properties  or 
functions  that  of  growth  or  multiphcation  falls  into  abeyance. 

The  analogy  between  organic  growth  and  chemical  autocatalysis 
is  close  enough  to  let  us  use,  or  try  to  use,  just  such  mathematics  as 
the  chemist  applies  to  his  reactions,  and  so  to  reduce  certain  curves 
of  growth  to  logarithmic  formulae.  This  has  been  done  by  many,  and 
with  no  httle  success  in  simple  cases.  So  have  we  done,  partially, 
in  the  case  of  yeast ;  so  the  statisticians  and  actuaries  do  with  human 
populations;  so  we  may  do  again,  borrowing  (for  illustration)  a 
certain  well-known  study  of  the  growing  sunflower  (Figs.  79,  80). 
Taking  our  mathematics  from  elementary  physical  chemistry,  we 
learn  that : 

The  velocity  of  a  reaction  depends  on  the  concentration  a  of 
the  substance  acted  on:  V  varies  as  a, 

V  =  Ka. 

The  concentration  continually  decreases,  so  that  at  time  t  (in  a 
monomolecular  reaction), 


in 


OF  AUTOCATALYSIS 


259 


4         5         6         7 
Time,  in  weeks 


10        II 


Fig.  79.     Growth  of  sunflower-stem :   observed  and  calculated  curves. 
From  Reed  and  Holland. 


cms. 

250 


200 


150 


100 


50 


■  -■   ■               .    * 

•  ^ — 

•  >^ 

/ 

/Calculauted  (autocaialytic) 

/                  curve 

•  / 

•/ 

y^\    1    1    1 

1        1        1        1        1        1        t        1 

Weeks  1 


10       li       12 


Fig.  80.     Growth  of  sunflower-stem :   calculated  (autocatalytic)  curve. 
After  Reed  and  Holland. 


260  THE  RATE  OF  GROWTH  [ch. 

But  if  the  substance  produced  exercise  a  catalytic  effect,  then  the 
velocity  will  vary  not  only  as  above  but  will  also  increase  as  x 
increases:   the  equation  becomes 

V  =  -T-  ^k'x(a  —  x), 

Cut 

which  is  the  elementary  equation  of  autoca.talysis.     Integrating, 

at       a  —  X 

In  our  growth-problem  it  is  sometimes  found  convenient  to  choose 
for  our  epoch,  t',  the  time  when  growth  is  half-completed,  as  the 
chemist  takes  the  time  at  which  his  reaction  is  half-way  through; 
and  we  may  then  write  (with  a  changed  constant) 


This  is  the  physico-chemical  formula  which  Reed  and  Holland 
apply  to  the  growing  sunflower-stem — a  simple  case*.  For  a  we 
take  the  maximum  height  attained,  viz.  254-5  cm. ;  for  t\  the  epoch 
when  one-half  of  that  height  was  reached,  viz.  (by  interpolation) 
about  34-2  days.  Taking  an  observation  at  random,  say  that  for 
the  56th  day,  when  the  stem  was  228-3  cm.  high,  we  have 

K  in  this  case  is  found  to  be  0-043,  and  the  mean  of  all  such 
determinations  t  is  not  far  difierent. 

Applying  this  formula  to  successive  epochs,  we  get  a  calculated 
curve  in  close  agreement  with  the  observed  one;  and  by  well- 
known  statistical  methods  we  confirm,  and  measure,  its  "closeness 
of  fit."  But  jtist  as  the  chemist  must  vary  and  develop  his  funda- 
mental formula  to  suit  the  course  of  more  and  more  comphcated 
reactions,  so  the  biologist  finds  that  only  the  simplest  of  his  curves 

*  H.  S.  Reed  and  R.  H.  Holland,  The  growth-rate  of  an  annual  plant,  Helianthus, 
Proc.  Nat.  Acad,  of  Sci.  (Washington),  v,  p.  135,  1919;  cf.  Lotka,  op.  cit.,  p.  74, 
A  sifnilar  case  is  that  of  a  gourd,  recorded  by  A.  P.  Anderson,  Bull.  Survey, 
Minnesota,  1895,  and  analysed  by  T.  B.  Robertson,  ibid.  pp.  72-75. 

t  Better  determined,  especially  in  more  complex  cases,  by  the  method  of  least 
squares. 


Ill]  ITS  CHEMICAL  ASPECT  261 

of  growth,  or  only  portions  of  the  rest,  can  be  fitted  to  this  simplest 
of  formulae.  In  a  Hfe-time  are  many  ages;  and  no  all-embracing 
formula  covers  the  infant  in  the  womb,  the  suckling  child,  the 
growing  schoolboy,  the  old  man  when  his  work  is  done.  Besides, 
we  need  such  a  formula  as  a  biologist  can  understand !  One  which 
gives  a  mere  coincidence  of  numbers  may  be  of  little  use  or  none, 
unless  it  go  some  way  to  depict  and  explain  the  modus  operandi  of 
growth.  As  d'Ancona  puts  it :  "II  importe  d'apphquer  des  formules 
qui  correspondent  non  seulement  au  point  de  vue  geometrique,  mais 
soient  representees  par  des  valeurs  de  signification  biologique." 
A  mere  curve-diagram  is  better  than  an  empirical  formula;  for  it 
gives  us  at  least  a  picture  of  the  phenomenon,  and  a  qualitative 
answer  to  the  problem. 

Growth  of  sunflower-stem.     (After  Reed  and  Holland) 


1st  diff. 

15-8 

24-4 

33-3 

39-2 

38-4 

31-6 

22-6 

14-4 

8-5 

4-9 

2-8 


The  chemical  aspect  of  growth 

As  soon  as  we  touch  on  such  matters  as  the  chemical  phenomenon 
of  catalysis  we  are  on  the  threshold  of  a  subject  which,  if  we  were 
able  to  pursue  it,  would  lead  us  far  into  the  special  domain  of 
physiology ;  and  there  it  would  be  necessary  to  follow  it  if  we  were 
dealing  with  growth  as  a  phenomenon  in  itself,  instead  of  mainly 
as  a  help  to  our  study  and  comprehension  of  form.  The  whole 
question  of  diet,  of  overfeeding  and  underfeeding*,  would  present 

*  For  example,  A.  S.  Parker  has  shewn  that  mice  suckled  by  rats,  and  conse- 
quently much  overfed,  grow  so  quickly  that  in  three  weeks  they  reach  double  their 
normal  weight;  but  their  development  is  not  accelerated;  Ann.  Appl.  Biol,  xvi, 
1929. 


Height  (( 

cm.) 

j^ 

>  (days) 

Observed 

Calculated 

7 

17-9 

21-9 

14 

34-4 

37-7 

21 

67-8 

62-1 

28 

981 

95-4 

35 

1310 

134-6 

42 

1690 

173-0 

49 

205-5 

204-6 

56 

228-3 

227-2 

63 

247-1 

241-6 

70 

250-5 

250-1 

77 

253-8 

255-0 

84 

254-5 

257-8 

262  THE  RATE  OF  GROWTH  [ch. 

itself  for  discussion*.  But  without  opening  up  this  large  subject, 
we  may  say  one  more  passing  word  on  the  remarkable  fact  that 
certain  chemical  substances,  or  certain  physiological  secretions, 
have  the  power  of  accelerating  or  of  retarding  or  in  some  way 
regulating  growth,  and  of  so  influencing  the  morphological  features 
of  the  organism. 

To  begin  with  there  are  numerous  elements,  such  as  boron, 
manganese,  cobalt,  arsenic,  which  serve  to  stimulate  growth,  or 
whose  complete  absence  impairs  or  hampers  it;  just  as  there  are 
a  few  others,  such  as  selenium,  whose  presence  in  the  minutest 
quantity  is  injurious  or  pernicious.  The  chemistry  of  the  hving 
body  is  more  complex  than  we  were  wont  to  suppose. 

Lecithin  was  shewn  long  ago  to  have  a  remarkable  power  of 
stimulating  growth  in  animals t,  and  accelerators  of  plant-growth, 
foretold  by  Sachs,  were  demonstrated  by  Bottomley  and  others  J; 
the  several  vitamins  are  either  accelerators  of  growth,  or  are  indis- 
pensable in  order  that  it  may  proceed. 

In  the  little  duckweed  of  our  ponds  and  ditches  [Lemna  minor)  the  botanists 
have  found  a  plant  in  which  growth  and  multiplication  are  reduced  to  very 
simple  terms.  For  it  multiplies  by  budding,  grows  a  rootlet  and  two  or  three 
leaves,  and  buds  again;  it  is  all  young  tissue,  it  carries  no  dead  load;  while 
the  sun  shines  it  has  no  lack  of  nourishment,  and  may  spread  to  the  limits  of 
the  pond.  In  one  of  Bottomley's  early  experiments,  duckweed  was  grown 
(1)  in  a  "culture  solution"  without  stint  of  space  or  food,  and  (2)  in  the  same, 
with  the  addition  of  a  little  bacterised  peat  or  "auximone."  In  both  cases  the 
little  plant  spread  freely,  as  in  the  first,  or  Malthusian,  phase  of  a  population 
curve;  but  the  peat  greatly  accelerated  the  rate,  which  was  not  slow  before. 
Without  the  auximone  the  population  doubled  in  nine  or  ten  days,  and  with 
it  in  five  or  six;  but  in  two  months  the  one  was  seventy-fold  the  other ! 

The  subject  has  grown  big  from  small  beginnings.  We  know 
certain  substances,  haematin  being  one,  which  stimulate  the  growth 
of  bacteria,  and  seem  to  act  on  them  as  true  catalysts.  An  obscure 
but  complex  body  known  as  "bios"  powerfully  stimulates  the 
growth  of  yeast;  and  the  so-called  auxins,  a  name  which  covers 
numerous  bodies  both  nitrogenous  and  non-nitrogenous,  serve  in 

*  For  a  brief  resume  of  this  subject  see  Morgan's  Experimental  Zoology,  chap.  xvi. 

t  Hatai,  Amer.  Joum.  Physiology,  x,  p.  57,  1904;  Danilewsky,  C.R.  cxxi,  cxxn, 
1895-96. 

X  W.  B.  Bottomley,  Proc.  R.S.  (B),  lxxxviii,  pp.  237-247,  1914,  and  other 
papers.     O.  Haberlandt,  Beitr.  z.  allgem.  Botanik,  1921. 


Ill]  OF  HORMONES  263 

minute  doses  to  accelerate  the  growth  of  the  higher  plants*.  Some 
of  these  "growth-substances"  have  been  extracted  from  moulds  or 
from  bacteria,  and  one  remarkable  one,  to  which  the  name  auxin 
is  especially  applied,  from  seedhng  oats.  This  last  is  no  enzyme 
but  a  stable  non-nitrogfenous  substance,  which  seems  to  act  by 
softening  the  cell- wall  and  so  facihtating  the  expansion  of  the  cell. 
Lastly  the  remarkable  discovery  has  been  made  that  certain  indol- 
compounds,  comparatively  simple  bodies,  act  to  all  intents  and 
purposes  in  the  same  way  as  the  growth-hormones  or  natural 
auxins,  and  one  of  these  "hetero-auxins."  an  indol-acetic  acidf, 
is  already  in  common  and  successful  use  to  promote  the  growth  and 
rooting  of  cuttings. 

Growth  of  duckweed,  with  and  without  peat-auodmone 

Without  With 


^ 

^ 

eeks              Obs. 

Calc. 

Obs. 

Calc. 

0                     20 

20 

20 

20 

1                      30 

33 

38 

55 

2                     52 

54 

102 

153 

.1                      77 

88 

326 

424 

4                    135 

155 

1,100 

1,173 

5                   211 

237 

3,064 

3,250 

6                   326 

390 

6,723 

8,980 

7                    550 

640 

19,763 

2,490 

8                   1052 

1048 

69,350 

68,800 

Percentage  increase, 

164  o/o 

2770/0 

per  week 

There  are  kindred  matters  not  less  interesting  to  the  morphologist. 
It  has  long  been  known  that  the  pituitary  body  produces,  in  its 
anterior  lobe,  a  substance  by  which  growth  is  increased  and  regulated. 
This  is  what  we  now  call  a  "hormone" — a  substance  produced  in 
one  organ  or  tissue  and  regulating  the  functions  of  another.  In  this 
case  atrophy  of  the  gland  leaves  the  subject  a  dwarf,  and  its  hyper- 

*  The  older  literature  is  summarised  by  Stark,  Ergebn.  d.  Biologies  n,  1906; 
the  later  by  N.  Nielsen,  Jh.  wiss.  Botan.  Lxxm,  1930;  by  Boyson  Jensen,  Die 
Wuchsstojftheorie,  1935;  by  F.  W.  Went  and  K.  V.  Thimann,  Phytohormones,  New 
York,  1937,  and  by  H.  L.  Pearse,  Plant  hormones  and  their  practical  importance. 
Imp.  Bureau  of  Horticulture,  1939.  Cf.  Went,  Bee.  d.  Trav.  Botan.  Neerl.  xxv,  p.  1, 
1928;  A.  N.  J.  Heyn,  ibid,  xxviii,  p.  113,  1931. 

t  Discovered  by  Kogl  and  Kostermans,  Ztschr.  f.  physiol.  Chem.  ccxxxv,  p.  201, 
1934.  Cf.  {int.  al.)  P.  W.  Zimmermann  and  F.  W.  Wilcox  in  Contrib.  Boyce- 
Thompson  Instil.  1935. 


264  THE  RATE  OF  GROWTH  264 

trophy  or  over-activity  goes  to  the  making  of  a  giant;  the  Umb- 
bones  of  the  giant  grow  longer,  their  epiphyses  get  thick  and  clumsy, 
and  the  deformity  known  as  "acromegaly"  ensues*.  This  has 
become  a  famihar  illustration  of  functional  regulation,  by  some 
glandular  or  "endocrinal"  secretion,  some  enzyme  or  harmozone 
as  Gley  called  it,  or  hormone'f  as  Bayliss  and  Starhng  called  it — in 
the  particular  case  where  the  function  to  be  regulated  is  growth, 
with  its  consequent  influence  on  form.  But  we  may  be  sure  that 
this  so-called  regulation  of  growth  is  no  simple  and  no  specific  thing, 
but  imphes  a  far-reaching  and  complicated  influence  on  the  bodily 
metabohsmj. 

Some  say  that  in  large  animals  the  pituitary  is.  apt  to  be  dispro- 
portionately large §;  and  the  giant  dinosaur  Branchiosaurus,  hugest 
of  land  animals,  is  reputed  to  have  the  largest  hypophyseal  recess 
(or  cavity  for  the  pituitary  body)  ever  observed. 

The  thyroid  also  has  its  part  to  play  in  growth,  as  Gudernatsch 
was  the  first  to  shew||;  perhaps  it  acts,  as  Uhlenhorth  suggests, 
by  releasing  the  pituitary  hormone.  In  a  curious  race  of  dwarf 
frogs  both  thyroid  and  pituitary  were  found  to  be  atrophied  ^.  When 
tadpoles  are  fed  on  thyroid  their  legs  grow  out  long  before  the  usual 
time;  on  the  other  hand  removal  of  the  thyroid  delays  metamor- 
phosis, and  the  tadpoles  remain  tadpoles  to  an  unusual  size**. 

The  great  American  bull-frog  (R.  Catesheiana)  fives  for  two  or 
three  years  in  tadpole  form;  but  a  diet  of  thyroid  turns  the  little 
tadpoles  into  bull-frogs  before  they  are  a  month  old  If-    The  converse 

*  Cf.  E.  A.  Schafer,  The  function  ofjthe  pituitary  body,  Proc.  R.S.  (B),  lxxxi, 
p.  442,  1904. 

t  It  is  not  easy  to  dtaw  a  line  between  enzyme  and  vitamin,  or  between  hormone 
and  enzyme. 

X  The -physiological  relations  between  insulin  and  the  pituitary  body  might 
seem  to  indicate  that  it  is  the  carbohydrate  metabolism  which  is  more  especially 
concerned.     Cf.  (e.g.)  Eric  Holmes,  Metabolism  of  the  Living  Tissue,  1937. 

§  Van  der  Horst  finds  this  to  be  the  case  in  Zalophus  and  in  the  ostrich,  compared 
with  smaller  seals  or  birds;   cf.  Ariens  Kappers,  Journ.  Anat.  lxiv,  p.  256,  1930. 

II   Gudernatsch,  in  Arch.  f.  Entw.  Mech.  xxxv,  1912. 

<;;  Eidmann,  ibid,  xlix,  pp.  510-537,  1921. 

**  Allen,  Journ.  Exp.  Zod.  xxiv,  p.  499,  1918.  Cf.  [int.  al.)  E.  Uhlenhuth, 
Experimental  production  of  gigantism,  Journ.  Gen.  Physiol,  ill,  p.  347;  iv,  p.  321, 
1921-22. 

tt  W.  W.  Swingle,  Journ.  Exp.  Zool.  xxiv,  1918;  xxxvn,  1923;  Journ.  Gen. 
Physiol.  I,  II,  1918-19;   etc! 


Ill]  THE  THYROID  GLAND  .       265 

experiment  has  been  performed  on  ordinary  tadpoles*;  with  their 
thyroids  removed  they  remain  normal  to  all  appearance,  but  the 
weeks  go  by  and  metamorphosis  does  not  take  place.  Gill-clefts 
and  tail  persist,  no  limbs  appear,  brain  and  gut  retain  their  larval 
features;  but  months  after,  or  apparently  at  any  time,  the  belated 
tadpoles  respond  to  a  diet  of  thyroid,  and  may  be  turned  into  frogs 
by  means  of  it.  The  Mexican  axolotl  is  a  grown-up  tadpole  which, 
when  the  ponds  dry  up  (as  they  seldom  do),  completes  its  growth 
and  turns  into  a  gill-less,  lung-breathing  newt  or  salamander!;  but 
feed  it  on  thyroid,  even  for  a  single  meal,  and  its  metamorphosis  is 
hastened  and  ensured  {. 

Much  has  been  done  since  these  pioneering  experiments,  all  going 
to  shew  that  the  thyroid  plays  its  active  part  in  the  tissue-changes 
which  accompany  and  constitute  metamorphosis.  It  looks  as  though 
more  thyroid  meant  more  respiratory  activity,  more  oxygen- 
consumption,  more  oxidative  metabohsm,  more  tissue-change,  hence/ 
earlier  bodily  development  §.  Pituitary  and  thyroid  are  very  different 
things;  the  one  enhances  growth,  the  other  retards  it.  Thyroid 
stimulates  metabolism  and  hastens  development,  but  the  tissues 
waste. 

It  is  a  curious  fact,  but  it  has  often  been  observed,  that  starvation 
or  inanition  has,  in  the  long  run,  a  similar  effect  of  hastening 
metamorphosis  II .     The  meaning  of  this  phenomenon  is  unknown. 

An  extremely  remarkable  case  is  that  of  the  "galls",  brought 
into  existence  on  various  plants  in  response  to  the  prick  of  a  small 
insect's  ovipositor.    One  tree,  an  oak  for  instance,  may  bear  galls 

*  Bennett  Allen,  Biol.  Bull,  xxxii,  1917;  Journ.  Exp.  Zool.  xxiv,  1918;  xxx, 
1920;   etc. 

t  Colorado  axolotls  are  much  more  apt  to  metamorphose  than  the  Mexican 
variety. 

J  Babak,  Ueber  die  Beziehung  der  Metamorphose . .  .  zur  inneren  Secretion, 
Centralbl  f.  Physiol,  x,  1913.  Cf.  Abderhalden,  Studien  iiber  die  von  einzelrien 
Organen  hervorgebrachten  Substanzen  mit  spezifischer  Wirkung,  Pfliiger's  Archiv, 
CLXii,  1915. 

§  Certain  experiments  by  M.  Morse  {Journ.  Biol.  Chem.  xix,  1915)  seeihed  to 
shew  that  the  effect  of  thyroid  on  metamorphosis  depended  on  iodine;  l;^t  the 
case  is  by  no  means  clear  (cf.  0.  Shinryo,  Sci.  Rep.  Tohoku  Univ.  iii,  1928,  and 
others).  The  axolotl  is  said  to  shew  little  response  to  experimental  iodine,  and 
its  ally  Necturus  none  at  all  (cf.  B.  M.  Allen,  in  Biol.  Reviews,  xiii,  1939). 

!|  Cf.  Krizensky,  Die  beschleunigende  Ein wirkung  des  Hungerns  auf  die  Meta- 
morphose, Biol.  Centralbl.  xi.iv,  1914.     Cf.  antea,  p.  170. 


266  THE  RATE  OF  GROWTH  [ch. 

of  many  kinds,  well-defined  and  widely  different,  each  caused  to 
grow  out  of  the  tissues  of  the  plant  by  a  chemical  stimulus  contri- 
buted by  the  insect,  in  very  minute  amount;  and  the  insects  are 
so  much  alike  that  the  galls  are  easier  to  distinguish  than  the  flies. 
The  same  insect  may  produce  the  same  gall  on  different  plants, 
for  instance  on  several  species  of  willow ;  or  sometimes  on  different 
parts,  or  tissues,  of  the  same  plant.  Small  pieces  of  a  dead  larva 
have  been  used  to  infect  a  plant,  and  a  gall  of  the  usual  kind  has 
resulted.  Beyerinck  killed  the  eggs  with  a  hot  wire  as  soon  as 
they  were  deposited  in  the  tree,  yet  the  galls  grew  as  usual.  Here, 
as  Needham  has  lately  pointed  out,  is  a  great  field  for  reflection 
and  future  experiment.  The  minute  drop  of  fluid  exuded  by  the 
insect  has  marvellous  properties.  It  is  not  only  a  stimulant  of 
growth,  like  any  ordinary  auxin  or  hormone;  it  causes  the  growth 
of  a  peculiar  tissue,  and  shapes  it  into  a  new  and  specific  form*. 

Among  other  illustrations  (which  are  plentiful)  of  the  subtle 
influence  of  some  substance  upon  growth,  we  have,  for  instance, 
the  growth  of  the  placental  decidua,  which  Loeb  shewed  to  be  due 
to  a  substance  given  off  by  the  corpus  luteum,  lending  to  the  uterine 
tissues  an  enhanced  capacity  for  growth,  to  be  called  into  action  by 
contact  with  the  ovum  or  even  of  a  foreign  body.  Various  sexual 
characters,  such  as  the  plumage,  comb  and  spurs  of  the  cock,  arise 
in  hke  manner  in  response  to  an  internal  secretion  or  "male 
hormone  " ;  and  when  castration  removes  the  source  of  the  secretion, 
well-known  morphological  changes  take  place.  When  a  converse 
change  takes  place  the  female  acquires,  in  greater  or  less  degree, 
characters  which  are  proper  to  the  male:  as  in  those  extreme  cases, 
known  from  time  immemorial,  when  an  old  and  barren  hen  assumes 
the  plumage  of  the  cockf. 

The  mane  of  the  lion,  the  antlers  of  the  stag,  the  tail  of  the  peacock, 
are  all  examples  of  intensifled  differential  growth,  or  localised  and 

*  Joseph  Needham,  Aspects  nouveaux  de  la  chimie  et  de  la  biologic  de  la  croia- 
sance  organisee.  Folia  Morphologica,  Warszawa,  viii,  p.  32,  1938.  On  galls,  see 
{int.  al.)  Cobbold,  Ross  und  Hedicke,  Die  Pflanzengallen,  Jena,  1927;  etc.  And 
on  their  "raorphogenic  stimulus",  cf.  Herbst,  Biolog.  Cblt.,  1894-5,  passim. 

t  The  hen  which  assumed  the  voice  and  plumage  of  the  male  was  a  portent  or 
omen — gallina  cecinit.  The  first  scientific  account  was  John  Hunter's  celebrated 
Account  of  an  extraordinary  pheasant,  and  Of  the  appearance  of  the  change 
of  sex  in  Lady  Tynte's  peahen,  Phil.  Trans,  lxx,  pp.  527,  534,  1780. 


Ill] 


THE  MALE  HORMONES 


267 


sex-linked  hypertrophy;  and  in  the  singular  and  striking  plumage 
of  innumerable  birds  we  may  easily  see  how  enhanced  growth  of  a 
tuft  of  feathers,  perhaps  exaggeration  of  a  single  plume,  is  at 
the  root  of  the  whole  matter.  Among  extreme  instances  we  may 
think  of  the  immensely  long  first  primary  of  the  pennant-winged 
nightjar;  of  the  long  feather  over  the  eye  in  Pteridophora  alberti, 


Fig.  81.     A  single  pair  of  hypertrophied  feathers  in  a  bird-of-paradise, 
Pteridophora  alberti. 


.-I 


Fig.  82.     Unequal  growth  in  the  three  pairs  of  tail-feathers  of  a  humming-bird 
{Loddigesia).     1,  rudimentafy:  2,  short  and  stiff;  3,  long  and  spathulate. 

or  the  Fix  long  plumes  over  or  behind  the  eye  in  the  six-shafted 
bird-of-paradise;  or  among  the  humming-birds,  of  the  long  outer 
rectrix  in  Lesbia,  the  second  outer  one  in  Aethusa,  or  of  the  extra- 
ordinary inequalities  of  the  tail-feathers  of  Loddigesia  mirabilis, 
some  rudimentary,  some  short  and  straight  and  stiff,  and  other  two 
immensely  elongated,  curved  and  spathulate.  The  sexual, hormones 
have  a  potent  influence  on  the  plumage  of  a>bird ;  they  serve,  somehow, 
to  orientate  and  regulate  the  rate  of  growth  from  one  feather-tract 
to  another,  and  from  one  end  to  another,  even  from  one  side  to  the 
other,  of  a  single  feather.     An  extreme  case  is  the  occasional  pheno- 


268  THE  RATE  OF  GROWTH  [ch. 

menon  of  a  "  gynandrous "  feather,  male  and  female  on  two  sides 
of  the  same  vane*. 

While  unequal  or  differential  growth  is  of  pecuhar  interest  to 
the  morphologist,  rate  of  growth  pure  and  simple,  with  all  the 
agencies  which  control  or  accelerate  it,  remains  of  deeper  importance 
to  the  practical  man.  The  live-stock  breeder  keeps  many  desirable 
quahties  in  view:  constitution,  fertility,  yield  and  quahty  of  milk 
or  wool  are  some  of  these;  but  rate  of  growth,  with  its  corollaries 
of  early  maturity  and  large  ultimate  size,  is  generally  more  important 
than  them  all.  The  inheritance  of  size  is  somewhat  complicated, 
and  limited  from  the  breeder's  point  of  view  by  the  mother's 
inability  to  nourish  and  bring  forth  a  crossbred  offspring  of  a  breed 
larger  than  her  own.  A  cart  mare,  covered  by  a  Shetland  sire, 
produces  a  good-sized  foal;  but  the  Shetland  mare,  crossed  with 
a  carthorse,  has  a  foal  a  little  bigger,  but  not  much  bigger,  than 
herself  (Fig.  83).  In  size  and  rate  of  growth,  as  in  other  qualities, 
our  farm  animals  differ  vastly  from  their  wild  progenitors,  or  from 
the  "  un-improved "  stock  in  days  before  Bake  well  and  the  other 
great  breeders  began.  The  improvement  has  been  brought  about 
by  "selection";  but  what  lies  behind?  Endocrine  secretions, 
especially  pituitary,  are  doubtless  at  work;  and  already  the  stock- 
raiser  and  the  biochemist  may  be  found  hand  in  hand. 

If  we  once  admit,  as  we  are  now  bound  to  do,  the  existence  of 
factors  which  by  their  physiological  activity,  and  apart  from  any 
direct  action  of  the  nervous  system,  tend  towards  the  acceleration 
of  growth  and  consequent  modification  of  form,  we  are  led  into  wide 
fields  of  speculation  by  an  easy  and  a  legitimate  pathway.  Professor 
Gley  carries  such  speculations  a  long,  long  way:  for  He  saysf  that 
by  these  chemical  influences  "Toute  une  partie  de  la  construction 
des  etres  parait  s'expliquer  d'une  fayon  toute  mecanique.  La  forteresse, 
si  longtemps  inaccessible,  du  vitahsme  est  entamee.  Car  la  notion 
morphogenique  etait,  suivant  le  mot  de  Dastre  J ,  comme  '  le  dernier 
reduit  de  la  force  vitale'." 

*  See  an  interesting  paper  by  Frank  R,  Lillie  and  Mary  Juhn,  on  The  physiology 
of  development  of  feathers:  I,  Growth-rate  and  pattern  in  the  individual  feather. 
Physiological  Zoology,  v,  pp.  124-184,  1932,  and  many  papers  quoted  therein. 

f  Le  Neo-vitalisme,  Revue  Scientifique,  March  1911. 

X  La  Vie  et  la  Mart,  1902,  p.  43. 


Ill] 


OF  INHERITANCE  OF  SIZE 


269 


The  physiological  speculations  we  need  not  discuss:  but,  to  take 
a  single  example  from  morphology,  we  begin  to  understand  the 
possibihty,  and  to  comprehend  the  probable  meaning,  of  the  all  but 
sudden  appearance  on  the  earth  of  such  exaggerated  and  almost 
monstrous  forms  as  those  of  the  great  secondary  reptiles  and  the 


500 


Pure  Shetland 


10  20 

Age,  in  months 


30 


40 


Fig.  83. 


Effect  of  cross-breeding  on  rate  of  growth  in  Shetland  ponies. 
From  Walton  and  Hammond's  data.* 


great  tertiary  mammals  f.  We  begin  to  see  that  it  is  in  order  to 
account  not  for  the  appearance  but  for  the  disappearance  of  such 
forms  as  these  that  natural  selection  must  be  invoked.  And  we 
then,  I  think,  draw  near  to  the  conclusion  that  what  is  true  of  these 
is  universally  true,  and  that  the  great  function  of  natural  selection 


*  Walton  and  Hammond,  Proc.  R.S.  (B),  No.  840,  p.  317,  1938. 
t  Cf.  also  Dendy,  Evolutionary  Biology,  1912,  p.  408. 


270  THE  RATE  OF  GROWTH  [ch. 

is  not  to  originate*  but  to  remove:  donee  ad  interitum  genus  id 
natura  redegitf. 

The  world  of  things  living,  like  the  world  of  things  inanimate, 
grows  of  itself,  and  pursues  its  ceaseless  course  of  creative  evolution. 
It  has  room,  wide  but  not  unbounded,  for  variety  of  living  form 
and  structure,  as  these  tend  towards  their  seemingly  endless  but 
yet  strictly  hmited  possibilities  of  permutation  and  degree^  it  has 
room  for  the  great,  and  for  the  small,  room  for  the  weak  and  for  the 
strong.  Environment  and  circumstance  do  not  always  make  a 
prison,  wherein  perforce  the  organism  must  either  live  or  die;  for 
the  ways  of  hfe  may  be  changed,  and  many  a  refuge  found,  before 
the  sentence  of  unfitness  is  pronounced  and  the  penalty  of  exter- 
mination paid.  But  there  comes  a  time  when  "variation,"  in  form, 
dimensions,  or  other  qualities  of  the  organism,  goes  further  than  is 
compatible  with  all  the  means  at  hand  of  health  and  welfare  for 
the  individual  and  the  stock;  when,  under  the  active  and  creative 
stimulus  of  forces  from  within  and  from  without,  the  active  and 
creative  energies  of  growth  pass  the  bounds  of  physical  and 
physiological  equilibrium:  and  so  reach  the  Hmits  which,  as  again 
Lucretius  tells  us,  natural  law  has  set  between  what  may  and  what 
may  not  be, 

et  quid  quaeque  queant  per  foedera  natural 
quid  porro  nequeant. 

Then,  at  last,  we  are  entitled  to  use  the  customary  metaphor,  and 
to  see  in  natural  selection  an  inexorable  force  whose  function  is  not 
to  create  but  to  destroy — to  weed,  to  prune,  to  cut  down  and  to 
cast  into  the  fire  J. 

*  So  said  Yves  Delage  {UherediU,  1903,  p.  397):  "La  selection  naturelle  est  un 
principe  admirable  et  parfaitement  juste.  Tout  le  monde  est  d'accord  sur  ce  point. 
Mais  ou  Ton  n'est  pas  d'accord,  c'est  sur  la  limite  de  sa  puissance  et  sur  la  question 
de  savoir  si  elle  pent  engendrer  des  formes  specifiques  nouvelles.  II  semble  bien 
demontre  aujourd'hui  qu'elle  ne  le  pent  pas.'' 

t  Lucret.  v,  875,  "Lucretius  nowhere  seems  to  recognise  the  possibility  of 
improvement  or  change  of  species  by  'natural  selection';  the  animals  remain  as 
they  were  at  the  first,  except  that  the  weaker  and  more  useless  kinds  have  been 
crushed  out.  Hence  he  stands  in  marked  contrast  with  modern  evolutionists." 
Kelsey's  note,  ad  loc. 

X  Even  after  we  have  so  narrowed  its  scope  and  sphere,  natural  selection  is 
still  a  hard  saying ;  for  the  causes  of  extinction  are  wellnigh  as  hard  to  understand 
as  are  those  of  the  origin  of  species.     If  we  assert  (as  has  been  lightly  and  too 


Ill]  OF  REGENERATIVE  GROWTH  271 

Of  regeneration,  or  growth  and  repair 

The  phenomenon  of  regeneration,  or  the  restoration  of  lost  or 
amputated  parts,  is  a  particular  case  of  growth  which  deserves 
separate  consideration.  It  is  a  property  manifested  in  a  high 
degree  among  invertebrates  and  many  cold-blooded  vertebrates, 
diminishing  as  we  ascend  the  scale,  until  it  lessens  down  in  the 
warm-blooded  animals  to  that  vis smedicatrix  which  heals  a  wound. 
Ever  since  the  days  of  Aristotle,  and  still  more  since  the  experiments 
of  Trembley,  Reaumur  and  Spallanzani  in  the  eighteenth  century, 
physiologist  and  psychologist  alike  have  recognised  that  the  pheno- 
menon is  both  perplexing  and  important.  "Its  discovery,"  said 
Spallanzani,  "  was  an  immense  addition  to  the  riches  of  organic  philo- 
sophy, and  an  inexhaustible  source  of  meditation  for  the  philosopher." 
The  general  phenomenon  is  amply  treated  of  elsewhere*,  and  we 
need  only  deal  with  it  in  its  immediate  relation  to  growth. 

Regeneration,  like  growth  in  other  cases,  proceeds  with  a  velocity 
which  varies  according  to  a  definite  law;  the  rate  varies  with  the 
time,  and  we  may  study  it  as  velocity  and  as  acceleration.  Let  us 
take,  as  an  instance,  Miss  M.  L.  Durbin's  measurements  of  the  rate 
.of  regeneration  of  tadpoles'  tails :  the  rate  being  measured  in  terms 
of  length,  or  longitudinal  increment  f.  From  a  number  of  tadpoles, 
whose  average  length  was  in  one  experiment  34  mm.,  and  in  another 
49  mm.,  about  half  the  tail  was  cut  off,  and  the  average  amounts 
regenerated  in  successive  periods  are  shewn  as  follows : 

Days                              3         5         7  10  12  14  17  18  24  28  30 

Amount  regenerated  (mm.): 

First  experiment      1-4      —  3-4  4-3  —  5-2  —  5-5  6-2  —  6o 

Second       „             0-9      2-2  3-7  5-2  60  6-4  7-1  —  7-6  8-2  8-4 

confidently  done)  that  Smilodon  perished  on  account  of  its  gigantic  tusks,  that 
Teleosaurus  was  handicapped  by  its  exaggerated  snout,  or  Stegosaurus  weighed 
down  by  its  intolerable  load  of  armour,  we  may  call  to  mind  kindred  forms  where 
similar  conditions  did  not  lead  to  rapid  extermination,  or  where  extinction  ensued 
apart  from  any  such  apparent  and  visible  disadvantages.  Cf.  F.  A.  Lucas,  On 
momentum  in  variation,  Amer.  Nat.  xli,  p.  46,  1907. 

*  See  Professor  T.  H.  Morgan's  Regeneration  (316  pp.),  1901,  for  a  full  account 
and  copious  bibliography.  The  early  experiments  on  regeneration,  by  V^allisneri, 
Dicquemare,  Spallanzani,  Reaumur,  Trembley,  Baster,  Bonnet  and  others,  are 
epitomised  by  Haller,  Elementa  Physiologiae,  viii,  pp.  156  seq. 

t  Journ.  Exper.  Zool.  vii,  p.  397,  1909. 


272 


THE  RATE  OF  GROWTH 


[CH. 


Both  experiments  give  us  fairly  smooth  curves  of  growth  within 
the  period  of  the  observations;  and,  with  a  shght  and  easy  extra- 
polation, both  curves  draw  to  the  base-hne  at  zero  (Fig.  84).    More- 


Fig.  84.     Curve  of  regenerative  growth  in  tadpoles'  tails. 
From  M.  L.  Durbin's  data. 


Fig.  85.     Tadpoles'  tails : ,  amount  regenerated  daily,  in  mm. 
(Smoothed  curve). 

over,  if  from  the  smoothed  curves  we  deduce  the  daily  increments, 
we  get  (Fig.  85)  a  bell-shaped  curve  similar  to  (or  to  all  appearance 
identical  with)  a  skew  curve  of  error.  In  point  of  fact,  this  instance 
of  regeneration  is   a  very  ordinary  example    of  growth,    with   its 


Ill] 


OF  REGENERATION 


273 


S-shaped  curve  of  integration  and  its  bell-shaped  differential  curve, 
just  as  we  have  seen  it  in  simple  cases,  or  simple  phases,  of  'the 
growth  of  a  population  or  an  individual. 

If  we  amputate  one  limb  of  a  pair  in  some  animal  with  rapid 
powers  of  regeneration,  we  may  compare  from  time  to  time  the 
dimensions  of  the  regenerating  hmb  with  those  of  its  uninjured 
fellow,  and  so  deal  with  a  relative  rather  than  an  absolute  velocity. 
The  legs  of  insect-larvae  are  easily  restored,  but  after  pupation  no 
further  growth  or  regeneration  takes  place.  An  easy  experiment, 
then,  is  to  remove  a  limb  in  larvae  of  various  ages,  and  to  compare 


100        120 


Fig.  86.     Regenerative  growth  in  mealworms'  legs. 

at  leisure  in  the  pupa  the  dimensions  of  the  new  Hmb  with  the  old. 
The  following  much-abbreviated  table  shews  the  gradual  increase 
of  a  regenerating  limb  in  a  mealworm,  up  to  final  equahty  with  the 
normal  Hmb,  the  rate  varying  according  to  the  usual  S-shaped 
curve*  (Fig.  86). 


Rate  of  regeneration  in  the  mealworm  (Tenebrio  moHtor,  larva) 


Days  after  amputation 
%  ratio  of  new  limb  to  old 


0      16      21       25      34      44      58      70      100      121 
0       7       11      20      29      42      71      83       91       100 


*  From  J.  Krizenecky,  Versuch  zur  statisch-graphischen  Untersuchung .  .  .der 
Regenerationsvorgange,  Arch.  f.  Entw.  Mech.  xxxix,  1914;   xlii,  1917. 


274  THE  RATE  OF  GROWTH  [ch. 

Some  writers  have  found  the  curve  of  regenerative  growth  to  be 
different  from  the  curve  of  ordinary  growth,  and  have  commented 
on  the  apparent  difference;  but  they  have  been  misled  (as  it  seems 
to  me)  by  the  fact  that  regeneration  is  seen  from  the  start  or  very 
nearly  so,  while  the  ordinary  curves  of  growth,  as  they  are  usually 
presented  to  us,  date  not  from  the  beginning  of  growth,  but  from 
the  comparatively  late,  and  unimportant,  and  even  fallacious  epoch 
of  birth.  A  complete  curve  of  growth,  starting  from  zero,  has  the 
same  essential  characteristics  as  the  regeneration  curve. 

Indeed  the  more  we  consider  the  phenomenon  of  regeneration, 
the  more  plainly  does  it  shew  itself  to  us  as  but  a  particular  case 
of  the  general  phenomenon  of  growth*,  following  the  same  lines, 
obeying  the  same  laws,  and  merely  started  into  activity  by  the 
special  stimulus,  direct  or  indirect,  caused  by  the  infliction  of  a 
wound.  Neither  more  nor  less  than  in  other  problems  of  physiology 
are  we  called  upon,  in  the  case  of  regeneration,  to  indulge  in 
metaphysical  speculation,  or  to  dwell  upon  the  beneficent  purpose 
which  seemingly  underhes  this  process  of  heahng  and  repair. 

It  is  a  very  general  rule,  though  not  a  universal  one,  that 
regeneration  tends  to  fall  somewhat  short  of  a  complete  restoration 
of  the  lost  part;  a  certain  percentage  only  of  the  lost  tissues  is 
restored.  This  fact  was  well  known  to  some  of  those  old  .investi- 
gators, who,  like  the  Abbe  Trembley  and  hke  Voltaire,  found  a 
fascination  in  the  study  of  artificial  injury  and  the  regeneration 
which  followed  it.  Sir  John  Graham  Dalyell,  for  instance,  says,  in 
the  course  of  an  admirable  paragraph  on  regeneration!:  "The 
reproductive  faculty ...  is  not  confined  to  one  pjortion,  but  may 
extend  over  many;  and  it  may  ensue  even  in  relation  to  the 
regenerated  portion  more  than  once.  Nevertheless,  the  faculty 
gradually  weakens,  so  that  in  general  every  successive  regeneration 
is  smaller  and  more  imperfect  than  the  organisation  preceding  it; 
and  at  length  it  is  exhausted." 

*  The  experiments  of  Loeb  on  the  growth  of  Tubularia  in  various  saline 
solutions,  referred  to  on  p.  24.5,  might  as  well  or  better  have  been  referred  to  under 
the  heading  of  regeneration,  as  they  were  performed  on  cut  pieces  of  the  zoophyte. 
(Cf.  Morgan,  op.  cit.  p.  35.) 

t  Powers  of  the  Creator,  i,  p.  7,  1851.  See  also  Rare  and  Remarkable  Animals, 
u,  pp.  17-19,  90,  1847. 


Ill]  OF  REGENERATION  275 

In  certain  minute  animals,  such  as  the  Infusoria,  in  which  the 
capacity  for  regeneration  is  so  great  that  the  entire  animal  may- 
be restored  from  a  mere  fragment,  it  becomes  of  great  interest  to 
discover  whether  there  be  some  definite  size  at  which  the  fragment 
ceases  to  display  this  power.  This  question  has  been  studied  by 
Lillie*,  who  found  that  in  Stentor,  while  still  smaller  fragments  were 
capable  of  surviving  for  days,  the  smallest  portions  capable  of 
regeneration  were  of  a  size  equal  to  a  sphere  of  about  80 /x  in 
diameter,  that  is  to  say  of  a  volume  equal  to  about  one  twenty- 
seventh  of  the  average  entire  animal.  He  arrives  at  the  remarkable 
conclusion  that  for  this,  and  for  all  other  species  of  animals,  there 
is  a  "minimal  organisation  mass,"  that  is  to  say  a  "minimal  mass 
of  definite  size  consisting  of  nucleus  and  cytoplasm  within  which 
the  organisation  of  the  species  can  just  find  its  latent  expression." 
And  in  like  manner,  Boverif  has  shewn  that  the  fragment  of  a  sea- 
urchin's  egg  capable  of  growing  up  into  a  new  embryo,  and  so 
discharging  the  complete  functions  of  an  entire  and  uninjured  ovum, 
reaches  its  limit  at  about  one-twentieth  of  the  original  egg — other 
writers  having  found  a  Hmit  at  about  one-fourth.  These  magnitudes, 
small  as  they  are,  represent  objects  easily  visible  under  a  low  power 
of  the  microscope,  and  so  stand  in  a  very  different  category  to  the 
minimal  magnitudes  in  which  fife  itself  can  be  manifested,  and 
which  we  have  discussed  in  another  chapter. 

The  Bermuda  "hfe-plant"  (Bryophyllum  calycinum)  has  so 
remarkable  a  power  of  regeneration  that  a  single  leaf,  kept  damp, 
sprouts  into  fresh  leaves  and  rootlets  which  only  need  nourishment 
to  grow  into  a  new  plant.  If  a  stem  bearing  two  opposite  leaves 
be  split  asunder,  the  two  co-equal  sister-leaves  will  produce  (as  we 
might  indeed  expect)  equal  masses  of  shoots  in  equal  times,  whether 
these  shoots  be  many  or  fe^^;  and,  if  one  leaf  of  the  pair  have  part 
cut  off  it  and  the  other  be  left  intact,  the  amount  of  new  growth 

*  F.  R.  Lillie,  The  smallest  parts  of  Stentor  capable  of  regeneration,  Journ. 
Morphology,  xii,  p.  239,  1897. 

t  -Boveri,  Entwicklungsfahigkeit  kernloser  Seeigeleier,  etc..  Arch.  /.  Entw.  Mech. 
u,  1895.  See  also  Morgan,  Studies  of  the  partial  larvae  of  Sphaerechinus,  ibid. 
1895;  J.  Loeb,  On  the  limits  of  divisibility  of  living  matter,  Biol.  Lectures,  1894; 
Pfluger's  Archiv,  lix,  1894,  etc.  Bonnet  studied  the  same  problem  a  hundred 
and  seventy  years  ago,  and  found  that  the  smallest  part  of  the  worm  Lumbriculus 
capable  of  regenerating  was  1^  lines  (3-4  mm.)  long.  For  other  references  and 
discussion  see  H.  Przibram,  Form  und  Formel,  1922,  ch.  v. 


276 


THE  RATE  OF  GROWTH 


[CH. 


will  be  in  direct  and  precise  proportion  to  the  mass  of  the  leaf  from 
which  it  grew.  The  leaf  is  all  the  while  a  living  tissue,  manu- 
facturing material  to  build  its  own  offshoots ;  and  we  have  a  simple 
case  of  the  law  of  mass  action  in  the  relation  between  the  mass  of 
the  leaf  with  its 'included  chlorophyll  and.  that  of  its  regenerated 
oifshoot*. 


16        18      20 
days 

Fig.  87.  Relation  between  the  percentage  amount  of  tail  removed,  the  percentage 
restored,  and  the  time  required  for  its  restoration.  Constructed  from  M.  M. 
Ellis's  data. 

A  number  of  phenomena  connected  with  the  linear  rate  of 
regeneration  are  illustrated  and  epitomised  in  the  accompanying 
diagram  (Fig.  87),  which  I  have  constructed  from  certain  data 
given  by  Ellis  in  a  paper  on  the  relation  of  the  amount  of  tail 
regenerated  to  the  amount  removed,  in  tadpoles.  These  data  are 
summarised  in  the  next  table.     The  tadpoles  were  all  very  much 


*  Jacques  Loeb,  The  law  controlling  the  quantity  and  rate  of  regeneration, 
Proc.  Nat.  Acad.  Sci.  iv,  pp.  117-121,  1918;  Journ.  Gen.  Physiol,  i,  pp.  81-96, 
1918;    Botan.  Oaz.  lxv,  pp.  150-174,  1918. 


Ill]  OF  REGENERATION  277 

of  a  size,  about  40  mm. ;  the  average  length  of  tail  was  very  near 
to  26  mm.,  or  65  per  cent,  of  the  whole  body-length;  and  in  four 
series  of  experiments  about  10,  20,  40  and  60  per  cent,  of  the  tail 
were  severally  removed.  The  amount  regenerated  in  successive 
intervals  of  three  days  is  shewn  in  our  table.  By  plotting  the 
actual  amounts  regenerated  against  these  three-day  intervals  of 
time,  we  may  interpolate  values  for  the  time  taken  to  regenerate 
definite  percentage  amounts,  5  per  cent.,  10  per  cent.,  etc.  of  the 
amount  removed;  and  my  diagram  is  constructed  from  the  four 
sets  of  values  thus  obtained,  that  is  to  say  from  the  four  sets  of 
experiments  which  differed  from  one  another  in  the  amount  of  tail 
amputated.  To  these  we  have  to  add  the  general  result  of  a  fifth 
series  of  experiments,  which  shewed  that  when  as  much  as  75  per 
cent,  of  the  tail  was  cut  off,  no  regeneration  took  place  at  all,  but 
the  animal  presently  died.  In  our  diagram,  then,  each  curve 
indicates  the  time  taken  to  regenerate  n  per  cent,  of  the  amount 
removed.  All  the  curves  converge  towards  infinity  of  time,  when 
the  amount  removed  approaches  75  per  cent,  of  the  whole;  and  all 
start  from  zero,  for  nothing  is  regenerated  where  nothing  had  been 
destroyed. 

The  rate  of  regenerative  growth  in  tadpoles'  tails 
(After  M.  M.  Ellis,  Journ.  Exp.  Zool.  vii,  ^.421,  1909) 


Body 

Tail 

Amount  Per  cent. 

/o 

amount  regenerated 

in  days 

length 

length 

removed 

of  tail 

r 

— ^ 

Series* 

mm. 

mm. 

mm. 

removed 

3 

6 

9 

12 

15 

18 

32 

0 

39-575 

25-895 

3-2 

12-36 

13 

31 

44 

44 

44 

44 

44 

P 

40-21 

26-13 

5-28 

20-20 

10 

29 

40 

44 

44 

44 

44 

R 

39-86 

25-70 

10-4 

40-50 

6 

20 

31 

40 

48 

48 

48 

S 

40-34 

2611 

14-8 

56-7 

0 

16 

33 

39 

45 

48 

48 

*  Each  series  gives  the  mean  of  20  experiments. 

The  amount  regenerated  varies  also  with  the  age  of  the  tadpole, 
and  with  other  factors  such  as  temperature;  in  short,  for  any  given 
age  or  size  of  tadpole,  and  for  various  temperatures,  and  doubtless 
for  other  varying  physical  conditions,  a  similar  diagram  might  be 
constructed!. 

The  power  of  reproducing,  or  regenerating,  a  lost  limb  is  par- 

t  Cf.  also  C.  Zeleny,  Factors  controlling  the  rate  of  regeneration,  Illinois  Biol- 
Monographs,  iii,  p.  1,  1916. 


278  THE   RATE   OF  GROWTH  [ch. 

ticularly  well  developed  in  arthropod  animals,  and  is  sometimes 
accompanied  by  remarkable  modification  of  the  form  of  the 
regenerated  limb.  A  case  in  point,  which  has  attracted  much 
attention,  occurs  in  connection  with  the  claws  of  certain  Crustacea*. 

In  many  of  these  we  have  an  asymmetry  of  the  great  claws, 
one  being  larger  than  the  other  and  also  more  or  less  different  in 
form.  For  instance  in  the  common  lobster,  one  claw,  the  larger 
of  the  two,  is  provided  with  a  few  great  "crushing"  teeth,  while 
the  smaller  claw  has  more  numerous  teeth,  small  and  serrated. 
Though  Aristotle  thought  otherwise,  it  appears  that  the  crushing- 
claw  may  be  on  the  right'  or  left  side,  indifferently ;  whether  it  be 
on  one  or  the  other  is  a  matter  of  "chance."  It  is  otherwise  in 
many  other  Crustacea,  where  the  larger  and  more  powerful  claw  is 
always  left  or  right,  as  the  case  may  be,  according  to  the  species: 
where,  in  other  words,  the  "probability"  of  the  large  or  the  small 
claw  being  left  or  being  right  is  tantamount  to  certainty  f. 

As  we  have  already  seen,  the  one  claw  is  the  larger  because  it 
has  grown  the  faster;  it  has  a  higher  "coefficient  of  growth,"  and 
accordingly,  as  age  advances,  the  disproportion  between  the  two 
claws  becomes  more  and  more  evident.  Moreover,  we  must  assume 
that  the  characteristic  form  of  the  claw  is  a  "function"  of  its 
magnitude ;  the  knobbiness  is  a  phenomenon  coincident  with 
growth,  and  we  never,  under  any  circumstances,  find  the  smaller 
claw  with  big  crushing  teeth  and  the  big  claw  with  Httle  serrate 
ones.  There  are  many  other  somewhat  similar  cases  where  size 
and  form  are  manifestly  correlated,  and  we  have  already  seen,  to 
some  extent,  how  the  phenomenon  of  growth  is  often  accompanied 
by  such  ratios  of  velocity  as  lead  inevitably  to  changes  of  form. 
Meanwhile,  then,  we  must  simply  assume  that  the  essential  difference 
between  the  two  claws  is  one  of  magnitude,  with  which  a  certain 
differentiation  of  form  is  inseparably  associated. 

*  Cf.  H.  Przibram,  Scheerenumkehr  bei  dekapoden  Crustaceen,  Arch.  f.  Entw. 
Mech.  XIX,  pp.  181-247,  1905;  xxv,  pp.  266-344,  1907;  Emmel,  ibid,  xxii,  p.  542, 
1906;  Regeneration  of  lost  parts  in  lobster,  Bep.  Comm.  Inland  Fisheries,  Rhode 
Island,  XXXV,  xxxvi,  1905-6;  Science  (N.S.),  xxvi,  pp.  83-87,  1907;  Zeleny, 
Compensatory  regulation,  Journ.  Exp.  Zool.  ii,  pp.  1-102,  347-369,  1905;   etc. 

t  Lobsters  are  occasionally  found  with  two  symmetrical  claws:  which  are  then 
usually  serrated,  sometimes  (but  very  rarely)  both  blunt-toothed.  Cf.  W.  T.  Caiman, 
P.Z.8.  1906,  pp.  633,  634,  and  reff. 


Ill]  OF  REGENERATION  279 

If  we  amputate  a  claw,  or  if,  as  often  happens,  the  crab  "casts 
it  off,"  it  undergoes  a  process  of  regeneration — it  grows  anew, 
and  does  so  with  an  accelerated  velocity  which  ceases  when 
equilibrium  of  the  parts  is  once  more  attained :  the  accelerated  velocity 
being  a  case  in  point  to  illustrate  that  vis  revulsionis  of  Haller  to 
which  we  have  already  referred. 

With  the  help  of  this  principle,  Przibram  accounts  for  certain 
curious  phenomena  which  accompany  the  process  of  regeneration. 
As  his  experiments  and  those  of  Morgan  shew,  if  the  large  or  knobby 
claw  (A)  be  removed,  there  are  certain  cases,  e.g.  the  common 
lobster,  where  it  is  directly  regenerated.  In  other  cases,  e.g. 
Alpheus*,  the  other  claw  (B)  assumes  the  size  and  form  of  that 
which  was  amputated,  while  the  latter  regenerates  itself  in  the 
form  of  the  lesser  and  weaker  one;  A  and  B  have  apparently 
changed  places.  In  a  third  case,  as  in  the  hermit-crabs,  the  A- 
claw  regenerates  itself  as  a  small  or  5-claw,  but  the  5-claw 
remains  for  a  time  unaltered,  though  slowly  and  in  the  course  of 
repeated  moults  it  later  on  assumes  the  large  and  heavily  toothed 
^-form. 

Much  has  been  written  on  this  phenomenon,  but  in  essence  it  is 
very  simple.  It  depends  upon  the  respective  rates  of  growth,  upon 
a  ratio  between  the  rate  of  regeneration  and  the  rate  of  growth  of 
the  uninjured  limb:  that  is  to  say,  on  the  familiar  phenomenon  of 
unequal  growth,  or,  as  it  has  been  called,  heterogony*.  It  is  com- 
plicated a  little,  however,  by  the  possibility  of  the  uninjured  limb 
growing  all  the  faster  for  a  time  after  the  animal  has  been  relieved 
of  the  other.  From  the  time  of  amputation,  say  of  A,  A  begins  to 
grow  from  zero,  with  a  high  "regenerative"  velocity;  while  B, 
starting  from  a  definite  magnitude,  continues  to  increase  with  its 
normal  or  perhaps  somewhat  accelerated  velocity.  The  ratio 
between  the  two  velocities  of  growth  will  determine  whether,  by  a 
given  time,  A  has  equalled,  outstripped,  or  still  fallen  short  of  the 
magnitude  of  B. 

That  this  is  the  gist  of  the  whole  problem  is  confirmed  (if  con- 
firmation be  necessary)  by  certain  experiments  of  Wilson's.     It  is 

*  E.'  B.  Wilson,  Reversal  of  symmetry  in  Alpheus  heterocheles,  Biol.  Bull,  iv, 
p.  197,  1903. 
t  See  p.  205. 


280  THE   RATE   OF  GROWTH  [ch. 

known  that  by  section  of  the  nerve  to  a  crab's  claw,  its  growth  is 
retarded,  and  as  the  general  growth  of  the  animal  proceeds  the  claw 
comes  to  appear  stunted  or  dwarfed.  Now  in  such  a  case  as  that 
of  Alpheus,  we  have  seen  that  the  rate  of  regenerative  growth  in  an 
amputated  large  claw  fails  to  let  it  reach  or  overtake  the  magnitude 
of  the  growing  little  claw:  which  latter,  in  short,  now  appears  as 
the  big  one.  But  if  at  the  same  time  as  we  amputate  the  big  claw 
we  also  sever  the  nerve  to  the  lesser  one,  we  so  far  slow  down  the 
latter's  growth  that  the  other  is  able  to  make  up  to  it,  and  in  this 
case  the  two  claws  continue  to  grow  at  approximately  equal  rates, 
or  in  other  words  continue  of  coequal  size. 

The  phenomenon  of  regeneration  goes  some  little  way  towards 
helping  us  to  comprehend  the  phenomenon  of  "multiplication  by 
fission,"  as  it  is  exemplified  in  its  simpler  cases  in  many  worms  and 
worm-like  animals.  For  physical  reasons  which  we  shall  have  to 
study  in  another  chapter,  there  is  a  natural  tendency  for  any  tube, 
if  it  have  the  properties  of  a  fluid  or  semi-fluid  substance,  to  break 
up  into  segments  after  it  comes  to  a  certain  length*;  and  nothing 
can  prevent  its  doing  so  except  the  presence  of  some  controlling 
force,  such  for  instance  as  may  be  due  to  the  pressure  of  some 
external  support,  or  some  superficial  thickening  or  other  intrinsic 
rigidity  of  its  own  substance.  If  we  add  to  this  natural  tendency 
towards  fission  of  a  cylindrical  or  tubular  worm,  the  ordinary 
phenomenon  of  regeneration,  we  have  all  that  is  essentially  implied 
in  "reproduction  by  fission."  And  in  so  far  as  the  process  rests 
upon  a  physical  principle,  or  natural  tendency,  we  may  account  for 
its  occurrence  in  a  great  variety  of  animals,  zoologically  dissimilar; 
and  for  its  presence  here  and  absence  there,  in  forms  which  are 
materially  different  in  a  physical  sense,  though  zoologically  speaking 
they  are  very  closely  allied. 

But  the  phenomena  of  regeneration,  like  all  the  other  phenomena 
of  growth,  soon  carry  us  far  afield,  and  we  must  draw  this  long 
discussion  to  a  close. 

*  A  morphological  polarity,  or  essential  difference  between  one  end  and  the  other 
of  a  segment,  is  important  even  in  so  simple  a  case  as  the  internode  of  a  hydroid 
zoophyte;  and  an  electrical  polarity  seems  always  to  accompany  it.  Cf.  A.  P. 
Matthews,  Amer.  Journ.  Physiology,  viii,  j).  294,  1903;  E.  J.  Lund,  Journ.  Exper. 
Zool.  xxxiy,  pp.  477-493;   xxxvi,  pp.  477^94,  1921-22. 


Ill]  THE  RATE  OF  GROWTH  281 

Summary  and  Conclusion 

For  the  main  features  which  appear  to  be  common  to  all  curves 
of  growth  we  may  hope  to  have,  some  day,  a  simple  explanation. 
In  particular  we  should  like  to  know  the  plain  meaning  of  that  point 
of  inflection,  or  abrupt  change  from  an  increasing  to  a  decreasing 
velocity  of  growth,  which  all  our  curves,  and  especially  our  accelera- 
tion curves,  demonstrate  the  existence  of,  provided  only  that  they 
include  the  initial- stages  of  the  whole  phenomenon:  just  as  we 
should  also  hke  to  have  a  full  physical  or  physiological  explanation 
of  the  gradually  diminishing  velocity  of  growth  which  follows,  and 
which  (though  subject  to  temporary  interruption  or  abeyance)  is 
on  the  whole  characteristic  of  growth  in  all  cases  whatsoever.  In 
short,  the  characteristic  form  of  the  curve  of  growth  in  length  (or 
any  other  linear  dimension)  is  a  phenomenon  which  we  are  at 
present  little  able  to  explain,  but  which  presents  us  with  a  definite 
and  attractive  problem  for  future  solution.  It  would  look  as 
though  the  abrupt  change  in  velocity  must  be  due,  either  to  a  change 
in  that  pressure  outwards  from  within  by  which  the  "forces  of 
growth"  make  themselves  manifest,  or  to  a  change  in  the  resistances 
against  which  they  act,  that  is  to  say  the  tension  of  the  surface; 
and  this  latter  force  we  do  not  by  any  means  limit  to  "surface- 
tension"  proper,  but  may  extend  to  the  development  of  a  more  or 
less  resistant  membrane  or  "skin,"  or  even  to  the  resistance  of  fibres 
or  other  histological  elements  binding  the  boundary  layers  to  the 
parts  within*.  I  take  it  that  the  sudden  arrest  of  velocity  is  much 
more  likely  to  be  due  to  a  sudden  increase  of  resistance  than  to  a 
sudden  diminution  of  internal  energies:  in  other  words,  I  suspect 
that  it  is  coincident  with  some  notable  event  of  histological 
differentiation,  such  as  the  rapid  formation  of  a  comparatively  firm 
skin ;  and  that  the  dwindling  of  velocities,  or  the  negative  accelera- 
tion, which  follows,  is  the  resultant  or  composite  effect  of  waning 
forces  of  growth  on  the  one  hand,  and  increasing  superficial  resistance 

*  It  is  natural  to  suppose  the  cell-wall  less  rigid,  or  more  plastic,  in  the  growing 
tissue  than  in  the  full-grown  or  resting  cell.  It  has  been  suggested  that  this  plasticity- 
is  due  to,  or  is  increased  by,  auxins,  whether  in  the  course  of  nature,  or  in  our 
stimulation  of  growth  by  the  use  of  these  bodies.  Cf.  H.  Soding,  Jahrb.  d.  wiss.  Bot. 
Lxxiv,  p.  127;  1931. 


282  THE   RATE   OF   GROWTH  [ch. 

on  the  other.  This  is  as  much  as  to  say  that  growth,  while  its  own 
energy  tends  to  increase,  leads  also,  after  a  while,  to  the  establish- 
ment of  resistances  which  check  its  own  further  increase. 

Our  knowledge  of  the  whole  complex  phenomenon  of  growth  is 
so  scanty  that  it  may  seem  rash  to  advance  even  this  tentative 
suggestion.  But  yet  there  are  one  or  two  known  facts  which  seem 
to  bear  upon  the  question,  and  to  indicate  at  least  the  manner  in 
which  a  varying  resistance  to  expansion  may  affect  the  velocity 
of  growth.  For  instance,  it  has  been  shewn  by  Frazee*  that 
electrical  stimulation  of  tadpoles,  with  small  current  density  and 
low  voltage,  increases  the  rate  of  regenerative  growth.  As  just 
such  an  electrification  would  tend  to  lower  the  surface-tension,  and 
accordingly  decrease  the  external  resistance,  the  experiment  would 
seem  to  support,  in  some  slight  degree,  the  suggestion  which  I  have 
made. 

To  another  important  aspect  of  regeneration  we  can  do  no  more 
than  allude.  The  Planarian  worms  rival  Hydra  itself  in  their  powers 
of  regeneration;  and  in  both  cases  even  small  bits  of  the  animal 
are  likely  to  include  endoderm  cells  capable  of  intracellular  digestion, 
whereby  the  fragment  is  enabled  to  live  and  to  grow.  Now  if  a 
Planarian  worm  be  cut  in  separate  pieces  and  these  be  suffered  to 
grow  and  regenerate,  they  do  so  in  a  definite  and  orderly  way ;  that 
part  of  a  shce  or  fragment  which  had  been  nearer  tq  the  original 
head  will  develop  a  head,  and  a  tail  will  be  regenerated  at  the 
opposite  end  of  the  same  fragment,  the  end  which  Had  been  tailward 
in  the  beginning;  the  amputated  fragments  possess  sides  and  ends, 
a  front  end  and  a  hind  end,  like  the  entire  worm;  in  short,  they 
retain  their  polarity.  This  remarkable  discovery  is  due  to  Child, 
who  has  amplified  and  extended  it  in  various  instructive  ways. 
The  existence  of  two  poles,  positive  and  negative,  implies  a 
"gradient"  between  them.  It  means  that  one  part  leads  and 
another  follows;  that  one  part  is  dominant,  or  prepotent  over  the 
rest,  whether  in  regenerative  growth  or  embryonic  development. 

We  may  summarise,  as  follows,  the  main  results  of  the  foregoing 
discussion : 

(1)   Except  in  certain  minute  organisms,  whose  form  (hke  that 

*  Journ.  Ezper.  ZooL  vii,  p.  457,  1909. 


Ill]  SUMMARY  AND  CONCLUSION  283 

of  a  drop  of  water)  is  due  to  the  direct  action  of  the  molecular  forces, 
we  may  look  upon  the  form  of  an  organism  as  a  "  function  of  growth," 
or  a  direct  consequence  of  growth  whose  rate  varies  in  its  different 
directions.  In  a  newer  language  we  might  call  the  form  of  an 
organism  an  "event  in  space-time,"  and  not  merely  a  "configuration 
in  space." 

(2)  Growth  varies  in  rate  in  an  orderly  way,  or  is  subject,  like 
other  physiological  activities,  to  definite  "laws."  The  rates  differ 
in  degree,  or  form  "gradients,"  from  one  point  of  an  organism  to 
another;  the  rates  in  different  parts  and  in  different  directions 
tend  to  maintain  more  or  less  constant  ratios  to  one  another  in 
each  organism ; ,  and  to  the  regularity  and  constancy  of  these  relative 
rates  of  growth  is  due  the  fact  that  the  form  of  the  organism  is  in 
general  regular  and  constant. 

(3)  Nevertheless,  the  ratio  of  velocities  in  different  directions  is 
not  absolutely  constant,  but  tends  to  alter  in  course  of  time,  or  to 
fluctuate  in  an  orderly  way;  and  to  these  progressive  changes  are 
due  the  changes  of  form  which  accompany  development,  and  the 
slower  changes  which  continue  perceptibly  in  after  hfe. 

(4)  Rate  of  growth  depends  on  the  age  of  the  organism.  It  has 
a  maximum  somewhat  early  in  hfe,  after  which  epoch  of  maximum 
it  slowly  declines. 

(5)  Rate  of  growth  is  directly  affected  by  temperature,  and  by 
other  physical  conditions:  the  influence  of  temperature  being 
notably  large  in  the  case  of  cold-blooded  or  "  poecilothermic " 
animals.  Growth  tends  in  these  latter  to  be  asymptotic,  becoming 
slower  but  never  ending  with  old  age. 

(6)  It  is  markedly  affected,  in  the  way  of  acceleration  or  retarda- 
tion, at  certain  physiological  epochs  of  hfe,  such  as  birth,  puberty 
or  metamorphosis. 

(7)  Under  certain  circumstances,  growth  may  be  negative,  the 
organism  growing  smaller;  and  such  negative  growth  is  a  common 
accompaniment  of  metamorphosis,  and  a  frequent  concomitant  of 
old  age. 

(8)  The  phenomenon  of  regeneration  is  associated  with  a  large 
transitory  increase  in  the  rate  of  growth  (or  acceleration  of  growth) 
in  the  region  of  injury;  in  other  respects  regenerative  growth  is 
similar  to  ordinary  growth  in  all  its  essential  phenomena. 


284  THE  RATE   OF  GROWTH  [ch. 

In  this  discussion  of  growth,  we  have  left  out  of  account  a  vast 
number  of  processes  or  phenomena  in  the  physiological  mechanism 
of  the  body,  by  which  growth  is  effected  and  controlled.  We  have 
dealt  with  growth  in  its  relation  to  magnitude,  and  to  that  relativity 
of  magnitudes  which  constitutes  form;  and  so  we  have  studied  it 
as  a  phenomenon  which  stands  at  the  beginning  of  a  morphological, 
rather  than  at  the  end  of  a  physiological  enquiry.  Under  these 
restrictions,  we  have  treated  it  as  far  as  possible,  or  in  such  fashion 
as  our  present  knowledge  permits,  on  strictly  physical  lines.  That 
is  to  say,  we  rule  "heredity"  or  any  such  concept  out  of  our  present 
account,  however  true,  however  important,  however  indispen- 
sable in  another  setting  of  the  story,  such  a  concept  may  be. 
In  physics  "on  admet  que  I'etat  actuel  du  monde  ne  depend  que  du 
passe  le.  plus  proche,  sans  etre  influence,  pour  ainsi  dire,  par  le 
souvenir  d'un  passe  lointain*."  This  is  the  concept  to  which  the 
differential  equation  gives  expression;  it  is  the  step  which  Newton 
took  when  he  left  Kepler  behind. 

In  all  its  aspects,  and  not  least  in  its  relation  to  form,  the  growth 
of  organisms  has  many  analogies,  some  close,  some  more  remote, 
among  inanimate  things.  As  the  waves  grow  when  the  winds  strive 
with  the  other  forces  which  govern  the  movements  of  the  surface 
of  the  sea,  as  the  heap  grows  when  we  pour  corn  out  of  a  sack,  as 
the  crystal  grows  when  from  the  surrounding  solution  the  proper 
molecules  fall  into  their  appropriate  places:  so  in  all  these  cases, 
very  much  as  in  the  organism  itself,  is  growth  accompanied  by 
change  of  form,  and  by  a  development  of  definite  shapes  and 
contours.  And  in  these  cases  (as  in  all  other  mechanical  phenomena), 
we  are  led  to  equate  our  various  magnitudes  with  time,  and  so  to 
recognise  that  growth  is  essentially  a  question  of  rate,  or  of  velocity. 

The  diiferences  of  form,  and  changes  of  form,  which  are  brought 
about  by  varying  rates  (or  "laws")  of  growth,  are  essentially  the 
same  phenomenon  whether  they  be  episodes  in  the  life-history  of 
the  individual,  or  manifest  themselves  as  the  distinctive  charac- 
teristics of  what  we  call  separate  species  of  the  race.  From  one 
form,  or  one  ratio  of  magnitude,  to  another  there  is  but  one  straight 
and  direct  road  of  transformation,  be  the  journey  taken  fast  or 

*  Cf.  H.  Poincare,  La  physique  generale  et  la  physique  mathematique,  Rev. 
gin.  des  Sciences,  xi,  p.  1167,  1900. 


Ill]  SUMMARY  AND  CONCLUSION  285 

slow;  and  if  the  transformation  take  place  at  all,  it  will  in  all 
likelihood  proceed  in  the  self-same  way,  whether  it  occur  within 
the  Ufetime  of  an  individual  or  during  the  long  ancestral  history  of 
a  race.  No  small  part  of  what  is  known  as  Wolff's  or  von  Baer*s 
law,  that  the  individual  organism  tends  to  pass  through  the  phases 
characteristic  of  its  ancestors,  or  that  the  life-history  of  the  individual 
tends  to  recapitulate  the  ancestral  history  of  its  race,  lies  wrapped 
up  in  this  simple  account  of  the  relation  between  growth  and  form. 
But  enough  of  this  discussion.  Let  us  leave  for  a  while  the 
subject  of  the  growth  of  the  organism,  and  attempt  to  study  the 
conformation,  within  and  without,  of  the  individual  cell. 


CHAPTER  IV 

ON    THE    INTERNAL    FORM    AND   STRUCTURE 
OF   THE    CELL 

In  the  early  days  of  the  cell-theory,  a  hundred  years  ago,  Goodsir 
was  wont  to  speak  of  cells  as  "centres  of  growth"  or  "centres  of 
nutrition,"  and  to  consider  them  as  essentially  "centres  of  force*". 
He  looked  forward  to  a  time  when  the  forces  connected  with  the 
cell  should  be  particularly  investigated :  when,  that  is  to  say,  minute 
anatomy  should  be  studied  in  its  dynamical  aspect.  "When  this 
branch  of  enquiry,"  he  says,  "shall  have  been  opened  up,  we  shall 
expect  to  have  a  science  of  organic  forces,  having  direct  relation 
to  anatomy,  the  science  of  organic  forms."  And  likewise,  long 
afterwards,  Giard  contemplated  a  science  of  morphodynatnique — but 
still  looked  upon  it  as  forming  so  guarded  and  hidden  a  "territoire 
scientifique,  que  la  plupart  des  naturalistes  de  nos  jours  ne  le  verront 
que  comme  Moise  vit  la  terre  promise,  seulement  de  loin  et  sans 
pouvoir  y  entrerf ." 

To  the  external  forms  of  cells,  and  to  the  forces  which  produce 
and  modify  these  forms,  we  shall  pay  attention  in  a  later  chapter. 
But  there  are  forms  and  configurations  of  matter  within  the  cell 
which  also  deserve  to  be  studied  with  due  regard  to  the  forces, 
known  or  unknown,  of  whose  resultant  they  are  the  visible 
expression. 

*  Anatomical  and  Pathological  Observations,  p.  3,  1845;  Anatomical  Memoirs, 
II,  p.  392,  1868.  This  was  a  notable  improvement  on  the  "kleine  wirkungsfahige 
Zentren  oder  Elementen"  of  the  Cellularpathologie.  Goodsir  seems  to  have  been 
seeking  an  analogy  between  the  living  cell  and  the  physical  atom,  which  Faraday, 
following  Boscovich,  had  been  speaking  of  as  a  centre  of  force  in  the  very  year 
before  Goodsir  published  his  Observations:  see  Faraday's  Speculations  concerning 
Electrical  Conductivity  and  the  Nature  of  Matter,  1844.  For  Newton's  "molecules" 
had  been  turned  by  his  successors  into  material  points;  and  it  was  Boscovich  (in 
1758)  who  first  regarded  these  material  points  as  mere  persistent  centres  of  force. 
It  was  the  same  fertile  conception  of  a  centre  of  force  which  led  Rutherford,  later 
on,  to  the  discovery  of  the  nucleus  of  the  atom. 

t  A.  Giard,  L'oeuf  et  les  debuts  de  revolution,  Bull.  Sci.  du  Nord  de  la  Fr.  vm, 
pp.  252-258,  1876. 


CH.  IV]  THE  CELL  THEORY      .  287 

In  the  long  interval  since  Goodsir's  day,  the  visible  structure, 
the  conformation  and  configuration,  of  the  cell,  has  been  studied 
far  more  abundantly  than  the  purely  dynamic  problems  which  are 
associated  therewith.  The  overwhelming  progress  of  microscopic 
observation  has  multipHed  our  knowledge  of  cellular  and  intra- 
cellular structure ;  and  to  the  multitude  of  visible  structures  it  has 
been  often  easier  to  attribute  virtues  than  to  ascribe  intelUgible 
functions  or  modes  of  action.  But  here  and  there  nevertheless, 
throughout  the  whole  hteratiire  of  the  subject,  we  find  recognition 
of  the  inevitable  fact  that  dynamical  problems  lie  behind  the 
morphological  problems  of  the  cell. 

Biitschli  pointed  out  sixty  years  ago,  with  emphatic  clearness, 
the  failure  of  morphological  methods  and  the  need  for  physical 
methods  if  we  were  to  penetrate  deeper  into  the  essential  nature  of 
the  cell*.  And  such  men  as  Loeb  and  Whitman,  Driesch  and  Roux, 
and  not  a  few  besides,  have  pursued  the  same  train  of  thought  and 
similar  methods  of  enquiry. 

Whitman t,  for  instance,  puts  the  case  in  a  nutshell  when,  in 
speaking  of  the  so-called  "  caryokinetic "  phenomena  of  nuclear 
division,  he  reminds  us  that  the  leading  idea  in  the  term  ''caryo- 
kinesis''  is  ynotion — "motion  viewed  as  an  exponent  of  forces 
residing  in,  or  acting  upon,  the  nucleus.  It  regards  the  nucleus 
as  a  seat  of  energy,  which  displays  itself  in  phenomena  of  motion X-^' 

In  short  it  would  seem  evident  that,  except  in  relation  to  a 
dynamical  investigation,  the  mere  study  of  cell  structure  has  but 

*  Entwickelungsvorgdnge  der  Eizelle,  1876;  Investigations  on  Microscopic  Foams 
and  Protoplasm,  p.  1,  1894. 

t  Journ.  Morphology,  i,  p.  229,  1887. 

t  While  it  has  been  very  common  to  look  upon  the  phenomena  of  mitosis  as 
sufficiently  explained  by  the  results  towards  which  they  seem  to  lead,  we  may  find 
here  and  there  a  strong  protest  against  this  mode  of  interpretation.  The  following 
is  a  case  in  point:  ''On  a  tente  d'etablir  dans  la  mitose  dite  primitive  plusieurs 
categories,  plusieurs  types  de  mitose.  On  a  choisi  le  plus  souvent  comme  base 
de  ces  systemes  des  concepts  abstraits  et  teleologiques :  repartition  plus  ou  moins 
exacte  de  la  chromatine  entre  les  deux  noyaux-fils  suivant  qu'il  y  a  ou  non  des 
chromosomes  (Dangeard),  distribution  particuliere  et  signification  dualiste  des 
substances  nucleaires  (substance  kinetique  et  substance  generative  ou  hereditaire, 
Ilartmann  et  ses  eleves),  etc.  Pour  moi  tous  ces  essais  sont  a  rejeter  categorique- 
ment  a  cause  de  leur  caractere  finaliste;  de  plus,  ils  sont  construits  sur  des  concepts 
non  demontres,  et  qui  parfois  representent  des  generalisations  absolument  erronees.'' 
A.  Alexeieflf,  Archiv  fiir  Protistenkunde,  xix,  p.  344,  1913. 


288  ON  THE  INTERNAL  FORM  [ch. 

little  value  of  its  own.  That  a  given  cell,  an  ovum  for  instance, 
contains  this  or  that  visible  substance  or  structure,  germinal  vesicle 
or  germinal  spot,  chromatin  or  achromatin,  chromosomes  or  centro- 
somes,  obviously  gives  no  explanation  of  the  activities  of  the  cell. 
And  in  all  such  hypotheses  as  that  of  "pangenesis,"  in  all  the 
theories  which  attribute  specific  properties  to  micellae,  chromosomes, 
idioplasts,  ids,  or  other  constituent  particles  of  protoplasm  or  of 
the  cell,  we  are  apt  to  fall  into  the  error  of  attributing  to  matter 
what  is  due  to  energy  and  is  manifested  in  force:  or,  more  strictly 
speaking,  of  attributing  to  material  particles  individually  what  is 
due  to  the  energy  of  their  collocation. 

The  tendency  is  a  very  natural  one,  as  knowledge  of  structure 
increases,  to  ascribe  particular  virtues  to  the  material  structures 
themselves,  and  the  error  is  one  into  which  the  disciple  is  Ukely 
to  fall  but  "of  which  we  need  not  suspect  the  master-mind.  The 
dynamical  aspect  of  the  case  was  in  all  probability  kept  well  in  view 
by  those  who,  Hke  Goodsir  himself,  first  attacked  the  problem  of 
the  cell  and  originated  our  conceptions  of  its  nature  and  functions*. 

If  we  speak,  as  Weismann  and  others  speak,  of  an  "hereditary 
sitbstance,''  a  substance  which  is  spHt  off  from  the  parent-body,  and 
which  hands  on  to  the  new  generation  the  characteristics  of  the  old, 
we  can  only  justify  our  mode  of  speech  by  the  assumption  that  that 
particular  portion  of  matter  is  the  essential  vehicle  of  a  particular 
charge  or  distribution  of  energy,  in  which  is  involved  the  capabihty 
of  producing  motion,  or  of  doing  "work."  For,  as  Newton  said, 
to  tell  us  that  a  thing  "is  endowed  with  an  occult  specific  quahtyf, 
by  which  it  acts  and  produces  manifest  effects,  is  to  tell  us  nothing ; 
but  to  derive  two  or  three  general  principles  of  motion  {   from 

*  See  also  {int.  al.)  R.  S.  Lillie's  papers  on  the  physiology  of  cell-division  in  the 
Journ.  Exper.  Physiology;  especially  No.  vi,  Rhythmical  changes  in  the  resistance 
of  the  dividing  sea-urchin  egg,  ibid,  xvi,  pp.  369-402,  1916. 

f  Such  as  the  vertu  donnitive  which  accounts  for  the  soporific  action  of  opium. 
We  are  now  more  apt.  as  Le  Dantec  says,  to  substitute  for  this  occult  quality  the 
hypothetical  substance  dormitin. 

X  This  is  the  old  philosophic  axiom  writ  large:  Ignorato  motu,  ignoratur  -natura; 
which  again  is  but  an  adaptation  of  Aristotle's  phrase,  17  dpx^  ^V^  Kivrjcrecxjs,  as 
equivalent  to  the  "Efiicient  Cause."  FitzGerald  holds  that  "all  explanation 
consists  in  a  description  of  underlying  motions"  {Scientific  Writings,  1902,  p.  385); 
and  Oliver  Lodge  remarked,  "You  can  move  Matter;  it  is  the  only  thing  you  can 
do  to  it." 


IV]  AND  STRUCTURE  OF  THE  CELL  289 

phenomena  would  be  a  very  great  step  in  philosophy,  though  the 
causes  of  those  principles  were  not  yet  discovered."  The  things 
which  we  see  in  the  cell  are  less  important  than  the  actions  which 
we  recognise  in  the  cell;  and  these  latter  we  must  especially, 
scrutinise,  in  the  hope  of  discovering  how  far  they  may  be  attributed 
to  the  simple  and  well-known  physical  forces,  and  how  far  they  be 
relevant  or  irrelevant  to  the  phenomena  which  we  associate  with, 
and  deem  essential  to,  the  manifestation  of  life.  It  may  be  that  in 
this  way  we  shall  in  time  draw  nigh  to  the  recognition  of  a  specific 
and  ultimate  residuum. 

And  lacking,  as  we  still  do  lack,  direct  knowledge  of  the 
actual  forces  inherent  in  the  cell,  we  may  yet  learn  something 
of  their  distribution,  if  not  also  of  their  nature,  from  the 
outward  and  inward  configuration  of  the  cell  and  from  the 
changes  taking  place  in  this  configuration;  that  is  to  say  from 
the  movements  of  matter,  the  kinetic  phenomena,  which  the  forces 
in  action  set  up. 

The  fact  that  the  germ-cell  develops  into  a  very  complex  structure 
is  no  absolute  proof  that  the  cell  itself  is  structurally  a  very  com- 
phcated  mechanism :  nor  yet  does  it  prove,  though  this  is  somewhat 
less  obvious,  that  the  forces  at  work  or  latent  within  it  are  especially 
numerous  and  complex.  If  we  blow  into  a  bowl  of  soapsuds  and 
raise  a  great  mass  of  many-hued  and  variously  shaped  bubbles,  if 
we  explode  a  rocket  and  watch  the  regular  and  beautiful  configura- 
tion of  its  falhng  streamers,  if  we  consider  the  wonders  of  a  Hmestone 
cavern  which  a  filtering  stream  has  filled  with  stalactites,  we  soon 
perceive  that  in  all  these  cases  we  have  begun  with  an  initial  system 
of  very  slight  complexity,  whose  structure  in  no  way  foreshadowed 
the  result,  and  whose  comparatively  simple  intrinsic  forces  only 
play  their  part  by  complex  interaction  with  the  equally  simple 
forces  of  the  surrounding  medium.  In  an  earlier  age,  men  sought 
for  the  visible  embryo,  even  for  the  homunculus,  within  the  repro- 
ductive cells;  and  to  this  day  we  scrutinise  these  cells  for  visible 
structure,  unable  to  free  ourselves  from  that  old  doctrine  of 
' '  pre-f ormation  * . " 

Moreover,  the  microscope  seemed  to  substantiate  the  idea  (which 

*  As  when  Nageli  concluded  that  the  organism  is,  in  a  certain  sense,  "vorge- 
bildet";   Beitr.  zur  wiss.  Botanik,  ii,  1860. 


290  ON  THE  INTERNAL  FORM  [ch. 

we  may  trace  back  to  Leibniz*  and  to  Hobbest),  that  there  is  no 
Hmit  to  the  mechanical  complexity  which  we  may  postulate  in  an 
organism,  and  no  limit,  therefore,  to  the  hypotheses  which  we  may 
rest  thereon.  But  no  microscopical  examination  of  a  stick  of  sealing- 
wax,  no  study  of  the  material  of  which  it  is  composed,  can  enlighten 
us  as  to  its  electrical  manifestations  or  properties.  Matter  of  itself 
has  no  power  to  do,  to  make,  or  to  become:  it  is  in  energy  that 
all  these  potentiaUties  reside,  energy  invisibly  associated  with  the 
material  system,  and  in  interaction  with  the  energies  of  the 
surrounding  universe. 

That  ''function  presupposes  structure"  has  been  declared  an 
accepted  axiom  of  biology.  Who  it  was  that  so  formula  ted  the 
aphorism  I  do  not  know;  but  as  regards  the  structure  of  the  cell 
it  harks  back  to  Briicke,  with  whose  demand  for  a  mechanism,  or 
an  organisation,  within  the  cell  histologists  have  ever  since  been 
trying  to  comply  J.  But  unless  we  mean  to  include  thereby 
invisible,  and  merely  chemicu,l  or  molecular,  structure,  we  come  at 
once  on  dangerous  ground.  For  we  have  seen  in  a  former  chapter 
that  organisms  are  known  of  magnitudes  so  nearly  approaching  the 
molecalar,  that  everything  which  the  morphologist  is  accustomed  to 
conceive  as  "structure"  has  become  physically  impossible;  and 
recent  research  tends  to  reduce,  rather  than  to  extend,  our  con- 
ceptions of  the  visible  structure  necessarily  inherent  in  living 
protoplasm §.     The  microscopic  structure  which  in  the  last  resort 

*  "La  matiere  arrangee  par  une  sagesse  divine  doit  etre  essentiellement  organisee 
partout. .  .il  y  a  machine  dans  les  parties  de  la  machine  naturelle  a  I'infini."  Siir  le 
principe  de  la  Vie,  p.  431  (Erdmann).  This  is  the  very  converse  of  the  doctrine 
of  the  Atomists,  who  could  not  conceive  a  condition  "w6/  dimidiae  partis  pars 
semper  hahehit  Dimidiam  partem,  nee  res  praefiniet  ulla.'' 

t  Cf.  an  interesting  passage  from  the  Elements  (i,  p.  445,  Molesworth's  edit.), 
quoted  by  Owen,  Hunterian  Lectures  on  the  Invertebrates,  2nd  ed.  pp.  40,  41,  1855. 

%  "Wir  miissen  deshalb  den  lebenden  Zellen,  abgesehen  von  der  Molekular- 
structur  der  organischen  Verbindungen  welche  sie  enthalt,  noch  eine  andere  und 
in  anderer  Weise  complicirte  Structur  zuschreiben,  und  diese  es  ist  welche  wir 
mit  dem  Namen  Organisation  bezeichnen,"  Briicke,  Die  Elementarorganismen, 
Wiener  Sitzungsber.  xliv,  1861,  p.  386;  quoted  by  Wilson,  The  Cell,  etc.,  p.  289. 
Cf.  also  Hardy,  Journ.  Physiol,  xxiv,  1899,  p.  159. 

§  The  term  protoplasm  was  first  used  by  Purkinje,  about  1839  or  1840  (cf. 
Reichert,  Arch.  f.  Anat.  u.  Physiol.  1841).  But  it  was  better  defined  and  more 
strictly  used  by  Hugo  von  Mohl  in  his  paper  Ueber  die  Saftbewegung  im  Inneren 
der  Zellen,  Botan.  Zeitung,  iv,  col.  73-78,  89-94,  1846. 


IV]  AND  STRUCTURE  OF  THE  CELL  291 

or  in  the  simplest  cases  it  seems  to  sheWj  is  that  of  a  more  or  less 
viscous  colloid,  or  rather  mixtm:e  of  colloids,  and  nothing  more. 
Now,  as  Clerk  Maxwell  puts  it  in  discussing  this  very  problem, 
"one  material  system  can  differ  from  another  only  in  the  configura- 
tion and  motion  which  it  has  at  a  given  instant*."  If  we  cannot 
assume  differences  in  structure  or  configuration,  we  must  assume 
differences  in  motion,  that  is  to  say  in  energy.  And  if  we  cannot 
do  this,  then  indeed  we  are  thrown  back  upon  modes  of  reasoning 
unauthorised  in  physical  science,  and  shall  find  ourselves  constrained 
to  assume,  or  to  "admit,  that  the  properties  of  a  germ  are  not  those 
of  a  purely  material  system." 

But  we  are  by  no  means  necessarily  in  this  dilemma.  For  though 
we  come  perilously  near  to  it  when  we  contemplate  the  lowest 
orders  of  magnitude  to  which  hfe  has  been  attributed,  yet  in  the 
case  of  the  ordinary  cell,  or  ordinary  egg  or  germ  which  is  going 
to  develop  into  a  complex  organism,  if  we  have  no  reason  to  assume 
or  to  beheve  that  it  comprises  an  intricate  "mechanism,"  we  mgiy 
be  quite  sure,  both  on  direct  and  indirect  evidence,  that,  hke  the 
powder  in  our  rocket,  it  is  very  heterogeneous  in  its  structure. 
It  is  a  mixture  of  substances  of  various  kinds,  more  or  less  fluid, 
more  or  less  mobile,  influenced  in  various  ways  by  chemical,  electrical, 
osmotic  and  other  forces,  and  in  their  admixture  separated  by  a 
multitude  of  surfaces  or  boundaries,  at  which  these  or  certain  of 
these  forces  are  made  manifest. 

Indeed,  such  an  arrangement  as  this  is  already  enough  to  con- 
stitute a  "mechanism";  for  we  must  be  very  careful  not  to  let  our 
physical  or  physiological  concept  of  mechanism  be  narrowed  to  an 
interpretation  of  the  term  derived  from  the  comphcated  contrivances 
of  himaan  skill.  From  the  physical  point  of  view,  we  understand 
by  a  "mechanism"  w^hatsoever  checks  or  controls,  and  guides  into 
determinate  paths,  the  workings  of  energy:  in  other  words,  what- 
soever leads  in  the  degradation  of  energy  to  its  manifestation  in 
some  form  of  work,  at  a  stage  short  of  that  ultimate  degradation 
which  lapses  in  uniformly  diffused  heat.  This,  as  Warburg  has  well 
explained,  is  the  general  effect  or  function  of  the  physiological 
machine,  and  in  particular  of  that  part  of  it  which  we  call  "cell- 

*  Precisely  as  in  the  Lueretian  concursus,  motus,  ordo,  positura,  figurae,  whereby 
bodies  mutato  ordine  mutant  naturam. 


292  ON  THE  INTERNAL  FORM  [ch. 

structure*."  The  normal  muscle-cell  is  something  which  turns 
energy,  derived  from  oxidation,  into  work;  it  is  a  mechanism  which 
arrests  and  utihses  the  chemical  energy  of  oxidation  in  its  downward 
course;  but  the  same  cell  when  injured  or  disintegrated  loses  its 
"usefulness,"  and  sets  free  a  greatly  increased  proportion  of  its 
energy  in  the  form  of  heat.  It  was  a  saying  of  Faraday's,  that 
"even  a  Hfe  is  but  a  chemical  act  prolonged.  If  death  occur,  the 
more  rapidly  oxygen  and  the  affinities  run  on  to  their  final  state  f." 
Very  great  and  wonderful  things  are  done  by  means  of  a 
mechanism  (whether  natural  or  artificial)  of  extreme  simphcity. 
A  pool  of  water,  by  virtue  of  its  surface,  is  an  admirable  mechanism 
for  the  making  of  waves ;  with  a  lump  of  ice  in  it,  it  becomes  an 
efficient  and  self-contained  mechanism  for  the  making  of  currents. 
Music  itself  is  made  of  simple  things — a  reed,  a  pipe,  a  string. 
The  great  cosmic  mechanisms  are  stupendous  in  their  simphcity; 
and,  in  point  of  fact,  every  great  or  little  aggregate  of  heterogeneous 
matter  (not  identical  in  "phase")  involves,  ipso  facto,  the  essentials 
of  a  mechanism.  Even  a  non-living  colloid,  from  its  intrinsic  hetero- 
geneity, is  in  this  sense  a  mechanism,  and  one  in  which  energy  is 
manifested  in  the  movement  and  ceaseless  rearrangement  of  the 
constituent  particles.  For  this  reason  Graham  speaks  somewhere 
or  other  of  the  colloid  state  as  "the  dynamic  state  of  matter";  in 
the  same  philosopher's' phrase,  it  possesses  ^' energia%.'^ 

Let  us  turn  then  to  consider,  briefly  and  diagrammatically,  the 
structure  of  the  cell,  a  fertilised  germ-cell  or  ovum  for  instance,  not 
in  any  vain  attempt  to  correlate  this  structure  with  the  structure 
or  properties  of  the  resulting  and  yet  distant  organism ;  but  merely 
to  see  how  far,  by  the  study  of  its  form  and  its  changing  internal 
configuration,  we  may  throw  hght  on  certain  forces  which  are  for 
the  time  being  at  work  within  it. 

We  may  say  at  once  that  we  can  scarcely  hope  to  learn  more  of 
these  forces,  in  the  first  instance,  than  a  few  facts  regarding  their 

*  Otto  Warburg,  Beitrage  zur  Physiologie  der  Zelle,  insbesondere  iiber  die 
Oxidationsgeschwindigkeit  in  Zellen;  in  Asher-Spiro's  Ergebnisse  der  Physiologie, 
XIV,  pp.  253-337,  1914  (see  p.  315). 

t  See  his  Life  by  Bence  Jones,  ii,  p.  299. 

X  Both  phrases  occur,  side  by  side,  in  Graham's  classical  paper  on  Liquid 
diffusion  applied  to  analysis,  Phil.  Trans,  cli,  p.  184,  1861;  Chem.  and  Phys. 
Researches  (ed.  Angus  Smith),  1876,  p.  554. 


iv]  AND  STRUCTURE  OF  THE  CELL  293 

direction  and  magnitude;  tli£  nature  and  specific  identity  of  the 
force  or  forces  is  a  very  different  matter.  This  latter  problem  is 
likely  to  be  difficult  of  elucidation,  for  the  reason,  among  others, 
that  very  different  forces  are  often  much  alike  in  their  outward  and 
visible  manifestations.  So  it  has  come  to  pass  that  we  have  a 
multitude  of  discordant  hypotheses  as  to  the  nature  of  the  forces 
acting  within  the  cell,  and  producing  in  cell  division  the  "caryo- 
kinetic"  figures  of  which  we  are  about  to  speak.  One  student  may, 
like  Rhumbler,  choose  to  account  for  them  by  an  hj^othesis  of 
mechanical  traction,  acting  on  a  reticular  web  of  protoplasm*; 
another,  hke  Leduc,  may  shew  us  how  in  many  of  their  most  striking 
features  they  may  be  admirably  simulated  by  salts  diffusing  in  a 
colloid  medium;  others,  hke  Lamb  and  Graham  Cannon,  have 
compared  them  to  the  stream-hnes  produced  and  the  field  of  force 
set  up  by  bodies  vibrating  in  a  fluid;  others,  like  Gallardof  and 
Rhumbler  in  his  earher  papers  J,  insisted  on  their  resemblance  to 
certain  phenomena  of  electricity  and  magnetism  §;  while  Hartog 
believed  that  the  force  in  question  is  only  analogous  to  these,  and 
has  a  specific  identity  of  its  own||.  All  these  conflicting  views  are 
of  secondary  importance,  so  long  as  we  seek  only  to  account  for 
certain  configurations  which  reveal  the  direction,  rather  than  the 
nature,  of  a  force.  One  and  the  same  system  of  lines  of  force  may 
appear  in  a  field  of  magnetic  or  of  electrical  energy,  of  the  osmotic 
energy  of  diffusion,  of  the  gravitational  energy  of  a  flowing  stream. 
In  short,  we  may  expect  to  learn  something  of  the  pure  or  abstract 
dynamics  long  before  we  can  deal  with  the  special  physics  of  the 

*  L.  Rhumbler,  Mechanische  Erklarung  der  Aehnlichkeit  zwischen  magne- 
tischen  Kraftliniensystemen  und  Zelltheilungsfiguren,  Arch.  f.  Entw.  Mech.  xv, 
p.  482,  1903. 

t  A.  Gallardo,  Essai  d'interpretation  des  figures  caryocinetiques,  Armies  del 
Museo  de  Buenos- Aires  (2),  ii,  1896;   Arch.  f.  Entw.  Mech.  xxvm,  1909,  etc. 

X  Arch.f.  Entw.  Mech.  ill,  iv,  1896-97. 

§  On  various  theories  of  the  mechanism  of  mitosis,  see  (e.g.)  Wilson,  The  Cell 
in  Development,  etc.;  Meves,  Zelltheilung,  in  Merkel  u.  Bonnet's  Ergehnisse  der 
Anatomic,  etc.,  vii,  viii,  1897-98;  Ida  H.  Hyde,  Amer.  Journ.  Physiol,  xii,  pp.  241- 
275,  1905;  and  especially  A.  Prenant,  Theories  et  interpretations  physiques  de 
la  mitose,  Journ.  de  VAnat.  et  Physiol,  xlvi,  pp.  511-578,  1910.  See  also  A.  Conard, 
Sur  le  mecanisme  de  la  division  cellulaire,  et  sur  les  bases,  morphologiques  de  la 
Cytologie,  Bruxelles,  1939:  a  work  which  I  find  hard  to  follow. 

II  M.  Hartog,  Une  force  nouvelle:  le  mitokinetisme,  C.R.  11  Juli  1910;  Arch.  f. 
Entw.  Mech.  xxvii,  pp.  141-145,  1909;    cf.  ibid,  xl,  pp.  33-64,  1914. 


294  ON  THE  INTERNAL  FORM  [ch. 

cell.  For  indeed,  just  as  uniform  expansion  about  a  single  centre, 
to  whatsoever  physical  cause  it  may  be  due,  will  lead  to  the  con- 
figuration of  a  sphere,  so  will  any  two  centres  or  foci  of  potential 
(of  whatsoever  kind)  lead  to  the  configurations  with  which  Faraday 
first  made  us  famihar  under  the  name  of  "lines  of  force*";  a^d 
this  is  as  much  as  to  say  that  the  phenomenon,  though  physical  in  the 
concrete,  is  in  the  abstract  purely  mathematical,  and  in  its  very  essence 
is  neither  more  nor  less  than  a  property  of  three-dimensional  space. 

But  as  a  matter  of  fact,  in  this  instance,  that  is  to  say  in  trying 
to  explain  the  leading  phenomena  of  the  caryokinetic  division  of 
the  cell,  we  shall  soon  perceive  that  any  explanation  which  is  based, 
like  Rhumbler's,  on  mere  mechanical  traction,  is  obviously  inade- 
quate, and  we  shall  find  ourselves  limited  to  the  hypothesis  of  some 
polarised  and  polarising  force,  such  as  we  deal  with,  for  instance, 
in  magnetism  or  electricity,  or  in  certain  less  familiar  phenomena 
of  hydrodynamics.  Let  us  speak  first  of  the  cell  itself,  as  it  appears 
in  a  state  of  rest,  and  let  us  proceed  afterwards  to  study  the  more 
active  phenomena  which  accompany  its  division. 

Our  typical  cell  is  a  spherical  body;  that  is  to  say,  the  uniform 
surface-tension  at  its  boundary  is  balanced  by  the  outward  resistance 
of  uniform  forces  within.  But  at  times  the  surface-tension  may  be 
a  fluctuating  quantity,  as  when  it  produces  the  rhythmical  con- 
tractions or  "Ransom's  waves "f  on  the  surface  of  a  trout's  egg;  or 
again,  the  surface-tension  may  be  locally  unequal  and  variable,  giving 
rise  to  an  amoeboid  figure,  as  in  the  egg  of  HydraX- 

Within  the  cell  is  a  nucleus  or  germinal  vesicle,  also  spherical, 

*  The  configurations,  as  obtained  by  the  usual  experimental  methods,  were 
of  course  known  long  before  Faraday's  day,  and  constituted  the  "convergent  and 
divergent  magnetic  curves"  of  eighteenth  century  mathematicians.  As  Leslie 
said,  in  1821,  they  were  "regarded  with  wonder  by  a  certain  class  of  dreaming 
philosophers,  whp  did  not  hesitate  to  consider  them  as  the  actual  traces  of  an 
invisible  fluid,  perpetually  circulating  between  the  poles  of  the  magnet."  Faraday's 
great  advance  was  to  interpret  them  as  indications  of  stress  in  a  medium — of. 
tension  or  attraction  along  the  lines,  and  of  repulsion  transverse  to  the  lines,  of  the 
diagram. 

t  W.  H.  Ransom,  On  the  ovum  of  osseous  fishes,  Phil.  Trans,  clvii,  pp.  431-502, 
1867  (vide  p.  463  et.  seq.)  (Ransom,  afterwards  a  Nottingham  physician,  was 
Huxley's  friend  and  class-fellow  at  University  College,  and  beat  him  for  the  medal 
in  Grant's  class  of  zoology.) 

X  Cf,  also  the  curiou.s  phenomenon  in  a  dividing  egg  described  as  "spinning" 
by  Mrs  G.  F.  Andrews,  Journ.  Morph.  xii,  pp.  367-389,  1897. 


IV]  AND  STRUCTURE  OF  THE  CELL  295 

and  consisting  of  portions  of  "chromatin,"  aggregated  together 
within  a  more  fluid  drop.  The  fact  has  often  been  commented 
upon  that,  in  cells  generally,  there  is  no  correlation  of  form 
(though  there  apparently  is  of  size)  between  the  nucleus  and  the 
"cytoplasm,"  or  main  body  of  the  cell.  So  Whitman*  remarks 
that  "except  during  the  process  of  division  the  nucleus  seldom 
departs  from  it^  cypical  spherical  form.  It  divides  and  sub-divides, 
ever  returning  to  the  same  round  or  oval  form. .  . .  How  different 
with  the  cell.  It  preserves  the  spherical  form  as  rarely  as  the 
nucleus  departs  from  it.  Variation  in  form  marks  the  beginning 
and  the  end  of  every  important  chapter  in  its  history."  On  simple 
dynamical  grounds,  the  contrast  is  easily  explained.  So  long  as 
the  fluid  substance  of  the  nucleus  is  qualitatively  different  from, 
and  incapable  of  mixing  with,  the  fluid  or  semi-fluid  protoplasm 
surrounding  it,  we  shall  expect  it  to  be,  as  it  almost  always  is,  of 
spherical  form.  For  on  the  one  hand,  it  has  a  surface  of  its  own 
whose  surface-tension  is  presumably  uniform,  and  on  the  other,  it 
is  immersed  in  a  medium  which  transmits  on  all  sides  a  uniform 
fluid  or  "hydrostatic"  pressure f;  thus  the  case  of  the  spherical 
nucleus  is  closely  akin  to  that  of  the  spherical  yolk  within  the 
bird's  egg.  Again,  for  a  similar  reason,  the  contractile  vacuole  of 
a  protozoon  is  spherical {.     It  is  just  a  drop  of  fluid,  bounded  by  a 

*  Whitman,  Journ.  Morph.  ii,  p.  40,  1889. 

t  "Souvent  il  n'y  a  qu'une  separation  physique  entre  le  cytoplasme  et  le  sue 
hucleaire,  comme  entre  deux  liquides  immiscibles,  etc.";  Alexeieff,  8ur  la  mitose 
dite  primitive,  Arch.  f.  Protistenk.  xxix,  p.  357,  1913. 

X  The  appearance  of  "  vacuolation "  is  a  result  of  endosmosis,  or  the  diffusion 
of  a  less  dense  fluid  into  the  denser  plasma  of  the  cell.  But  while  water  is  probably 
taken  up  at  the  surface  of  the  cell  by  purely  passive  osmotic  intake,  a  definite 
"vacuole"  appears  at  a  place  where  osmotic  work  is  being  actively  done.  A  higher 
osmotic  pressure  than  that  of  the  external  medium  is  maintained  within  the  cell, 
but  as  a  "steady  state"  rather  than  a  condition  of  equilibrium,  in  other  words  by 
the  continual  expenditure  of  energy;  and  the  difference  of  pressure  is  at  best  small. 
The  "contractile  vacuole"  bursts  when  it  touches  the  surface  of  the  cell,  and 
bursting  may  be  delayed  by  manipulating  the  vacuole  towards  the  interior.  It 
may  sometimes  burst  towards  the  interior  of  the  cell  through  inequalities  in  its 
own  surface-tension,  and  the  collapsing  vacuole  is  then  apt  to  shew  a  star-shaped 
figure.  The  cause  of  the  higher  osmotic  pressure  within  the  cell  is  a  matter  for 
the  colloid  chemist,  and  cannot  be  discussed  here.  On  the  physiology  of  the 
contractile  vacuole,  see  {int.  al.)  H.  Z.  Gow,  Arch.  f.  Protistenk.  lxxxvii,  pp.  185- 
212,  1936;  J.  Spek,  Einfluss  der  Salze  auf  die  Plasmkolloide  von  Actinosphaerium, 
Acta  Zool.  1921;   J.  A,  Kitching,  Journ.  Exp.  Biology,  xi,  xiii,  xv,  1934-38. 


296  ON  THE  INTERNAL  FORM  [ch. 

uniform  surface-tension,  and  through  whose  boundary-film  diffusion 
is  taking  place;  but  here,  owing  to  the  small  difference  between  the 
fluid  constituting  and  that  surrounding  the  drop,  the  surface-tension 
equihbrium  is  somewhat  unstable;  it  is  apt  to  vanish,  and  the 
rounded  outhne  of  the  drop  disappears,  Uke  a  burst  bubble,  in  a 
moment. 

If,  on  the  other  hand,  the  substance  of  the  cell  acquire  a  greater 
soUdity,  as  for  instance  in  a  muscle-cell,  or  by  reason  of  mucous 
accumulations  in  an  epithehum  cell,  then  the  laws  of  fluid  pressure 
no  longer  apply,  the  pressure  on  the  nucleus  tends  to  become 
unsymmetrical,  and  its  shape  is  modified  accordingly.  Amoeboid 
movements  may  be  set  up  in  the  nucleus  by  anything  which  disturbs 
the  symmetry  of  its  own  surface-tension;  and  where  "nuclear 
material"  is  scattered  in  small  portions  throughout  the  cell  as  in 
many  Rhizopods,  instead  of  being  aggregated  in  a  single  nucleus, 
the  simple  explanation  probably  is  that  the  "phase  difference"  (as 
the  chemists  say)  between  the  nuclear  and  the  protoplasmic  substance 
is  comparatively  shght,  and  the  surface-tension  which  tends  to  keep 
them  separate  is  correspondingly  small*. 

Apart  from  that  invisible  or  ultra-microscopic  heterogeneity 
which  is  inseparable  from  our  notion  of  a  "colloid,"  there  is  a 
visible  heterogeneity  of  structure  within  both  the  nucleus  and  the 
outer  protoplasm.  The  former  contains,  for  instance,  a  rounded 
nucleolus  or  "germinal  spot,"  certain  conspicuous  granules  or 
strands  of  the  peculiar  substance  called  chromatin*)*,  and  a  coarse 
mesh  work  of  a  protoplasmic  material  known  as  "linin"  or  achro- 
matin;  the  outer  protoplasm,  or  cytoplasm,  is  generally  believed 
to  consist  throughout  of  a  sponge-work,  or  rather  alveolar  mesh- 
work,  of  more  and  less  fluid  substances;  it  may  contain  "mito- 
chondria," appearing  in  tissue-cultures  as  small  amoeboid  bodies; 
and  lastly,  there  are  generally  to  be  detected  (in  the  animal,  rarely 
in  the  vegetable  kingdom)  one  or  more  very  minute  bodies,  usually 
in  the  cytoplasm  sometimes  within  the  nucleus,  known  as  the 
centrosome  or  centrosomes. 

*  The  elongated  or  curved  "macronucleus"  of  an  Infusorian  is  to  be  looked 
upon  as  a  single  mass  of  chromatin,  rather  than  as  an  aggregation  of  particles  in 
a  fluid  drop,  as  in  the  case  described.  It  has  a  shape  of  its  own,  in  which  ordinary 
surface-tension  plays  a  very  subordinate  part. 

•j-  First  so-called  by  W.  Flemming,  in  his  Zellsubstanz,  Kern  und  Zelltheilung,  1882. 


IV]  AND  STRUCTURE  OF  THE  CELL  297 

The  morphologist  is  accustomed  to  speak  of  a  "polarity"  of  the 
cell,  meaning  thereby  a  symmetry  of  visible  structure  about  a 
particular  axis.  For  instance,  whenever  we  can  recognise  in  a  cell 
both  a  nucleus  and  a  centrosome,  we  may  consider  a  hne  drawn 
through  the  two  as  the  morphological  axis  of  polarity ;  an  epitheUum 
cell  is  morphologically  symmetrical  about  a  median  axis  passing 
from  its  free  surface  to  its  attached  base.  Again,  by  an  extension 
of  the  term  polarity,  as  is  customary  in  dynamics,  we  may  have 
a  "radial"  polarity,  between  centre  and  periphery;  and  lastly,  we 
may  have  several  apparently  independent  centres  of  polarity  within 
the  single  cell.  Only  in  cells  of  quite  irregular  or  amoeboid  form 
do  we  fail  to  recognise  a  definite  and  symmetrical  polarity.  The 
morphological  polarity  is  accompanied  by,  and  is  but  the  outward 
expression  (or  part  of  it)  of  a  true  dynamical  polarity,  or  distribution 
of  forces;  and  the  hues  of  force  are,  or  may  be,  rendered  visible 
by  concatenation  of  particles  of  matter,  such  as  come  under  the 
influence  of  the  forces  in  action. 

When  hnes  of  force  stream  inwards  from  the  periphery  towards 
a  point  in  the  interior  of  the  cell,  particles  susceptible  of  attraction 
either  crowd  towards  the  surface  of  the  cell  or,  when  retarded  by 
friction,  are  seen  forming  lines  or  "fibrillae"  which  radiate  outwards 
from  the  centre.  In  the  cells  of  columnar  or  cihated  epithehum, 
where  the  sides  of  the  cell  are  symmetrically  disposed  to  their 
neighbours  but  the  free  and  attached  surfaces  are  very  diverse  from 
one  another  in  their  external  relations,  it  is  these  latter  surfaces 
which  constitute  the  opposite  poles;  and  in  accordance  with  the 
parallel  lines  of  force  so  set  up,  we  very  frequently  see  parallel  lines 
of  granules  which  have  ranged  themselves  perpendicularly  to  the 
free  surface  of  the  cell  (cf.  Fig.  149). 

A  simple  manifestation  of  polarity  may  be  well  illustrated  by 
the  phenomenon  of  diffusion,  where  we  may  conceive,  and  may 
automatically  reproduce,  a  field  of  force,  with  its  poles  and  its 
visible  lines  of  equipotential,  very  much  as  in  Faraday's  conception 
of  the  field  of  force  of  a  magnetic  system.  Thus,  in  one  of  Leduc's 
experiments*,  if  we  spread  a  layer  of  salt  solution  over  a  level 
plate  of  glass,  and  let  fall  into  the  middle  of  it  a  drop  of  indian 
ink,  or  of  blood,  we  shall  find  the  coloured  particles  travelling 
*  Thtorie  physico-chimique  de  la  Vie,  1910,  p.  73. 


298  ON  THE  INTERNAL  FORM  [ch 

outwards  from  the  central  "pole  of  concentration"  along  the  lines 
of  diffusive  force,  and  so  mapping  out  for  us  a  "monopolar  field" 
of  diffusion :  and  if  we  set  two  such  drops  side  by  side,  their  fines 
of  diffusion  will  oppose  and  repel  one  another.  Or,  instead  of  the 
uniform  layer  of  salt  solution,  we  may  place  at  a  little  distance 
from  one  another  a  grain  of  salt  and  a  drop  of  blood,  representing 
two  opposite  poles:  and  so  obtain  a  picture  of  a  "bipolar  field" 
of  diffusion.  In  either  case,  we  obtain  results  closely  analogous  to 
the  morphological,  but  really  dynamical,  polarity  of  the  organic 
cell.  But  in  all  probability,  the  dynamical  polarity  or  asymmetry 
of  the  cell  is  a  very  complicated  phenomenon:  for  the  obvious 
reason  that,  in  any  system,  one  asymmetry  will  tend  to  beget 
another.  A  chemical  asymmetry  will  induce  an  inequafity  of 
surface-tension,  which  will  lead  directly  to  a  modification  of  form ; 
the  chemical  asymmetry  may  in  turn  be  due  to  a  process  of 
electrolysis  in  a  polarised  electrical  field;  and  again  the  chemical 
heterogeneity  may  be  intensified  into  a  chemical  polarity,  by  the 
tendency  of  certain  substances  to  seek  a  locus  of  gi^eater  or  less 
surface-energy.  We  need  not  attempt  to  grapple  with  a  subject  so 
compHcated,  and  leading  to  so  many  problems  which  lie  beyond 
the  sphere  of  interest  of  the  morphologist.  But  yet  the  morpho- 
logist,  in  his  study  of  the  cell,  cannot  quite  evade  these  important 
issues;  and  we  shall  return  to  them  again  when  we  have  dealt 
somewhat  with  the  form  of  the  cell,  and  have  taken  account  of 
some  of  its  simpler  phenomena. 

We  are  now  ready,  and  in  some  measure  prepared,  to  study  the 
numerous  and  complex  phenomena  which  accompany  the  division 
of  the  cell,  for  instance  of  the  fertilised  egg.  But  it  is  no  easy  task 
to  epitomise  the  facts  of  the  case,  and  none  the  easier  that  of  late 
new  methods  have  shewn  us  new  things,  and  have  cast  doubt  on 
not  a  little  that  we  have  been  accustomed  to  believe. 

Division  of  the  cell  is  of  necessity  accompanied,  or  preceded,  by 
a  change  from  a  radial  or  monopolar  to  a  definitely  bipolar  sym- 
metry. In  the  hitherto  quiescent  or  apparently  quiescent  cell,  we 
perceive  certain  movements,  which  correspond  precisely  to  what 
must  accompany  and  result  from  a  polarisation  of  forces  within: 
of  forces   which,   whatever  be  their  specific   nature,   are  at  least 


IV]  AND  STRUCTURE  OF  THE  CELL  299 

capable  of  polarisation,  and  of  producing  consequent  attraction  or 
repulsion  between  charged  particles.  The  opposing  forces  which 
are  distributed  in  equilibrium  throughout  the  cell  become  focused 
in  two  "centrosomes*,"  which  may  or  may  not  be  already  visible. 
It  generally  happens  that,  in  the  egg,  one  of  these  centrosomes  is 
near  to  and  the  other  far  from  the  "animal  pole,"  which  is  both 
visibly  and  chemically  different  from  the  other,  and  is  where  the 
more  conspicuous  developmental  changes  will  presently  begin. 

Between  the  two  centrosomes,  in  stained  preparations,  a  spindle- 
shaped  figure  appears  (Fig.  88),  whose  striking  resemblance  to  the 


Fig.  8».     Caryokinetic  ligure  in  a  dividing  cell  (or  blastomere)  of  a  trout  s  egg. 
After  Prenant,  from  a  preparation  by  Prof.  Bouin. 

lines  of  force  made  visible  by  iron-filings  between  the  poles  of  a 
magnet  was  at  once  recognised  by  Hermann  Fol,  in  1873,  when  he 
witnessed  the  phenomenon  for  the  first  timef.  On  the  farther 
side  of  the  centrosomes  are  seen  star-like  figures,  or  "asters,"  in 
which  we  se^m  to  recognise  the  broken  lines  of  force  which  run 
externally  to  those  stronger  lines  which  lie  nearer  to  the  axis  and 
constitute  the  "spindle."  The  lines  of  force  are  rendered  visible, 
or  materialised,  just  as  in  the  experiment  of  the  iron-fihngs,  by  the 
fact  that,  in  the  heterogeneous  substance  of  the  cell,  certain  portions 

*  These  centrosomes  are  the  two  halves  of  a  single  granule,  and  are  said  (by 
Boveri)  to  come  from  the  middle  piece  of  the  original  spermatozoon. 

t  He  did  so  in  the  egg  of  a  medusa  {Geryon),  Jen.  Zeitschr.  vii,  p.  476,  1873. 
Similar  ideas  have  been  expressed  by  Strasbiirger,  Henneguy,  Van  Beneden, 
Errera,  Ziegler,  Gallardo  and  others. 


300  ON  THE  INTERNAL  FORM  [ch. 

of  matter  are  more  "permeable"  to  the  acting  force  than  others, 
become  themselves  polarised  after  the  fashion  of  a  magnetic  or 
"paramagnetic"  body,  arrange  themselves  in  an  orderly  way 
between  the  two  poles  of  the  field  of  force,  seem  to  cling  to  one 
another  as  it  were  in  threads*,  and  are  only  prevented  by  the 
friction  of  the  surrounding  medium  from  approaching  and  con- 
gregating around  the  adjacent  poles. 

As  the  field  of  force  strengthens,  the  more  will  the  lines  of  force 
be  drawn  in  towards  the  interpolar  axis,  and  the  less  evident  will 
be  those  remoter  lines  which  constitute  the  terminal,  or  extrapolar, 
asters:  a  clear  space,  free  from  materialised  fines  of  force,  may 
thus  tend  to  be  set  up  on  either  side  of  the  spindle,  the  so-called 
''Biitschfi  space"  of  the  histologistsl.  On  the  other  hand,  the  lines 
of  force  constituting  the  spindle  will  be  less  concentrated  if  they 
find  a  path  of  less  resistance  at  the  periphery  of  the  ceU :  as  happens 
in  our  experiment  of  the  iron-filings,  when  we  encircle  the  field  of 
force  with  an  iron  ring.  On  this  principle,  the  differences  observed 
between  cells  in  which  the  spindle  is  well  developed  and  the  asters 
small,  and  others  in  which  the  spindle  is  weak  and  the  asters  greatly 
developed,  might  easily  be  explained  by  variations  in  the  potential 
of  the  field,  the  large,  conspicuous  asters  being  correlated  in  turn 
with  a  marked  permeability  of  the  surface  of  the  cell. 

The  visible  field  of  force,  though  often  called  the  "nuclear 
spindle,"  is  formed  outside  of,  but  usually  near  to,  the  nucleus. 


*  Whence  the  name  "mitosis"  (Greek  /ziros,  a  thread),  applied  first  by  Flemming 
to  the  whole  phenomenon.  Kolimann  (Biol.  Centralbl.  ii,  p.  107,  1882)  called  it 
divisio  per  fila,  or  divisio  laqueis  implicata.  Many  of  the  earlier  students,  such  as 
Van  Beneden  (Rech.  sur  la  maturation  de  I'oeuf,  Arch,  de  Biol,  iv,  1883),  and 
Hermann  Fol  (Zur  Lehre  v.  d.  Entstehung  d.  karyokinetischen  Spindel,  Arch.  f. 
mikrosk.  Anat.  x,xxvii,  1891)  thought  they  recognised  actual  muscular  threads, 
drawing  the  nuclear  material  asunder  towards  the  respective  foci  or  poles;  and 
some  such  view  of  Zugkrdfte  was  long  maintained  by  other  writers,  by  Heidenhain 
especially,  by  Boveri,  Flemming,  R.  Hertwig,  Rhumbler,  and  many  more.  In  fact, 
the  existence  of  contractile  threads,  or  the  ascription  to  the  spindle  rather  than  to 
the  poles  or  centrosomes  of  the  active  forces  concerned  in  nuclear  division,  formed 
the  main  tenet  of  all  those  who  declined  to  go  beyond  the  "contractile  properties 
of  protoplasm"  for  an  explanation  of  the  phenomenon  (cf.  J.  W.  Jenkinson, 
Q.J. M.S.  XLViii,  p.  471,  1904.  See  also  J.  Spek's  historical  account  of  the  theories 
of  cell-division.  Arch.  f.  Entw.  Mech.  xliv,  pp.  5-29,  1918). 

t  Cf.  0,  Biitschli,  Ueber  die  kiinstliche  Nachahmung  der  karyokinetischen 
Figur,  Verh.  Med.  Nat.  Ver.  Heidelberg,  v,  pp.  28-41  (1892),  1897. 


IV 


AND  STRUCTURE  OF  THE  CELL 


301 


Let  us  look  a  little  more  closely  into  the  structure  of  this  body, 
and  into  the  changes  which  it  presently  undergoes. 

Within  its  spherical  outHne  (Fig.  89  a),  it  contains  an  ''alveolar" 
meshwork  (often  described,  from  its  appearance  in  optical  section, 
as  a  "reticulum"),  consisting  of  more  sohd  substances  with  more 
fluid  matter  filling  up  the  interalveolar  spaces.  This  phenomenon, 
familiar  to  the  colloid  chemist,  is  what  he  calls  a  "two-phase 
system,"  one  substance  or  "phase"  forming  a  continuum  through 
which  the  other  is  dispersed;   it  is  closely  alhed  to  what  we  call  in 


atr  racHon  -  sphere 
1   ,C€ntro3omes 


Fig.  89  A. 


Fig.  89  B. 


ordinary  language  a.  froth  ot'sl  foam*,  save  that  in  these  latter  the 
disperse  phase  is  represented  by  air.  It  is  a  surface-tension  pheno- 
menon, due  to  the  interaction  of  two  intermixed  fluids  not  very 
different  in  density,  as  they  strive  to  separate.  Of  precisely  the 
same  kind  (as  Biitschli  was  the  first  to  shew)  are  the  minute  alveolar 
networks  which  are  to  be  discerned  in  the  cytoplasm  of  the  cellf, 

*  Froth  and  foam  have  been  much  studied  of  late  years  for  technical  reasons, 
and  other  factors  than  surface-tension  are  foiind  to  be  concerned  in  their  existence 
and  their  stability.  See  (int.  al.)  Freundlich's  Capillarchemie,  and  various  papers 
by  Sasaki,  in  Bull.  Chem.  Soc.  of  Japan,  1936-39. 

t  Biitschli,  Untersuchungen  iiber  mikroskopische  Schdume  und  das  Protoplasma, 
1892;  Untersuchungen  uber  Strukturen,  etc.,  1898;  L.  Rhumbler,  Protoplasma  als 
physikalisches  System,  Ergehn.  d.  Physiologie,  1914;  H.  Giersberg,  Plasmabau 
der  Amoben,  im  Hinblick  auf  die  Wabentheorie,  Arch.  f.  Entw.  Mech.  li,  pp.  150-250, 
1922;   etc. 


302  ON  THE  INTERNAL  FORM  [ch. 

and  which  we  now  know  to  be  not  inherent  in  the  nature  of  proto- 
plasm nor  of  Hving  matter  in  general,  but  to  be  due  to  various 
causes,  natural  as  well  as  artificial*.  The  microscopic  honeycomb 
structure  of  cast  metal  under  various  conditions  of  coohng  is  an 
example  of  similar  surface-tension  phenomena. 

Such  then,  in  briefest  outhne,  is  the  typical  structure  commonly 
ascribed  to  a  cell  when  its  latent  energies  are  about  to  manifest 
themselves  in  the  phenomenon  of  cell-division.  The  account  is 
based  on  observation  not  of  the  hving  cell  but  of  the  dead :  on  the 
assumption,  that  is  to  say,  that  fixed  and  stained  material  gives  a 
true  picture  of  reahty.  But  in  Robert  Chambers's  method  of  micro- 
dissection f,  the  hving  cell  is  manipulated  with  fine  glass  needles 
under  a  high  magnification,  and  shews  us  many  interesting  things. 
Chambers  assures  us  that  the  spindle  fibres  never  make  their 
appearance  as  visible  structures  until  coagulation  has  set  in;  and 
that  astral  rays  are,  or  appear  to  be,  channels  in  which  the  more 
fluid  content  of  the  cell  flows  towards  a  centrosome:|:.  Within  the 
bounds  to  which  we  are  at  present  keeping,  these  things  are  of  no 
great  moment;  for  whether  the  spindle  appear  early  or  late,  it  still 
bears  witness  to  the  fact  that  matter  has  arranged  itself  along 
bipolar  fines  of  force;  and  even  if  the  astral  rays  be  only  streams 
or  currents,  on  lines  of  force  they  still  approximately  he.  Yet  the 
change  from  the  old  story  to  the  new  is  important,  and  may  make 
a  world  of  diff'erence  when  we  attempt  to  define  the  forces  concerned. 
All  our  descriptions,  all  our  interpretations,  are  bound  to  be 
influenced  by  our  conception  of  the  mechanism  before  us;    and  he 

*  Arrhenius,  in  describing  a  typical  colloid  precipitate,  does  so  in  terms  that 
are  very  closely  applicable  to  the  ordinary  microscopic  appearance  of  the  protoplasm 
of  the  cell.  The  precipitate  consists,  he  says,  "en  un  reseau  d'une  substance  solide 
contenant  peu  d'eau,  dans  les  mailles  duquel  est  inclus  un  fluide  contenant  un  peu 
de  colloide  dans  beaucoup  d'eau. . . .  Evidemment  cette  structure  se  forme  a  cause 
de  la  petite  difference  de  poids  specifique  des  deux  phases,  et  de  la  consistance 
gluante  des  particules  separees,  qui  s'attachent  en  forme  de  reseau  "  {Rev.  Scientifique, 
Feb.  1911).  This,  however,  is  far  from  being  the  whole  story:  cf.  (e.g.)  S.  C. 
Bradford,  On  the  theory  of  gels,  Biochem.  Journ.  xvii,  p.  230,  1925;  W.  Seifritz, 
The  alveolar  structure  of  protoplasm,  Protoplasma,  ix,  p.  198,  1930;  and  A.  Frey- 
Wissling,  Submikroskopische  Morphologie  des  Protoplasmas,  Berlin,  1938. 

t  See  R.  Chambers,  An  apparatus. .  .for  the  dissection  and  injection  of  living 
cells,  Anatom.  Record,  xxiv,  19  pp.,  1922. 

X  This  centripetal  flow  of  fluid  was  announced  by  Biitschli  in  his  early  papers, 
and  confirmed  by  Rhumbler,  though  attributed  to  another  cause. 


IV]  AND  STRUCTURE  OF  THE  CELL  303 

who  sees  threads  where  another  sees  channels  is  hkely  to  tell  a 
different  story  about  neighbouring  and  associated  things. 

It  has  also  been  suggested  that  the  spindle  is  somehow  due  to  a  re-arrange- 
ment of  protein  macromolecules  or  micelles ;  that  such  changes  of  orientation 
of  large  colloid  particles  may  be  a  widespread  phenomenon;  and  that  coagu- 
lation itself  is  but  a  polymerisation  of  larger  and  larger  macromolecules*. 

But  here  we  have  touched  the  brink  of  a  subject  so  important  that  we  must 
not  pass  it  by  without  a  word,  and  yet  so  contentious  that  we  must  not  enter 
into  its  details.  The  question  involved  is  simply  whether  the  great  mass  of 
recorded  observations  and  accepted  beliefs  with  regard  to  the  visible  structure 
of  protoplasm  and  of  the  cell  constitute  a  fair  picture  of  the  actual  living  cell, 
or  be  based  on  appearances  which  are  incident  to  death  itself  and  to  the 
artificial  treatment  which  the  microscopist  is  accustomed  to  apply.  The  great 
bulk  of  histological  work  is  done  by  methods  which  involve  the  sudden  killing 
of  the  cell  or  organism  by  strong  reagents,  the  assumption  being  that  death 
is  so  rapid  that  the  visible  phenomena  exhibited  during  life  are  retained  or 
"fixed"  in  our  preparations. 

Hermann  Fol  struck  a  warning  note  full  sixty  years  ago:  "II  importe  a 
I'avenir  de  I'histologie  de  combattre  la  tendance  a  tirer  des  conclusions  des 
images  obtenues  par  des  moyens  artificiels  et  a  leur  donner  une  valeur  intrin- 
seque,  sans  que  ces  images  aient  ete  controlees  sur  le  vivantf."  Fol  was 
thinking  especially  of  cell-membranes  and  the  delimitation  of  cells;  but  still 
more  difficult  and  precarious  is  the  interpretation  of  the  minute  internal  net- 
works, granules,  etc.,  which  represent  the  alleged  structure  of  protoplasm. 
A  colloid  body,  or  colloid  solution,  is  ipso  facto  heterogeneous;  it  has  after 
some  fashion  a  structure  of  its  own.  And  this  structure  chemical  action, 
under  the  microscope,  may  demonstrate,  or  emphasise,  or  alter  and  disguise. 
As  Hardy  put  it,  "It  is  notorious  that  the  various  fixing  reagents  are  co- 
agulants of  organic  colloids,  and  that  the  figure  varies  according  to  the  reagent 
used." 

A  case  in  point  is  that  of  the  vitreous  humour,  to  which  some  histologists 
have  ascribed  a  fairly  complex  structure,  seeing  in  it  a  framework  of  fibres 
with  the  meshes  filled  with  fluid.  But  it  is  really  a  true  gel,  without  any 
structure  in  the  usual  sense  of  the  word.  The  "fibres"  seen  in  ordinary 
microscopic  preparations  are  due  to  the  coagulation  of  micellae  by  the  fixative 
employed.  Under  the  ultra -microscope  the  vitreous  is  optically  empty  to 
begin  with;  then  innumerable  minute  fibrillae  appear  in  the  beam  of  light, 
criss-crossing  one  another.     Soon  these  break  down  into  strings  of  beads,  and 

*  Cf.  J.  D.  Bernal,  on  Molecular  architecture  of  biological  systems,  Proc.  Boy. 
Inst.,  1938;    H.  Staiidinger,  Nature,  Aug.  I,  1939. 

t  H.  Fol,  Becherches  sur  la  fecondation  et  le  commencement  de  VMnogenie  chez 
divers  animaux,  Geneve,  1879,  pp.  241-242.  Cf.  A.  Daleq,  in  Biol.  Beviews,  iii, 
p.  24,  1928:  "II  serait  desirable  de  nous  debarrasser  de  I'idee  que  tout  ce  qu'il 
y  a  d'important  dans  la  cellule  serait  providentiellement  colorable  par  I'hematoxy- 
line,  la  safranine  ou  le  violet  de  gentiane." 


304  ON  THE  INTERNAL  FORM  [ch. 

finally  only  separate  dots  are  seen*.  Other  sources  of  error  arise  from  the 
optical  principles  concerned  in  microscopic  vision;  for  the  diffraction-pattern 
which  we  call  the  "image"  may,  under  certain  circumstances,  be  very  different 
from  the  actual  object  f.  Furthermore,  the  optical  properties  of  living  proto- 
plasm are  especially  complicated  and  imperfectly  known,  as  in  general  those 
of  colloids  may  be  said  to  be;  the  minute  aggregates  of  the  "disperse  phase" 
of  gels  produce  a  scattering  action  on  light,' leading  to  appearances  of  turbidity 
etc.,  with  no  other  or  more  real  basis  J. 

So  it  comes  to  pass  that  some  writers  have  altogether  denied  the  existence 
in  the  living  cell-protoplasm  of  a  network  or  alveolar  "foam";  others  have 
cast  doubts  on  the  main  tenets  of  recent  histology  regarding  nuclear  structure ; 
and  Hardy,  discussing  the  structure  of  certain  gland-cells,  declared  that 
"there  is  no  evidence  that  the  structure  discoverable  in  the  cell-substance  of 
these  cells  after  fixation  has  any  counterpart  in  the  cell  when  living."  "A 
large  part  of  it"  he  went  on  to  say  "is  an  artefact.  The  profound  difference 
in  the  minute  structure  of  a  secretory  cell  of  a  mucous  gland  according  to  the 
reagent  which  is  used  to  fix  it  would,  it  seems  to  me,  almost  suffice  to  establish 
this  statement  in  the  absence  of  other  evidence  §." 

Nevertheless,  histological  study  proceeds,  especially  on  the  part  of  the 
morphologists,  with  but  little  change  in  theory  or  in  method,  in  spite  of  these 
and  many  other  warnings.  That  certain  visible  structures,  nucleus,  vacuoles, 
"attraction-spheres"  or  centrosomes,  etc.,  are  actually  present  in  the  living 
cell  we  know  for  certain;  and  to  this  class  belong  the  majority  of  structures 
with  which  we  are  at  present  concerned.  That  many  other  alleged  structures 
are  artificial  has  also  been  placed  beyond  a  doubt;  but  where  to  draw  the 
dividing  line  we  often  do  not  know. 

The  following  is  a  brief  epitome  of  the  visible  changes  undergone 
by  a  t3rpical  cell,  subsequent  to  the  resting  stage,  leading  up  to  the 
act  of  segmentation,  and  constituting  the  phenomenon  of  mitosis 
or  caryokinetic  division.  In  the  fertilised  egg  of  a  sea-urchin  we 
see  with  almost  diagrammatic  completeness,  in  fixed  and  stained 
specimens,  what  is  set  forth  here||. 

*  W.  S.  Duke-Elder,  Journ.  Physiol,  lxviii,  pp.  1.54-165,  1930;  of.  Baurmann, 
Arch.f.  Ophthalm.  1923,  1926;  etc. 

t  Abbe,  Arch.f.  mikrosk.  Anat.  ix,  p.  413,  1874;  Gesammelte  Ahhandl.  i,  p.  45, 
1904. 

X  Cf.  Rayleigh,  On  the  light  from  the  sky,  Phil.  Mag.  (4)  xli,  p.  107,  1871. 

§  W.  B.  Hardy,  On  the  structure  of  cell  protoplasm,  Journ.  Physiol,  xxiv, 
pp.  158-207,  1889;  also  Hober,  Physikalische  Chemie  der  Zelle  und  der  Gewebe, 
1902;  W.  Berg,  Beitrage  zur  Theorie  der  Fixation,  etc.,  Arch.f.  mikr.  Anat.  LXii,' 
pp.  367-440,  1903,  Cf.  {int.  al.)  Flemming,  Zellsubstanz,  Kern  und  Zelltheilung, 
1882,  p.  51;  etc. 

II  My  description  and  diagrams  (Figs.  89-93)  are  mostly  based  on  those  of 
the  late  Professor  E.  B.  Wilson. 


IV] 


AND  STRUCTURE  OF  THE  CELL 


305 


1.  The  chromatin,  which  to  begin  with  had  been  dimly  seen  as 
granules  on  a  vague  achromatic  reticulum  (Figs.  89,  90) — perhaps  no 
more  than  an  histological  artefact — concentrates  to  form  a  skein  or 
spireme,  often  looked  on  as  a  continuous  thread,  but  perhaps 
discontinuous  or  fragmented  from  the  first.  It,  or  its  several 
fragments,  will  presently  spht  asunder;  for  it  is  essentially  double, 
and  may  even  be  seen  as  a  double  thread,  or  pair  of  chromatids,  from 
an  early  stage.  The  chromosomss  are  portions  of  this  double  thread, 
which  shorten  down  to  form  httle  rods,'  straight  or  curved,  often 


chromoaome* 


Fig.  90  A. 


Fig.  90  B. 


bent  into  a  V,  sometimes  ovoid,  round  or  even  annular,  and  which 
in  the  living  cell  are  frequently  seen  in  active,  writhing  movement, 
*"hke  eels  in  a  box"*;  they  keep  apart  from  one  another,  as  by 
some  repulsion,  and  tend  to  move  outward  towards  the  nuclear 
membrane.  Certain  deeply  staining  masses,  the  nucleoh,  may  be 
present  in  the  resting  nucleus,  but  take  no  part  (at  least  as  a  rule) 
in  the  formation  of  the  chromosomes;  they  are  either  cast  out  of 
the  nucleus  and  dissolved  in  the  cytoplasm,  or  else  fade  away  in  situ. 

*  T.  S.  Strangeways,  Proc.  E.S.  (B),  xciv,  p.  139,  1922.  The  tendency  of  the 
chromatin  to  form  spirals,  large  or  small,  while  the  nucleus  is  issuing  from  its 
resting-stage,  is  very  remarkable.  The  tensions  to  which  it  is  due  may  be  overcome, 
and  the  chromosomes  made  to  uncoil,  by  treatment  with  ammonia  or  acetic  acid 
vapour.  See  Y.  Kuwada,  Botan.  Mag.  Tokyo,  xlvi,  p.  307,  1932;  and  C.  D. 
Darlington,  Mechanical  aspects  of  nuclear  division,  Sci.  Journ.  B.  Coll,  of  Sci. 
TV,  p.  94,  1934. 


306  ON  THE  INTERNAL  FORM  [ch. 

But  this  rule  does  not  always  hold;  for  they  persist  in  many 
protozoa,  and  now  and  then  the  nucleolus  remains  and  becomes 
itself  a  chromosome,  as  in  the  spermogonia  of  certain  insects. 

2.  Meanwhile  a  certain  deeply  staining  granule  (here  extra- 
nuclear),  known  as  the  centrosome*,  has  divided  into  two.  It  is  all 
but  universally  visible,  save  in  the  higher  plants;  perhaps  less  stress 
is  laid  on  it  than  at  one  time,  but  Bovery  called  i-t  the  "dynamic 
centre"  of  the  cellf.  The  two  resulting  granules  travel  to 
opposite  poles  Df  the  nucleus,  and  there  eacii  becomes  surrounded 
by  a  starhke  figure,  the  aster,  of  which  we  have  sptken  already; 
immediately  around  the  centrosome  is  a  clear  space,  the  centro- 
sphere.  Between  the  two  centrosomes,  or  the  two  asters,  stretches 
the  spindle.  It  lies  in  the  long  axis,  if  there  be  one,  of  the  cell,  a 
rule  laid  down  nearly  sixty  years  ago,  and  still  remembered  as 
"Hertwig's  Law"  J;  but  the  rule  is  as  much  and  no  more  than  to 
say  that  the  spindle  sets  in  the  direction  of  least  resistance.  Where 
the  egg  is  laden  with  food-yolk,  as  often  happens,  the  latter  is 
heavier  than  the  cytoplasm;  and  gravity,  by  orienting  the  egg 
itself,  thus  influences,  though  only  indirectly,  the  first  planes  of 
segmentation  §. 

3.  The  definite  nuclear  outhne  is  soon  lost;  for  the  chemical 
"phase-difference"  between  nucleus  and  cytoplasm  has  broken 
down,  and  where  the  nucleus  was,  the  chromosomes  now  he  (Figs. 
90,  91).  The  lines  of  the  spindle  become  visible,  the  chromosomes 
arrange  themselves  midway  between  its  poles,  to  form  the  equatorial 
plate,  and  are  spaced  out  evenly  around  the  central  spindle,  again 
a  simple  result  of  mutual  repulsion. 

4.  Each  chromosome  separates  longitudinally  into  two|| :  usually 
at  this  stage — but  it  is  to  be  noted  that  the  spHtting  may  have  taken 
place  as  early  as  the  spireme  stage  (Fig.  92). 

*  The  centrosome  has  a  curious  history  of  its  own,  none  too  well  ascertained. 
The  ovum  has  a  centrosome,  and  in  self- fertilised  eggs  this  is  retained;  but  when 
a  sperm-cell  enters  the  egg  the  original  centrosome  degenerates,  and  its  place  is 
taken  by  the  "middle-piece"  of  the  spermatozoon, 

f  The  stages  1,  2,  5  and  6  are  called  by  embryologista  the  prophase,  metaphase, 
anaphase  and  telophase. 

X  C.  Hertwig,  Jenaische  Ztschr.  xviii,  1884. 

§  See  James  Gray,  The  effect  of  gravity  on  the  eggs  of  Echinus,  Jl.  Exp.  Zool.  v, 
pp.  102-11,  1927. 

II  A  fundamental  fact,  first  seen  by  Flemming  in  1880. 


IV] 


AND  STRUCTURE  OF  THE  CELL 


307 


5.  The  halves  of  the  spht  chromosomes  now  separate  from  and 
apparently  repel  one  another,  travelUng  in  opposite  directions 
towards  the  two  poles*  (Fig.  92  b),  for  all  the  world  as  though  they 
were  being  pulled  asunder  by  actual  threads. 


Fig.  91  A. 


Fig.  91  B. 


central  spindle 

mantle 'fibres 


split  chromosome* 


Fig.  92  A. 


Fig.  92  B. 


6.     Presently  the  spindle  itself  changes  shape,  lengthens  and  con- 
tracts, and  seems  as  it  were  to  push  the  two  groups  of  daughter- 


*  Cf.  K.  Belar,  Beitrage  zur  Causalanalyse  der  Mitose,  Ztschr.  f.  Zellforschung, 
X,  pp.  73-124,  1929. 


308 


ON  THE  INTERNAL  FORM 


[CH. 


chromosomes  into  their  new  places*  (Figs.  92,  93);  and  its  chromo- 
somes form  once  more  an  alveolar  reticulum  and  may  occasionally 
form  another  spireme  at  this  stage.  A  boundary-surface,  or  at  least  a 
recognisable  phase-difference,  now  develops  round  each  reconstructed 
nuclear  mass,  and  the  spindle  disappears  (Fig.  93  b).  The  centrosome 
remains,  as  a  rule,  outside  the  nucleus. 

7.  On  the  central  spindle,  in  the  position  of  the  equatorial  plate, 
a  "cell-plate,"  consisting  of  deeply  staining  thickenings,  has  made 
its  appearance  during  the  migration  of  the  chromosomes.  This  cell- 
plate  is  more  conspicuous  in  plant-cells. 


otfrottron  spncft 


,d>iopptQr,ng  spmifte 


Htcon^rucffd  dough/trnuei^t 


Fig.  93  B. 


8.  Meanwhile  a  constriction  has  appeared  in  the  cytoplasm,  and 
the  cell  divides  through  the  equatorial  plane.  In  plant-cells  the 
line  of  this  division  is  foreshadowed  by  the  "cell-plate,"  which 
extends  from  the  spindle  across  the  entire  cell,  and  spHts  into  two 
layers,  between  which  appears  the  membrane  by  which  the  daughter- 
cells  are  cleft  asunder.  In  animal  cells  the  cell-plate  does  not  attain 
such  dimensions,  and  no  cell-wall  is  formed. 

The  whole  process  takes  from  half-an-hour  to  an  hour;  and  this 
extreme  slowness  is  not.  the  least  remarkable  part  of  the  pheno- 
menon, from  a  physical  point  of  view.     The  two  halves  of  the 

*  The  spindle  has  no  actual  threads  or  fibres,  for  Robert  Chambers's  micro- 
needles pass  freely  through  it  without  disturbing  the  chromosomes:  nor  is  it 
visible  at  all  in  living  cells  in  vitro.  It  seems  to  be  due  to  partial  gelation  of  the 
cytoplasm,  under  conditions  which,  whether  they  be  mechanical  or  chemical,  are 
not  easy  to  understand. 


IV]  AND  STRUCTURE  OF  THE  CELL  309 

dividing  centrosome,  while  moving  apart,  take  some  twenty  minutes 
to  travel  a  distance  of  20 /x,  or  at  the  rate,  say,  of  two  years  to  a 
yard.  It  is  a  question  of  inertia,  and  the  inertia  of  the  system  must 
be  very  large. 

The  beautiful  technique  of  cell-culture  in  vitro  has  of  late  years 
let  this  whole  succession  of  phenomena,  once  only  to  be  deduced 
from  sections,  be  easily  followed  as  it  proceeds  within  the  living 
tissue  or  cell.  The  vivid  accounts  which  have  been  given  of  this 
spectacle  add  little  to  the  older  account  as  we  have  related  it: 
save  that,  when  the  equatorial  constriction  begins  and  the  halves 
of  the  split  chromosomes  drift  apart,  the  protoplasm  begins  to  show 
a  curious  and  even  violent  activity.  The  cytoplasm  is  thrust  in 
and  out  in  bulging  pustules  or  "balloons";  and  the  granules  and 
fat-globules  stream  in  and  out  as  the  pustules  rise  and  fall  away. 
At  length  the  turmoil  dies  down;  and  now  each  half  of  the  cell 
(not  an  ovum  but  a  tissue-cell  or  "fibroplast")  pushes  out  large 
pseudopodia,  flattens  into  an  amoeboid  phase,  the  connecting  thread 
of  protoplasm  snaps  in  the  divided  cell,  and  the  daughter-cells  fall 
apart  and  crawl  away.  The  two  groups  of  chromosomes,  on  reaching 
the  poles  of  the  spindle,  turn  into  bunches  of  short  thick  rods;  these 
grow  diffuse,  and  form  a  network  of  chromatin  within  a  nucleus; 
and  at  last  the  chromosomes,  having  lost  their  identity,  disappear 
entirely,  and  two  or  more  nucleoH  are  all  that  is  to  be  seen  within 
the  cell. 

The  whole,  or  very  nearly  the  whole,  of  these  nuclear  phenomena 
may  be  brought  into  relation  with  some  such  polarisation  of  forces 
in  the  cell  as  a  whole  as  is  indicated  by  the  "spindle"  and  "asters" 
of  which  we  have  already  spoken:  certain  particular  phenomena, 
directly  attributable  to  surface-tension  and  diffusion,  taking  place 
in  more  or  less  obvious  and  inevitable  dependence  upon  the  polar 
system.  At  the  same  time,  in  attempting  to  explain  the  phenomena, 
we  cannot  say  too  clearly,  or  too  often,  that  all  that  we  are  meanwhile 
justified  in  doing  is  to  try  to  shew  that  such  and  such  actions  He 
within  the  range  of  known  physical  actions  and  phenomena,  or  that 
known  physical  phenomena  produce  effects  similar  to  them.  We 
feel  that  the  whole  phenomenon  is  iiot  sui  generis,  but  is  some- 
how or  other  capable  of  being  referred  to  dynamical  laws,  and  to 


310  ON  THE  INTERNAL  FORM  [ch. 

the  general  principles  of  physical  science.  But  when  we  speak  of 
some  particular  force  or  mode  of  action,  using  it  as  an  illustrative 
hypothesis,  we  stop  far  short  of  the  implication  that  this  or  that 
force  is  necessarily  the  very  one  which  is  actually  at  work  within 
the  living  cell;  and  certainly  we  need  not  attempt  the  formidable 
task  of  trying  to  reconcile,  or  to  choose  between,  the  various 
hypotheses  which  have  already  been  enunciated,  or  the  several 
assumptions  on  which  they  depend. 

Many  other  things  happen  within  the  cell,  especiall)^  in  the  germ- 
cell  both  before  and  after  fertilisation.  They  also  have  a  physical 
element,  or  a  mechanical  aspect,  like  the  phenomena  of  cell- 
division  which  we  are  speaking  of;  but  the  narrow  bounds  to  which 
we  are  keeping  hold  difficulties  enough*. 

Any  region  of  space  within  which  action  is  manifested  is  a  field 
of  force;  and  a  simple  example  is  a  bipolar  field,  in  which  the 
action  is  symmetrical  with  reference  to  the  fine  joining  two  points, 
or  poles,  and  with  reference  also  to  the  "equatorial"  plane  equi- 
distant from  both.  We  have  such  a  field  of  force  in  the  neigh- 
bourhood of  the  centrosome  of  the  ripe  cell  or  ovum,  when  it  is 
about  to  divide;  and  by  the  time  the  centrosome  has  divided,  the 
field  is  definitely  a  bipolar  one. 

The  quality  of  a  medium  filling  the  field  of  force  may  be  uniform, 
or  it  may  vary  from  point  to  point.  In  particular,  it  may  depend 
upon  the  magnitude  of  the  field;  and  the  quality  of  one  medium 
may  differ  from  that  of  another.  Such  variation  of  quality,  within 
one  medium,  or  from  one  medium  to  another,  is  capable  of  diagram- 
matic representation  by  a  variation  of  the  direction  or  the  strength 
of  the  field  (other  conditions  being  the  same)  from  the  state 
manifested  in  some  uniform  medium  taken  as  a  standard.  The 
medium  is  said  to  be  permeable  to  the  force,  in  greater  or  less  degree 
than  the  standard  medium,  according  as  the  variation  of  the  density 
of  the  lines  of  force  from  the  standard  case,  under  otherwise  identical 
conditions,  is  in  excess  or  defect.  A  body  placed  in  the  medium  will 
tend  to  move  towards  regions  of  greater  or  less  force  according  as  its 

*  Cf.  C.  D.  Darlington,  JieretU  Advances  in  Cytology,  1932,  and  other  well-known 
works. 


IV]  AND  STRUCTURE  OF  THE  CELL  311 

penneability  is  greater  or  less  than  that  of  the  surrounding  medium'^. 
In  the  common  experiment  of  placing  iron-filings  between  the  two 
poles  of  a  magnetic  field,  the  filings  have  a  very  high  permeability; 
and  not  only  do  they  themselves  become  polarised  so  as  to  attract 
one  another,  but  they  tend  to  be  attracted  from  the  weaker  to  the 
stronger  parts  of  the  field,  and  as  we  have  seen,  they  would  soon 
gather  together  around  the  nearest  pole  were  it  not  for  friction 
or  some  other  resistance.  But  if  we  "repeat  the  same  experiment 
with  such  a  metal  as  bismuth,  which  is  very  little  permeable  to  the 
magnetic  force,  then  the  conditions  are  reversed,  and  the  particles, 
being  repelled  from  the  stronger  to  the  weaker  parts  of  the  field, 
tend  to  take  up  their  position  as  far  from  the  poles  as  possible. 
The  particles  have  become  polarised,  but  in  a  sense  opposite  to  that 
of  the  surrounding,  or  adjacent,  field. 

Now,  in  the  field  of  force  whose  opposite  poles  are  marked  by 
the  centrosomes,  we  may  imagine  the  nucleus  to  act  as  a  more  or 
less  permeable  body,  as  a  body  more  permeable  than  the  surrounding 
medium,  that  is  to  say  the  "  cytoplasm  "  of  the  cell.  It  is  accordingly 
attracted  by,  and  drawn  into,  the  field  of  force,  and  tries,  as  it 
were,  to  set  itself  between  the  poles  and  as  far  as  possible  from  both 
of  them.  In  other  words,'  the  centrosome-foci  will  be  apparently 
drawn  over  its  surface,  until  the  nucleus  as  a  whole  is  involved 
within  the  field  of  force  which  is  visibly  marked  out  by  the  "spindle" 
(Fig.  90  b). 

If  the  field  of  force  be  electrical,  or  act  in  a  fashion  analogous 
to  an  electrical  field,  the  charged  nucleus  will  have  its  surface- 
tensions  diminished  f:  with  the  double  result  that  the  inner  alveolar 
mesh  work  will  be  broken  up  (par.  1),  and  that  the  spherical 
boundary  of  tKe  whole  nucleus  will  disappear  (par.  2).  The  break- 
up of  the  alveoli  (by  thinning  and  rupture  of  their  partition  walls) 


*  If  the  word  penneability  be  deemed  too  directly  suggestive  of  the  phenomena 
of  magnetism,  we  may  replace  it  by  the  more  general  term  of  specific  iyidiictive 
capacity.  This  would  cover  the  particular  case,  which  is  by  no  means  an  improbable 
one,  of  our  phenomena  being  due  to  a  "surface  charge"  borne  by  the  nucleus 
itself  and  also  by  the  chromosomes:  this  surface  charge  being  in  turn  the  result 
of  a  difference  in  inductive  capacity  between  the  body  or  particle  and  its  surrounding 
medium. 

t  On  the  effect  of  electrical  influences  in  altering  the  surface-tensions  of  the 
colloid  particles,  see  Bredig,  Anorganische  Fermente,  pp.  15,  16,  1901. 


312  ON  THE  INTERNAL  FORM  [ch. 

leads  to  the  formation  of  a  net,  and  the  further  break-up  of  the  net 
may  lead  to  the  unravelling  of  a  thread  or  "spireme". 

Here  there  comes  into  play  a  fundamental  principle  which,  in 
so  far  as  we  require  to  understand  it,  can  be  explained  in  simple 
words.  The  eifect  (and  we  might  even  say  the  object)  of  drawing 
the  more  permeable  body  in  between  the  poles  is  to  obtain  an 
"easier  path"  by  which  the  Hues  of  force  may  travel;  but  it  is 
obvious  that  a  longer  route  through  the  more  permeable  body  may 
at  length  be  found  less  advantageous  than  a  shorter  route  through 
the  less  permeable  medium.  That  is  to  say,  the  more  permeable 
body  will  only  tend  to  be  drawn  into  the  field  of  force  until  a  point 
is  reached  where  (so  to  speak)  the  way  round  and  the  way  through 
are  equally  advantageous.  We  should  accordingly  expect  that  (on 
our  hjrpothesis)  there  would  be  found  cases  in  which  the  nucleus 
was  wholly,  and  others  in  which  it  was  only  partially,  and  in  greater 
or  less  degree,  drawn  in  to  the  field  between  the  centrosomes.  This 
is  precisely  what  is  found  to  occur  in  actual  fact.  Figs.  90  a  and  b 
represent  two  so-called  "types,"  of  a  phase  which  follows  that 
represented  in  Fig.  89.  According  to  the  usual  descriptions  we  are 
told  that,  in  such  a  case  as  Fig.  90b,  the  "primary  spindle" 
disappears*  and  the  centrosomes  diverge  to  opposite  poles  of  the 
nucleus;  such  a  condition  being  found  in  many  plant-cells,  and  in 
the  cleavage-stages  of  many  eggs.  In  Fig.  90  a,  on  the  other  hand, 
the  primary  spindle  persists,  and  subsequently  comes  to  form  the 
main  or  "central"  spindle;  while  at  the  same  time  we  see  the 
fading  away  of  the  nuclear  membrane,  the  breaking  up  of  the 
spireme  into  separate  chromosomes,  and  an  ingrowth  into  the  nu- 
clear area  of  the  "astral  rays" — all  as  in  Fig.  91  a,  which  represents 
the  next  succeeding  phase  of  Fig.  90  b.  This  condition,  of  Fig.  91  a, 
occurs  in  a  variety  of  cases;  it  is  well  seen  in  the  epidermal  cells 
of  the  salamander,  and  is  also  on  the  whole  characteristic  of  the 
mode  of  formation  of  the  "polar  bodies t."  It  is  clear  and  obv;ous 
that  the  two  "types"  correspond  to  mere  differences  of  degree, 

*  The  spindle  is  potentially  there,  even  though  (as  Chambers  assures  us)  it  only 
becomes  visible  after  post-mortem  coagulation.  It  is  also  said  to  become  visible 
under  crossed  nicols:  W.  J.  Schmidt,  Biodynamica,  xxii,  1936. 

t  These  v/ere  first  observed  in  the  egg  of  a  pond-snail  (Limnaea)  by  B.  Dumortier, 
Mim.  sur  Vemhryoginie  des  mollusques,  Bruxelles,  1837. 


IV]  AND  STRUCTURE  OF  THE  CELL  313 

and  are  such  as  would  naturally  be  brought  about  by  differences 
in  the  relative  permeabilities  of  the  nuclear  mass  and  of  the 
surrounding  cytoplasm,  or  even  by  differences  in  the  magnitude  of 
the  former  body. 

But  now  an  important  change  takes  place,  or  rather  an  important 
difference  appears;  for,  whereas  the  nucleus  as  a  whole  tended  to 
be  drawn  in  to  the  stronger  parts  of  the  field,  when  it  comes  to  break 
up  we  find,  on  the  contrary,  that  its  contained  spireme-thread  or 
separate  chromosomes  tend  to  be  repelled  to  the  weaker  parts. 
Whatever  this  difference  may  be  due  to — whether,  for  instance,  to 
actual  differences  of  permeability,  or  possibly  to  differences  in 
"surface-charge"  or  to  other  causes — the  fact  is  that  the  chromatin 
substance  now  behaves  after  the  fashion  of  a  "diamagnetic''  body, 
and  is  repelled  from  the  stronger  to  the  weaker  parts  of  the  field. 
In  other  words,  its  particles,  lying  in  the  inter-polar  field,  tend  to 
travel  towards  the  equatorial  plane  thereof  (Figs.  91,  92),  and 
further  tend  to  move  outwards  towards  the  periphery  of  that  plane, 
towards  what  the  histologist  calls  the  "mantle-fibres,"  or  outermost 
of  the  lines  of  force  of  which  the  spindle  is  made  up  (par.  5,  Fig.  91  b). 
And  if  this  comparatively  non-permeable  chromatin  substance  come 
to  consist  of  separate  portions,  more  or  less  elongated  in  form, 
these  portions,  or  separate  "chromosomes,"  will  adjust  themselves 
longitudinally,  in  a  peripheral  equatorial  circle  (Figs.  92  a,  b).  This 
is  precisely  what  actually  takes  place.  Moreover,  before  the  breaking 
up  of  the  nucleus,  long  before  the  chromatin  material  has  broken 
up  into  separate  chromosomes,  and  at  the  v^ry  time  when  it  is 
being  fashioned  into  a  "spireme,"  this  body  already  lies  in  a  polar 
field,  and  must  already  have  a  tendency  to  set  itself  in  the  equatorial 
plane  thereof.  But  the  long,  continuous  spireme  thread  is  unable, 
so  long  as  the  nucleus  retains  its  spherical  boundary  wall,  to  adjust 
itself  in  a  simple  equatorial  annulus;  in  striving  to  do  so,  it  must 
tend  to  coil  and  "kink"  itself,  and  in  so  doing  (if  all  this  be  so), 
it  must  t^nd  to  assume  the  characteristic  convolutions  of  the 
"spireme." 

After  the  spireme  has  broken  up  into  separate  chromosomes, 
these  bodies  come  to  rest  in  the  equatorial  plane,  somewhere  near 
its  periphery ;  and  here  they  tend  to  set  themselves  in  a  symmetrical 
arrangement  (Fig.  94),  such  as  makes  for  still  better  equihbrium. 


314 


ON  THE  INTERNAL  FORM 


[CH. 


The  particles  ir^y  be  rounded  or  linear,  straight  or  bent,  sometimes 
annular;  they  may  be  all  alike,  or  one  or  more  may  differ  from 
the  rest.  Lying  as  they  do  in  a  semi-fluid  medium,  and  subject 
(doubtless)  to  some  symmetrical  play  of  forces,  it  is  not  to  be 
wondered  at  that  they  arrange  themselves  in  a  symmetrical  con- 
figuration; and  the  field  of  force  seems  simple  enough  to  let  us 
predict,  to  some  extent,  the  symmetries  open  to  them.  We  do  not 
know,  we  cannot  safely  surmise,  the  nature  of  the  forces  involved. 
In  discussing  Brauer's  observations  on  the  sphtting  of  the  chromatic 
filament,  and  on  the  symmetrical  arrangement  of  the  separate 
granules,  in  Ascaris  megalocephala,  LiUie*  remarks:  "This  behaviour 


Fig.  94.     Chromosomes,  undergoing  splitting  and  separation. 
After  Hatsehek  and  Flemming,  diagrammatised. 


is  strongly  suggestive  of  the  division  of  a  colloidal  particle  under 
the  influence  of  its  surface  electrical  charge,  and  of  the  effects  of 
mutual  repulsion  in  keeping  the  products  of  division  apart."  It  is 
probable  that  surface-tensions  between  the  particles  and  the  sur- 
rounding protoplasm  would  bring  about  an  identical  result,  and 
would  sufficiently  account  for  the  obvious,  and  at  first  sight  very 
curious  symmetry.  If  we  float  a  couple  of  matches  in  water,  we 
know  that  they  tend  to  approach  one  another  till  they  He  close 
together,  side  by  side;  and  if  we  lay  upon  a  smooth  wet  plate 
four  matches,  half  broken  across,  a  similar  attraction  brings  the 
four  matches  together  in  the  form  of  a  symmetrical  cross.  Whether 
one  of  these,  or  yet  another,  be  the  explanation  of  the  phenomenon, 


*  R.  S.  Lillie,  Conditions  determining  the  disposition  of  the  chromatic  filaments, 
etc.,  in  mitosis;   Biol.  Bulletin,  viii,  1905. 


IV]  AND  STRUCTURE  OF  THE  CELL  315 

it  is  at  least  plain  that  by  some  physical  cause,  some  mutual 
attraotion  or  common  repulsion  of  the  particles,  we  must  seek  to 
account  for  the  symmetry  of  the  so-called  "tetrads,"  and  other 
more  or  less  familiar  configurations.  The  remarkable  annular 
chromosomes,  shewn  in  Fig.  95,  can  be  closely  imitated  by  loops 
of  thread  upon  a  soapy  film,  when  the  film  within  the  annulus  is 
broken  or  its  tension  reduced ;  the  balance  of  forces  is  here  a  simple 
one,  between  the  uniform  capillary  tension  which  tends  to  widen  out 
the  ring  and  the  uniform  cohesion  of  its  particles  which  keeps  it 
together. 

We  may  find  other  cases,  at  once  simpler  and  more  varied,  where 
the  chromosomes  are  bodies  of  rounded  form  and  more  or  less 


Fig.  95.     Annular  chromosomes,  formed  in  the  spermatogenesis  of  the 
mole-cric'iict.     From  Wilson,  after  Vom  Rath. 

uniform  size.  These  also  find  their  way  to.  an  equatorial  plate; 
we  gather  (and  Lamb  assures  us)  that  they  are  repelled  from  the 
centrosomes.  They  may  go  near  the  equatorial  periphery,  but  they 
are  not  driven  there;  and  we  infer  that  some  bond  of  mutual 
attraction  holds  them  together.  If  they  be  free  to  move  in  a  fluid 
medium,  subject  both  to  some  common  repulsion  and  some  mutual 
attraction,  then  their  circumstances  are  much  like  those  of  Mayer's 
well-known  experiment  of  the  floating  magnets.  A  number  of 
magnetised  needles  stuck  in  corks,  all  with  like  poles  upwards,  are 
set  afloat  in  a  basin;  they  repel  one  another,  and  scatter  away  to 
the  sides.  But  bring  a  strong  magnet  (of  unlike  pole)  overhead, 
and  the  little  magnets  gather  in  under  its  common  attraction,  while 
still  keeping  asunder  through  their  own  mutual  repulsion.  The 
symmetry  of  forces  leads  to  a  symmetrical  configuration,  which  is 


316  ON  THE  INTERNAL  FORM  [ch. 

the  mathematical  expression  of  a  physical  equiUbrium — and  is  the 
not  too  remote  counterpart  of  the  arrangement  of  the  electrons  in 
an  atom.  Be  that  as  it  may,  it  is  found  that  a  group  of  three, 
four  or  five  Httle  magnets  arrange  themselves  at  the  corners  of  an 
equilateral  triangle,  square  or  pentagon;  but  a  sixth  passes  within 
the  ring,  and  comes  to  rest  in  the  centre  of  symmetry  of  the 
pentagon.  If  there  be  seven  magnets,  six  form  the  ring,  and  the 
seventh  occupies  the  centre ;  if  there  be  ten,  there  is  a  ring  of  eight 
and  two  within  it ;   and  so  on,  as  follows  * : 


Number  of  magnets 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

Do.  in  outer  ring 
Do.  in  inner  ring 

5 
0 

5 

1 

6 

1 

7 
1 

8 

1 

8 
2 

8 
3 

9 
3 

10 
3 

10 

4 

10 
5 

11 
5 

When  we  choose  from  the  published  figures  cases  where  the 
chromosomes  are  as  nearly  as  possible  alike  in  size  and  form — the 
condition  necessary  for  our  parallel  to  hold — then,  as  LilHe  pre- 
dicted and  as  Doncaster  and  Graham  Cannon  have  shewn,  their 
congruent  arrangement  agrees,  even  to  a  surprising  degree,  with 
what  we  are  led  to  expect  by  theory  and  analogy  (Fig.  96). 

The  break-up  of  the  nucleus,  already  referred  to  and  ascribed 
to  a  diminution  of  its  surface-tension,  is  accompanied  by  certain 
diffusion  phenomena  which  are  sometimes  visible  to  the  eye;  and 
we  are  reminded  of  Lord  Kelvin's  view  that  diffusion  is  implicitly 
associated  with  surface-tension  changes,  of  which  the  first  step  is 
a  minute  puckering  of  the  surface-skin,  a  sort  of  interdigitation  with 
the  surrounding  medium.  For  instance,  Schewiakoff  has  observed 
in  Euglyphaf  that,  just  before  the  break-up  of  the  nucleus,  a  system 
of  rays  appears,  concentred  about  it,  but  having  nothing  to  do  with 
the  polar  asters:  and  during  the  existence  of  this  striation  the 
nucleus  enlarges  very  considerably,  evidently  by  imbibition  of  fluid 
from  the  surrounding  protoplasm.  In  short,  diffusion  is  at  work, 
hand  in  hand  with,  and  as  it  were  in  opposition  to,  the  surface- 
tensions  which  define  the  nucleus.  By  diffusion,  hand  in  hand  with 
surface-tension,  the  alveoli  of  the  nuclear  meshwork  are  formed, 
enlarged  and  finally  ruptured:   diffusion  sets  up   the   movements 

*  H.  Graham  Cannon,  On  the  nature  of  the  centrosomal  force,  Journ.  Genetics, 
XIII,  p.  55,  1923. 

t  Schewiakoff,  Ueber  die  karyokinetische  Kerntheilung  der  Euglypha  alveolata, 
Morph.  Jahrb.  xiii,  pp.  193-258,  1888  (see  p.  216). 


IV]  AND  STRUCTURE  OF  THE  CELL  317 

which  give  rise  to  the  appearance  of  rays,  or  striae,  around  the 
nucleus:  and  through  increasing  diffusion  and  weakening  surface- 
tension  the  rounded  outHne  of  the  nucleus  finally  disappears. 

As  we  study  these  manifold  phenomena  in  the  individual  cases 
of  particular  plants  and  animals,  we  recognise  a  close  identity  of 
type  coupled  with  almost  endless  variation  of  specific  detail;    and 


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Fig.  96.  Various  numbers  of  chromosomes  in  the  equatorial  plate:    the  ring- 
diagrams  give  the  arrangements  predicted  by  theory.     From  Graham  Cannon. 

in  particular,  the  order  of  succession  in  which  certain  of  the  pheno- 
mena occur  is  variable  and  irregular.  The  precise  order  of  the 
phenomena,  the  tiiiie  of  longitudinal  and  of  transverse  fission  of 
the  chromatin  thread,  of  the  break-up  of  the  nuclear  wall,  and  so 
forth,  will  depend  upon  various  minor  contingencies  and  ''inter- 
ferences." And  it  is  worthy  of  particular  note  that  these  variations 
in  the  order  of  events  and  in  other  subordinate  details,   while 


318  ON  THE  INTERNAL  FORM  [ch. 

doubtless  attributable  to  specific  physical  conditions,  would  seem 
to  be  without  any  obvious  classificatory  meaning  or  other  biological 
significance. 

So  far  as  we  have  now  gone,  there  is  no  great  difficulty  in  pointing 
to  simple  and  familiar  examples  of  a  field  of  force  which  are 
similar,  or  comparable,  to  the  phenomena  which  we  witness  within 
the  cell.  But  among  these  latter  phenomena  there  are  others  for 
which  it  is  not  so  easy  to  suggest,  in  accordance  with  known  laws, 
a  simple  mode  of  physical  causation.  It  is  not  at  once  obvious 
how,  in  any  system  of  symmetrical  forces,  the  chromosomes,  which 
had  at  first  been  apparently  repelled  from  the  poles  towards  the 
equatorial  plane,  should  then  be  spht  asunder,  and  should  presently 
be  attracted  in  opposite  directions,  some  to  one  pole  and  some  to 
the  other.  Remembering  that  it  is  not  our  purpose  to  assert  that 
some  one  particular  mode  of  action  is  at  work,  but  merely  to  shew 
that  there  do  exist  physical  forces,  or  distributions  of  force,  which 
are  capable  of  producing  the  required  result,  I  give  the  following 
suggestive  hypothesis,  which  I  owe  to  my  colleague  Professor  W. 
Peddie. 

As  we  have  begun  by  supposing  that  the  nuclear  or  chromosomal 
matter  differs  in  permeability  from  the  medium,  that  is  to  say  the 
cytoplasm,  in  which  it  hes,  let  us  now  make  the  further  assumption 
that  its  permeabihty  is  variable,  and  depends  upon  the  strength  of 
the  field. 

In  Fig.  97,  we  have  a  field  of  force  (representing  our  cell),  con- 
sisting of  a  homogeneous  medium,  and  including  two  opposite 
poles :  lines  of  force  are  indicated  by  full  lines,  and  loci  of  constant 
magnitude  of  force  are  shewn  by  dotted  lines,  these  latter  being  what 
are  known  as  Cayley's  equipotential  curves*. 

Let  us  now  consider  a  body  whose  permeabihty  (/a)  depends  on 
the  strength  of  the  field  F.  At  two  field -strengths,  such  as  F^,  F^, 
let  the  permeability  of  the  body  be  equal  to  that  of  the  medium, 
and  let  the  curved  line  in  Fig.  98  represent  generally  its  permeabihty 
at  other  field-strengths;  and  let  the  outer  and  inner  dotted  curves 
in  Fig.  97  represent  respectively  the  loci  of  the  field-strengths  F^, 

*  Phil.  Trans,  xiv,  p.  142,  1857.  Cf.  also  F.  G.  Teixeira,  TraiU  des  Courhes, 
I,  p.  372,  Coimbra'.  1908. 


IV]  AND  STRUCTURE  OF  THE  CELL  319 

and  Fa.  The  body  if  it  be  placed  in  the  medium  within  either 
branch  of  the  inner  curve,  or  outside  the  outer  curve,  will  tend  to 
move  into  the  neighbourhood  of  the  adjacent  pole.     If  it  be  placed 


Fb 


Fig.  97. 


Fig.  98. 


in  the  region  intermediate  to  the  two  dotted  curves,  it  will  tend  to 
move  towards  regions  of  weaker  field-strength. 

The  locus  Fly  is  therefore  a  locus  of  stable  position,  towards  which 
the  body  tends  to  move ;  the  locus  F^  is  a  locus  of  unstable  position, 
from  which  it  tends  to  move.     If  the  body  were  placed  across  F^^, 


320  ON  THE  INTERNAL  FORM  [ch. 

it  might  be  torn  asunder  into  two  portions,  the  split  coinciding 
with  the  locus  F^. 

Suppose  a  number  of  such  bodies  to  be  scattered  throughout  the 
medium.  Let  at  first  the  regions  ¥„,  and  F^  be  entirely  outside  the 
space  where  the  bodies  are  situated:  and,  in  making  this  supposition 
we  may,  if  we  please,  suppose  that  the  loci  which  we  are  calhng 
Fa  and  F^,  are  meanwhile  situated  somewhat  farther  from  the  axis 
than  in  our  figure,  that  (for  instance)  F^  is  situated  where  we  have 
drawn  F^,  and  that  F^  is  still  farther  out.  The  bodies  then  tend 
towards  the  poles;  but  the  tendency  may  be  very  small  if,  in 
Fig.  98,  the  curve  and  its  intersecting  straight  hne  do  not  diverge 
very  far  from  one  another  beyond  Fa\    in  other  words,  if,  when 


Fig.  99. 

situated  in  this  region,  the  permeability  of  the  bodies  is  not  very 
much  in  excess  of  that  of  the  medium. 

Let  the  poles  now  tend  to  separate  farther  and  farther  from  one 
another,  the  strength  of  each  pole  remaining  unaltered;  in  other 
words,  let  the  centrosome-foci  recede  from  one  another,-  as  they 
actually  do,  drawing  out  the  spindle-threads  between  them.  The 
loci  Fa,  F^  will  close  in  to  nearer  relative  distances  from  the  poles. 
In  doing  so,  when  the  locus  F^  crosses  one  of  the  bodies,  the  body 
may  be  torn  asunder;  if  the  body  be  of  elongated  shape,  and  be 
crossed  at  more  points  than  one,  the  forces  at  work  will  tend  to 
exaggerate  its  foldings,  and  the  tendency  to  rupture  is  greatest 
when  Fa  is  in  some  median  position  (Fig.  99). 

When  the  locus  Fa  has  passed  entirely  over  the  body,  the  body 
tends  to  move  towards  regions  of  weaker  force;  but  when,  in  turn, 
the  locus  Fi,  has  crossed  it,  then  the  body  again  moves  towards 
regions  of  stronger  force,  that  is  to  say,  towards  the  nearest  pole. 


IV]  AND  STRUCTURE  OF  THE  CELL  321 

And,  in  thus  moving  towards  the  pole,  it  will  do  so,  as  appears 
actually  to  be  the  case  in  the  dividing  cell,  along  the  course  of  the 
outer  hues  of  force,  the  so-called  "mantle-fibres"  of  the  histologist*. 

Such  considerations  as  these  give  general  results,  easily  open  to 
modification  in  detail  by  a  change  of  any  of  the  arbitrary  postulates 
which  have  been  made  for  the  sake  of  simpHcity.  Doubtless  there 
are  other  assumptions  which  would  meet  the  case;  for  instance, 
that  during  the  active  phase  of  the  chromatin  molecule  (when  it  de- 
composes and  sets  free  nucleic  acid)  it  carries  a  charge  opposite  to 
that  which  it  bears  during  its  resting,  or  alkahne  phase ;  and  that  it 
would  accordingly  move  towards  different  poles  under  the  influetice 
of  a  current,  wandering  with  its  negative  charge  in  an  alkahne  fluid 
during  its  acid  phase  to  the  anode,  and  to  the  kathode  during  its 
alkahne  phase.  A  whole  field  of  speculation  is  opened  up  when  we 
begin  to  consider  the  cell  not  merely  as  a  polarised  electrical  field, 
but  also  as  an  electrolytic  field,  full  of  wandering  ions.  Indeed  it 
is  high  time  we  reminded  ourselves  that  we  have  perhaps  been 
deahng  too  much  with  ordinary  physical  analogies:  and  that  our 
whole  field  of  force  within  the  cell  is  of  an  order  of  magnitude  where 
these  grosser  analogies  may  fail  to  serve  us,  and  might  even  play 
us  false,  or  lead  us  astray.  But  our  sole  object  meanwhile,  as  I 
have  said  more  than  once,  is  to  demonstrate,  by  such  illustrations 
as  these,  that,  whatever  be  the  actual  and  as  yet  unknown  modus 
operandi,  there  are  physical  conditions  and  distributions  of  force 
which  could  produce  just  such  phenomena  of  movement  as  we  see 
taking  place  within  the  living  cell.  This,  and  no  more,  is  precisely 
what  Descartes  is  said  to  have  claimed  for  his  description  of  the 
human  body  as  a  "  mechanism  f." 

While  it  can  scarcely  be  too  often  repeated  that  our  enquiry  is 
not  directed  towards  the  solution  of  physiological  problems,  save 
only  in  so  far  as  they  are  inseparable  from  the  problems  presented 
by  the  visible  configurations  of  form  and  structure,  and  while  we 
try,  as  far  as  possible,  to  evade,  the  difficult  question  of  what 

*  •  We  have  not  taken  account  in  the  above  paragraphs  of  the  obvious  fact  that 
the  supposed  symmetrical  field  of  force  is  distorted  by  the  presence  in  it  of  the 
more  or  less  permeable  bodies;  nor  is  it  necessary  for  us  to  do  so,  for  to  that 
distorted  field  the  above  argument  continues  to  apply,  word  for  word. 

t  Michael  Foster,  Lectures  on  the  History  of  Physiology,  1901,  p.  62. 


322  ON  THE  INTERNAL  FORM  [ch. 

particular  forces  are  at  work  when  the  mere  visible  forms  produced 
are  such  as  to  leave  this  an  6pen  question,  yet  in  this  particular 
case  we  have  been  drawn  into  the  use  of  electrical  analogies,  and 
we  are  bound  to  justify,  if  possible,  our  resort  to  this  particular 
mode  of  physical  action.  There  is  an  important  paper  by  R.  S.  LiUie, 
on  the  "Electrical  convection  of  certain  free  cells  and  nuclei*," 
which,  while  I  cannot  quote  it  in  direct  support  of  the  suggestions 
which  I  have  made,  yet  gives  just  the  evidence  we  need  in  order 
to  shew  that  electrical  forces  act  upon  the  constituents  of  the  cell, 
and  that  their  action  discriminates  between  the  two  species  of 
colloids  represented  by  the  cytoplasm  and  the  nuclear  chromatin. 
And  the  difference  is  such  that,  in  the  presence  of  an  electrical 
current,  the  cell  substance  and  the  nuclei  (including  sperm-cells) 
tend  to  migrate,  the  former  on  the  whole  with  the  positive,  the 
latter  with  the  negative  stream :  a  difference  of  electrical  potential 
being  thus  indicated  between  the  particle  and  the  surrounding 
medium,  just  as  in  the  case  of  minute  suspended  particles  of  various 
kinds  in  various  feebly  conducting  media  j*.  And  the  electrical 
difference  is  doubtless  greatest,  in  the  case  of  the  cell  constituents, 
just  at  the  period  of  mitosis:  when  the  chromatin  is  invariably 
in  its  most  deeply  staining,  most  strongly  acid,  and  therefore, 
presumably,  in  its  most  electrically  negative  phase.  In  short,  LilHe 
comes  easily  to  the  conclusion  that  "electrical  theories  of  mitosis 
are  entitled  to  more  careful  consideration  than  they  have  hitherto 
received." 

*  Amer.  J.  Physiol,  viii,  pp.  273-283,  1903  {vide  supra,  p.  314);  cf.  ibid,  xv, 
pp.  46-84,  1905;  xxii,  p.  106,  1910;  xxvii,  p.  289,  1911;  Journ.  Exp.  Zool.  xv, 
p.  23,  1913;   etc. 

t  In  like  manner  Hardy  shewed  that  colloid  particles  migrate  with  the  negative 
stream  if  the  reaction  of  the  surrounding  fluid  be  alkahne,  and  vice  versa.  The 
whole  subject  is  much  wider  than  these  brief  allusions  suggest,  and  is  essentially 
part  of  Quincke's  theory  of  Electrical  Diffusion  or  Endosmosis:  according  to 
which  the  particles  and  the  fluid  in  which  they  float  (or  the  fluid  and  the  capillary 
wall  through  which  it  flows)  each  carry  a  charge:  there  being  a  discontinuity  of 
potential  at  the  surface  of  contact  and  hence  a  field  of  force  leading  to  powerful 
tangential  or  shearing  stresses,  communicating  to  the  particles  a  velocity  which 
varies  with  the  density  per  unit  area  of  the  surface  charge.  See  W.  B.  Hardy's 
paper  on  Coagulation  by  electricity,  Journ.  Physiol,  xxrv,  pp.  288-304,  1899; 
also  Hardy  and  H.  W.  Harvey,  Surface  electric  charges  of  living  cells,  Proc.  E.S. 
(B),  Lxxxiv,  pp.  217-226,  191 1,  and  papers  quoted  therein.  Cf.  also  E.  N.  Harvey's 
observations  on  the  convection  of  unicellular  organisms  in  an  electric  field  (Studies 
on  the  permeability  of  cells,  Journ.  Exp.  Zool.  x,  pp.  508-556,  1911). 


IV] 


AND  STRUCTURE  OF  THE  CELL 


323 


Among  other  investigations  all  leading  towards  the  same  general 
conclusion,  namely  that  differences  of  electric  potential  play  their 
part  in  the  phenomena  of  cell  division,  I  would  mention  a  note- 
worthy paper  by  Ida  H.  Hyde*,  in  which  the  writer  shews  (among 
other  important  observations)  that  not  only  is  there  a  measurable 
difference  of  potential  between  the  animal  and  vegetative  poles  of 
a  fertilised  egg  (Fundulus,  toad,  turtle,  etc.),  but  also  that  this 
difference  fluctuates,  or  actually  reverses  its  direction,  periodically, 
at  epochs  coinciding  with  successive  acts  of  segmentation  or  other 


Fig.  100.  Final  stage  in  the  first  seg- 
mentation of  the  egg  of  Cerebra- 
tidus.  From  Prenant,  after  Coe*. 


Fig.  101.     Diagram  of  field  of  force 
with  two  similar  poles. 


important  phases  in  the  development  of  the'  eggt;  just  as  other 
physical  rhythms,  for  instance,  in  the  production  of  COg ,  had  already 
been  shewn  to  do.  Hence  we  need  not  be  surprised  to  find  that  th^ 
"materialised"  hues  of  force,  which  in  the  earlier  stages  form  the 


*  On  differences  in  electrical  potential  in  developing  eggs,  Amer.  Journ.  Physiol. 
XII,  pp.  241-275,  1905.  This  paper  contains  an  excellent  summary,  for  the  time 
being,  of  physical  theories  of  the  segmentation  of  the  cell. 

t  Gray  has  demonstrated  a  temporary  increase  of  electrical  conductivity  in 
sea-urchin  eggs  during  the  process  of  fertilisation,  and  ascribes  the  changes  in 
resistance  to  polarisation  of  the  surface:  Electrical  conductivity  of  echinoderm 
eggs,  etc.,  Phil.  Trans.  (B),  ccvii,  pp.  481-529,  1916. 


324 


ON  THE  INTERNAL  FORM 


[CH. 


convergent  curves  of  the  spindle,  are  replaced  in  the  later  phases  of 
caryokinesis  by  divergent  curves,  indicating  that  the  two  foci,  which 
are  marked  out  in  the  field  by  the  divided  and  reconstituted  nuclei, 
are  now  ahke  in  their  polarity*  (Figs.  100,  101). 

The  foregoing  account  is  based  on  the  provisional  assumption 
that  the  phenomena  of  caryokinesis  are  analogous  to  those  of  a 
bipolar  electrical  field — a  comparison  which  seems  to  offer  a  helpful 
and  instructive  series  of  analogies.  But  there  are  other  forces  which 
lead  to  similar  configurations.  For  instance,  some  of  Leduc's 
diffusion-experiments  offer  very  remarkable  analogies  to  the  dia- 
grammatic  phenomena   of   caryokinesis,  as   shewn   in   Fig.   102  f. 


Fig.  102.     Artificial  caryokinesis  (after  Leduc),  for  comparison  with  Fig.  88,  p.  299. 

Here  we  have  two  identical  (not  opposite)  poles  of  osmotic  con- 
centration, formed  by  placing  a  drop  of  Indian  ink  in  salt  water, 
and  then  on  either  side  of  this  central  drop,  a  hypertonic  drop  of 
salt  solution  more  lightly  coloured.  On  either  side  the  pigment  of 
the  central  drop  has  been  drawn  towards  the  focus  nearest  to  it; 
but  in  the  middle  line,  the  pigment  is  drawn  in  opposite  directions 
by  equal  forces,  and  so  tends  to  remain  undisturbed,  in  the  form  of 
an  "equatorial  plate." 

To  account  for  the  same  mitotic  phenomena  an  elegant  hypothesis 
has  been  put  forward  by  A.  B.  Lamb  J,  and  developed  by  Graham 

*  W.  R.  Coe,  Maturation  and  fertilisation  of  the  egg  of  Cerebratulus,  Zool. 
Jahrbiicher  {Anat.  Abth.),  xii,  pp,  425^76,  1899. 

t  Op.  cit.  pp.  110  and  91. 

t  A.  B.  Lamb,  A  new  explanation  of  the  mechanism  of  mitosis,  Journ.  Exp.  Zool. 
V,  pp.  27-33.  1908. 


IV]  AND  STRUCTURE  OF  THE  CELL  325 

Cannon*.  It  depends  on  certain  investigations  of  the  Bjerknes, 
father  and  sonf,  which  prove  that  bodies  pulsating  or  oscillating  J 
in  a  fluid  set  up  a  field  of  force  precisely  comparable  with  the  lines 
of  force  in  a  magnetic  field.  Certain  old  and  even  familiar  observa- 
tions had  pointed  towards  this  phenomenon.  Guyot  had  noticed 
that  bits  of  paper  were  attracted  towards  a  vibrating  tuning-fork; 
and  Schellbach  found  that  a  sounding-board  so  acts  on  bodies  in  its 
neighbourhood  as  to  attract  those  which  are  heavier  and  repel  those 
which  are  lighter  than  the  surrounding  medium;  in  air  bits  of 
paper  are  attracted  and  a  gas-flame  is  repelled.  To  explain  these 
simple  observations,  Bjerknes  experimented  with  Httle  drums 
attached  to  an  automatic  bellows.  He  found  that  two  bodies  in 
a  fluid  field,  synchronously  pulsating  or  synchronously  oscillating, 
repel  one  another  when  their  oscillations  are  in  the  same  phase,  or 
their  pulsations  are  in  opposite  phase;  and  vice  versa:  while  other 
particles,  floating  passively  in  the  same  fluid,  tend  (as  Schellbach 
had  observed  before)  to  be  attracted  or  repulsed  according  as  they 
are  heavier  or  lighter  than  the  fluid  medium.  The  two  bodies 
behave  towards  one  another  like  two  electrified  bodies,  or  like  two 
poles  of  a  magnet;  we  are  entitled  to  speak  of  them  as  "hydro- 
dynamic  poles,"  we  might  even  call  them  " hydrodynamic  magnets" ; 
and  pursuing  the  analogy,  we  may  call  the  heavy  bodies  para- 
magnetic, and  the  light  ones  diamagnetic  with  regard  to  them. 
Lamb's  hypothesis  then,  and  Cannon's,  is  that  the  centrosomes  act 
as  "hydrodynamic  magnets."  The  explanation  depends  on  oscilla- 
tions which  have  never  been  seen,  in  centrosomes  which  are  not 
always  to  be  discovered.  But  it  brings  together  certain  curious 
analogies,  and  these,  where  we  know  so  little,  may  be  worth 
reflecting  on. 

If  we  assume  that  each  centrosome  is  endowed  with  a  vibratory 
motion  as  it  floats  in  the  semi-fluid  colloids,  or  hydrosols  (to  use 
Graham's  word)  of  the  cell,  we  ma;^  take  it  that  the  visible  intra- 
cellular  phenomena   will  be   much   the   same  as   those   we   have 

*  Op.  cit.  Cf.  also  Gertrud  Woken,  Zur  Physik  der  Kernteilung,  Z.  f.  allg,  Physiol. 
XVIII,  pp.  39-57,  1918. 

t  V.  Bjerknes,  Vorlesungen  liber  hydrodynamische  Fernkrdjte,  nach  C.  A.  Bjerknes^ 
Theorie,  Leipzig,  1900. 

J  A  body  is  said  to  pulsate  when  it  undergoes  a  rhythmic  change  of  volume; 
it  oscillates  when  it  undergoes  a  rhythmic  change  of  place. 


326  ON  THE  INTERNAL  FORM  [ch. 

described  under  an  electrical  hypothesis;  the  lines  of  force  will 
have  the  same  distribution,  and  such  movements  as  the  chromo- 
somes undergo,  and  such  symmetrical  configurations  as  they  assume, 
may  be  accounted  for  under  the  one  hypothesis  pretty  much  as 
under  the  other.  There  are  however  other  phenomena  accompanying 
mitosis,  such  as  Chambers's  astral  currents  and  certain  local  changes 
in  the  viscosity  of  the  egg,  which  are  more  easily  explained  by  the 
hydrodynamic  theory. 

We  may  assume  that  the  cytoplasm,  however  complex  it  may  be, 
is  but  a  sort  of  microscopically  homogeneous  emulsion  of  high 
dispersion,  that  is  to  say  one  in  which  the  minute  particles  of  one 
phase  are  widely  scattered  throughout,  and  freely  mobile  in,  the 
other;  and  this  indeed  is  what  is  meant  by  caUing  it  a  hydrosol. 
Let  us  assume  also  that  the  particles  are  a  little  less  dense  than  the 
continuous  phase  in  which  they  are  dispersed;  and  assume  lastly 
(it  is  not  the  easiest  of  our  assumptions)  that  these  ultra-minute 
particles  will  be  affected,  just  as  are  the  grosser  ones,  by  the  forces 
of  the  hydrodynamic  field. 

All  this  being  so,  the  disperse  particles  will  be  repelled  from  the 
oscillating  centrosome,  with  a  force  which  falls  off  very  rapidly,  for 
Bjerknes  tells  us  that  it  varies  inversely  as  the  seventh  power  of 
the  distance;  a  round  clear  field,  hke  a  drop  or  a  bubble,  will  be 
formed  round  the  centrosome ;  and  the  disperse  particles,  expelled 
from  this  region,  will  tend  to  accumulate  in  a  crowded  spherica- 
zone  immediately  beyond  it.  Outside  of  this  again  they  will  con- 
tinue to  be  repulsed,  but  slowly,  and  we  may  expect  a  second  and 
lesser  concentration  at  the  periphery  of  the  cell.  A  clear  central 
mass,  or  "centrosphere,"  will  thus  come  into  being;  and  the 
surrounding  cytoplasm  will  be  rendered  denser  and  more  viscous, 
especially  close  around  the  centrosphere  and  again  peripherally,  by 
condensation  of  the  disperse  particles.  Moreover,  all  outward 
movements  of  these  lighter  particles  entail  inward  movements  of 
the  heavier,  which  (by  hypothesis)  are  also  the  more  fluid ;  stream- 
lines or  visible  currents  will  flow  towards  the  centre,  giving  rise  to 
the  star-shaped  "aster,"  and  the  best  accounts  of  the  sea-urchin's 
egg*  tally  well  with  what  is  thus  deduced  from  the  hydrodynamic 

*  Cf.  R.  Chambers,  in  Journ.  Exp.  Zool.  xxni,  p.  483,  1917;  Trans.  R.S.  Canada, 
XII,  1918;   Journ.  Gen.  Physiol,  ii,  1919. 


IV]  AND  STRUCTURE  OF  THE  CELL  327 

hypothesis.  The  round  drop  of  clear  fluid  which  forms  the  centre 
of  the  aster  grows  as  the  aster  grows,  fluid  streaming  towards  it 
from  all  parts  of  the  cell  along  the  channels  of  the  astral  rays. 
The  cytoplasni  between  the  rays  is  in  the  gel  state,  but  gradually 
passes  into  a  sol  beyond  the  confines  of  the  aster.  Seifritz  asserts 
that  the  substance  of  the  centrosphere  is  *'not  much  more  viscous 
than  water,"  but  that  the  wedges  of  cytoplasm  between  the  inwardly 
directed  streams  are  stiff  and  viscous*. 

After  the  centrosome  divides  we  have  two  oscillating  bodies 
instead  of  one;  they  tend  to  repel  one  another,  and  pass  easily 
through  the  fluid  centrosphere  to  the  denser  layer  around.  But 
now  the  new  centrosomes,  on  opposite  sides  of  the  centrosphere, 
repel,  each  on  its  own  side,  the  disperse  particles  of  the  denser  zone ; 
and  two  new  asters  are  formed,  their  rays  marked  by  the  streams 
coursing  inwards  to  the  centrosome-foci.  Thus  the  amphiaster 
comes  into  being;  it  is  not  that  the  old  aster  divides,  as  a  definite 
entity;  but  the  old  aster  ceases  to  exist  when  its  focus  is  disturbed, 
and  about  the  new  foci  new  asters  are  necessarily  and  automatically 
developed.  Again  this  hypothetic  account  taUies  well  with  Chambers's 
description. 

The  same  attractions  and  repulsions  should  be  manifested,  perhaps 
better  still,  in  whatsoever  bodies  he  or  float  within  the  cell,  whether 
liquid  or  solid,  oil-globules,  yolk-particlea,  mitochondria,  chromo- 
somes or  what  not.  A  zoned,  concentric  arrangement  of  yolk- 
globules  is  often  seen  in  the  egg,  with  the  centrosome  as  focus; 
and  in  certain  sea-urchin  eggs  the  mitochondria  gather  around 
the  centrosome  while  the  amphiaster  is  forming,  collecting  together 
in  that  very  zone  to  which  Chambers  ascribes  a  semi-rigid  or  viscous 
consistency!.  The  Golgi  bodies  found  in  various  germ-cells  are  at 
first  black  rod-hke  bodies  embedded  in  the  centrosphere ;  they 
undergo  changes  and  complex  movements,  now  scattering  through 
the  cytoplasm  and  anon  crowding  again  around  the  centrosome. 
Some  periodic  change  in  the  density  of  these  bodies  compared  with 

♦  Cf.  W.  Seifritz,  Some  physical  properties  of  protoplasm,  Ann.  Bot.  xxxv, 
1921.  Wo.  Ostwald  and  M.  H.  Fischer  had  thought  that  the  astral  rays  were 
due  to  local  changes  of  the  plasma-sol  into  a  gel,  Zur  physikal.  chem.  Theorie  der 
Befruchtung,  Pfluger's  Archiv,  cvi,  pp.  2^3-266,  1905. 

t  Cf.  F.  Vejdovsky  and  A.  Mrazek,  Umbildung  des  Cytoplasma  wahrend  der 
Befruchtung  und  Zelltheilung,  Arch.  f.  mikr.  Anat.  LXii.  431-579,  1903. 


328  ON  THE  INTERNAL  FORM    .  [ch. 

that  of  the  medium  in  which  they  He  seems  all  that  is  required 
to  account  for  their  excursions;  and  such  changes  of  density  are 
not  only  of  likely  occurrence  during  the  active  chemical  operations 
associated  with  fertilisation  and  division,  but  are  in  all  probability 
inseparable  from  the  changes  in  viscosity  which  are  known  to 
occur*.  The  movements  and  arrangements  of  the  chromosomes, 
already  described,  may  be  easily  accounted  for  if  we  postulate,  in 
addition  to  their  repulsion  from  the  oscillating  centrosomes,  induced 
oscillations  in  themselves  such  as  to  cause  them  to  attract  one  another. 

The  well-defined  length  of  the  spindle  and  the  position  of  equili- 
brium in  which  it  comes  to  rest  may  be  conceived  as  resultants  of 
the  several  mutual  repulsions  of  the  centrosomes  by  one  another, 
by  the  chromosomes  or  other  lighter  material  of  the  equatorial  plate, 
and  again  by  such  lighter  material  as  may  have  accumulated  at  the 
periphery  of  the  egg ;  the  first  two  of  these  will  tend  to  lengthen  the 
spindle,  the  last  to  shorten  it;  and  the  last  will  especially  affect  its 
position  and  direction.  When  Chambers  amputated  part  of  an 
amphiastral  egg,  the  remains  of  the  amphiaster  disappeared,  and 
then  came  into  being  again  in  a  new  and  more  symmetrical  position ; 
it  or  its  centrosomal  focus  had  been  symmetrically  repelled,  we  may 
suppose,  by  the  fresh  surface.  Hertwig's  law  that  the  spindle-axis 
tends  to  lie  in  the  direction  of  the  largest  mass  of  protoplasm,  in 
other  words  to  point  where  the  cell-surface  lies  farthest  off  and  its 
repulsion  is  least  felt,  may  likewise  find  its  easy  explanation. 

Between  these  hypotheses  we  may  choose  one  or  other  (if  we 
choose  at  all),  according  to  our  judgment.  As  Henri  Poincare  tells 
us,  we  never  know  that  any  one  physical  hypothesis  is  true,  we  take 
the  simplest  we  can  find;  and  this  we  call  the  guiding  principle  of 
simphcity !  In  this  case,  the  hydrodynamic  hypothesis  is  a  simple 
one;  but  it  all  rests  on  a  hypothetic  oscillation  of  the  centrosomes, 
which  has  never  been  witnessed.  Bayliss  has  shewn  that  precisely 
such  reversible  states  of  gelation  as  we  have  been  speaking  of  as 

*  Cf.  G.  Odquist,  Viscositatsanderungen  des  Zellplasmas  wahrend  der  ersten 
Entwicklungsstufen  des  Froscheies,  Arch.  f.  Entw.  Mech.  ia,  pp.  610-624,  1922; 
A.  Gurwitsch,  Pramissen  und  anstossgebende  Faktoren  der  Furchung  und 
Zelltheilung,  Arch.  f.  Zellforsch.  n,  pp.  495-548,  1909;  L.  V.  Heilbrunn, 
Protoplasmic  viscosity-changes  during  mitosis,  Journ.  Exp.  Zool.  xxxiv,  pp.  417-447, 
1921 ;  ibid,  xliv,  pp.  255-278,  1926;  E.  Leblond,  Passage  de  I'etat  de  gel  a  I'etat  de  sol 
dans  le  protoplasme  vivant,  C.R.  Soc.  Biol,  lxxxii,  p.  1150;   of.  ibid.  p.  1220;  etc. 


rv]  AND  STRUCTURE  OF  THE  CELL  329 

"periodic  changes  in  viscosity"  may  be  induced  in  living  protoplasm 
by  electrical  stimulation*.  On  the  other  hand,  the  fact  that  the 
hydrodynamic  forces  fall  off  as  fast  as  they  do  with  increasing 
distance  Hmits  their  efficacy ;  and  the  minute  disperse  particles 
must,  under  Stokes's  law,  be  slow  to  move.  Lastly,  it  may  well  be 
(as  Lillie  has  urged)  that  such  work  as  his  own,  or  Ida  Hyde's,  or 
Gray's,  on  change  of  potential  in  developing  eggs,  taken  together 
with  that  of  many  others  on  the  behaviour  of  colloid  particles  in  an 
electrical  field,  has  not  yet  been  followed  out  in  all  its  consequences, 
either  on  the  physical  or  the  physiological  side  of  the  problem. 

But  to  return  to  our  general  discussion. 

As  regards  the  actual  mechanical  division  of  the  cell  into  two 
halves,  we  shall  see  presently  that,  in  certain  cases,  such  as  that 
of  a  long  cylindrical  filament,  surface-tension,  and  what  is  known 
as  the  principle  of  "minimal  areas,"  go  a  long  way  to  explain  the 
mechanical  process  of  division;  and  in  all  cells  whatsoever,  the 
process  of  division  must  somehow  be  explained  as  the  result  of  a 
conflict  between  surface-tension  and  its  opposing  forces.  But  in 
such  a  case  as  our  spherical  cell,  it  is  none  too  easy  to  see  what 
physical  cause  is  at  work  to  disturb  its  equiUbrium  and  its  integrity. 

The  fact  that  when  actual  division  of  the  cell  takes  place,  it  does 
so  at  right  angles  to  the  polar  axis  and  precisely  in  the  direction 
of  the  equatorial  plane,  would  lead  us  to  suspect  that  the  new 
surface  formed  in  the  equatorial  plane  sets  up  an  annular  tension, 
directed  inwards,  where  it  meets  the  outer  surface  layer  of  the  cell 
itself.  But  at  this  point  the  problem  becomes  more  comphcated. 
Before  we  can  hope  to  comprehend  it,  we  shall  have  not  only  to 
enquire  into  the  potential  distribution  at  the  surface  of  the  cell  in 
relation  to  that  which  we  have  seen  to  exist  in  its  interior,  but  also 
to  take  account  of  the  differences  of  potential  which  the  material 
arrangements  along  the  lines  of  force  must  themselves  tend  to 
produce.  Only  thus  can  we  approach  a  comprehension  of  the 
balance  of  forces  which  cohesion,  friction,  capillarity  and  electrical 
distribution  combine  to  set  up. 

The  manner  in  which  we  regard  the  phenomenon  would  seem  to 

*  W.  M.  Bayliss,  Reversible  gelation  in  living  protoplasm,  Proc.  R.S.  (B),  xci, 
pp.  196-201,  1920. 


PO  ON  THE  INTERNAL  FORM  [ch. 

turn,  in  great  measure,  upon  whether  or  no  we  are  justified  in 
assuming  that,  in  the  hquid  surface-film  of  a  minute  spherical  cell, 
local  and  symmetrically  localised  differences  of  surface-tension  are 
likely  to  occur.  If  not,  then  changes  in  the  conformation  of  the 
cell  such  as  lead  immediately  to  its  division  must  be  ascribed  not 
to  local  changes  in  its  surface-tension,  but  rather  to  direct  changes 
in  internal  pressure,  or  to  mechanical  forces  due  to  an  induced 
surface-distribution  of  electrical  potential.  We  have  little  reason  to 
be  sceptical ;  in  fact  we  now  know  that  the  cell  is  so  far  from  being 
chemically  and  physically  homogeneous  that  local  variations  in  its 
surface-tension  are  more  than  likely,  they  are  certain  to  occur. 

Biitschh  suggested  more  than  sixty  years  ago  that  cell-division 
was  brought  about  by  an  increase  of  surface-tension  in  the  equatorial 
region  of  the  cell ;  and  the  suggestion  was  the  more  remarkable  that 
it  was  (I  beheve)  the  very  first  attempt  to  invoke  surface-tension 
as  a  factor  in  the  physical  causation  of  a  biological  phenomenon*. 
An  increase  of  equatorial  tension  would  cause  the  surface-area  there 
to  diminish,  and  the  equator  to  be  pinched  in,  but  the  total  surface- 
area  of  the  cell  would  be  increased  thereby,  and  the  two  effects 
would  strike  a  balance  f.  But,  as  Biitschh  knew  very  well,  the 
surface-tension  change  would  not  stand  alone;  it  would  bring  other 
phenomena  in  its  train,  currents  would  tend  to  be  set  up,  and 
tangential  strains  would  be  imposed  on  the  cell-membrane  or  cell- 
surface  as  a  whole.  The  secondary  if  not  the  direct  effects  of 
increased  equatorial  tension  might,  after  all,  suffice  for  the  division  of 
the  cell.  It  was  Loeb,  in  1895,  who  first  shewed  that  streaming  went 
on  from  the  equator  towards  the  divided  nuclei.  To  the  violence 
of  these  streaming  movements  he  attributed  the  phenomenon  of 
division,  and  many  other  physiologists  have  adopted  this  hypo- 
thesis J.     The  currents  of  which  Loeb  spoke  call  for  counter-currents 

*  0.  Butschli,  tJber  die  ersten  Entwicklungsvorgange  der  Eizelle,  Abh. 
Senckenberg.  naturf.  Gesellsch.  x,  1876;  Uber  Plasmastrqmungen  bei  der  Zell- 
theilung,  Arch.  f.  Entw.  Mech.  x,  p.  52,  1900.  Ryder  ascribed  the  earyokinetic 
figures  to  surface-tension  in  his  Dynamics  in  Evolution,  1894. 

t  A  relative,  not  positive,  increase  of  surface-tension,  was  part  of  Giardina's 
hypothesis:   Note  sul  mecanismo  della  divisione  cellulare,  Anat.  Anz.  xxi.  1902. 

J  J.  Loeb,  Amer.  Journ.  Physiol,  vi,  p.  432,  1902;  E.  G.  Conklin,  Protoplasmic 
movements  as  a  factor  in  differentiation,  Wood's  Hole  Biol.  Lectures,  p.  69,  etc., 
1898-99;  J.  Spek,  Oberflachenspannungsdifferenzen  als  eine  Ursache  der  Zell- 
teilung,  Arch.f.  Entw.  Mech.  xliv,  pp.  54-73,  1918. 


IV]  AND  STRUCTURE  OF  THE  CELL  331 

towards  the  equator,  in  or  near  the  surface  of  the  cell;  and  theory 
and  observation  both  indicate  that  precisely  such  currents  are  bound 
to  be  set  up  by  the  surface-energy  involved  in  the  increase  of 
equatorial  tension. 

An  opposite  view  has  been  held  by  some,  and  especially  by 
T.  B.  Robertson*.  Quincke  had  shewn  that  the  formation  of  soap 
at  the  surface  of  an  oil-droplet  lowers  the  surface-tension  of  the 
latter,  and  that  if  the  saponification  be  local,  that  part  of  the  surface 
tends  to  enlarge  and  spread  out  accordingly.  Robertson,  in  a  very 
curious  experiment,  found  that  by  laying  a  thread,  moistened  with 
dilute  caustic  alkali  or  merely  smeared  with  soap,  across  a  drop  of 
olive  oil  afloat  in  water,  the  drop  at  once  divided  into  two.  A 
vast  amount  of  controversy  has  arisen  over  this  experiment,  but 
Spek  seems  to  have  shewn  conclusively  that  it  is  an  exceptional 
case. 

In  a  drop  of  olive-oil,  balanced  in  water f  and  touched  anywhere 
with  an  alkaH,  there  is  so  copious  a  formation  of  lighter  soaps  that 
di^erences  of  density  tend  to  drag  the  drop  in  two.  But  in  the 
case  of  other  oils  (and  especially  the  thinner  oils,  such  as  oil  of 
bergamot)  the  saponified  portion  bulges,  as  theory  directs;  and 
when  the  alkali  is  applied  to  two  opposite  poles  the  equatorial 
region  is  pinched  in,  as  McClendonJ,  in  opposition  to  Robertson, 
had  found  it  to  do.  Conversely,  if  an  alkaline  thread  be  looped 
around  the  drop,  the  zone  of  contact  bulges,  and  instead  of  dividing 
at  the  equator  the  drop  assumes  a  lens-like  form. 

We  may  take  it  then  as  proven  that  a  relative  increase  of  equatorial 
surface-tension,  whether  in  oil-drops,  mercury-globules  or  living 
cells,  does  lead,  or  tend  to  lead,  to  an  equatorial  constriction.  In 
all  cases  a  system  of  surface-currents  is  set  up  among  the  fluid  drops 
towards  the  zone  of  increased  tension ;  and  an  axial  counter-current 
flows  towards  the  pole  or  poles  of  lowered  tension.  Precisely  such 
currents  have  been  observed  to  run  in  various  eggs  (especially  of 

♦  T.  B.  Robertson,  Note  on  the  chemical  mechanics  of  cell-division,  Arch.  f. 
Entw.  Mech.  xxvn,  p.  29,  1909;  xxxii,  p.  308,  1911;  xxxv,  p.  402,  1913.  Cf. 
R.  S.  Lillie,  Joum.-Exp.  Zool.  xxi,  pp.  369^02,  1916;   McClendon,  loc.  cit.;   etc. 

t  In  these  experiments,  and  in  many  of  Quincke's,  a  little  chloroform  is  added 
to  the  oil,  in  order  to  bring  its  density  as  near  as  may  be  to  that  of  water. 

X  J.  F.  McClendon,  Note  on  the  mechanics  of  cell -division,  Arch.  f.  Entw.  Mech. 
xxxrv,  pp.  263-266,  1912. 


332  ON  THE  INTERNAL  FORM  [ch. 

certain  Nematodes)  during  division  of  the  cell;  but  if  the  })rocess 
be  slow,  more  than  7  or  8  minutes  long,  the  slow  currents  become 
hard  to  see.  Various  contents  of  the  cell  are  transported  by  these 
currents,  and  clear,  yolk-free  polar  caps  and  equatorial  accumula- 
tions of  yolk  and  pigment  are  among  the  various  manifestations  of 
the  phenomenon.  The  extrusion  of  a  polar  body,  at  a  small  and 
sharply  defined  region  of  lowered  tension,  is  a  particular  case  of  the 
same  principle*. 

But  purely  chemical  changes  are  not  of  necessity  the  fundamental 
cause  of  alteration  in  the  surface-tension  of  the  egg,  for  the  action 
of  electrolytes  on  surface-tension  is  now  well  known  and  easily 
demonstrated.  So,  according  to  other  views  than  those  with  which 
we  have  been  dealing,  electrical  charges  are  sufficient  in  themselves 
to  account  for  alterations  of  surface-tension,  and  in  turn  for  that 
protoplasmic  streaming  which,  as  so  many  investigators  agree, 
initiates  the  segmentation  of  the  eggf.  A  great  part  of  our  difficulty 
arises  from  the  fact  that  in  such  a  case  as  this  the  various  pheno- 
mena are  so  entangled  and  apparently  concurrent  that  it  is  hard 
to  say  which  initiates  another,  and  to  which  this  or  that  secondary 
phenomenon  may  be  considered  due.  Of  recent  years  the  pheno- 
menon of  adsorption  has  been  adduced  (as  we  have  already  briefly 
said)  in  order  to  account  for  many  of  the  events  and  appearances 
which  are  associated  with  the  asymmetry,  and  lead  towards  the 
division,  of  the  cell.  But  our  short  discussion  of  this  phenomenon 
may  be  reserved  for  another  chapter. 

However,  we  are  not  directly  concerned  here  with  the  phenomena 
of  segmentation  or  cell-division  in  themselves,  except  only  in  so  far 
as  visible  changes  of  form  are  capable  of  easy  and  obvious  correla- 
tion with  the  play  of  force.  The  very  fact  of  "development" 
indicates  that,  while  it  lasts,  the  equihbrium  of  the  egg  is  never 
complete  J.  And  the  gist  of  the  matter  is  that,  if  you  have  caryo- 
kinetic  figures  developing  inside  the  cell,  that  of  itself  indicates  that 
the  dynamic  system  and  the  localised  forces  arising  from  it  are  in 

*  J.  Spek,  loc.  cit.  pp.  108-109. 

t  Cf.  D'Arsonval,  Relation  entre  la  tension  superficielle  et  certains  phenomenes 
electriques  d'origine  animale.  Arch,  de  Physiol,  i,  pp.  460-472,  1889;  Ida  H.  Hyde, 
op.  cit.  p.  242. 

I  Cf.  Plateau's  remarks  (Statique  des  liquides,  ii,  p.  154)  on  the  tendency  towards 
equilibrium,  rather  than  actual  equilibrium,  in  many  of  his  systems  of  soap-films. 


IV]  AND  STRUCTURE  OF  THE  CELL  333 

gradual  alteration;  and  changes  in   the  outward  configuration  of 
the  system  are  bound,  consequently,  to  take  place. 

Perhaps  we  may  simplify  the  case  still  more.  We  have  learned 
many  things  about  cell-division,  but  we  do  not  know  much  in  the 
end.  We  have  dealt,  perhaps,  with  too  many  related  phenomena, 
and  failed  because  we  tried  to  combine  and  account  for  them  all. 
A  physical  problem,  still  more  a  mathematical  one,  wants  reducing 
to  its  simplest  terms,  and  Dr  Rashevsky  has  simplified  and  general- 
ised the  problem  of  cell-division  (or  division  of  a  drop)  in  a  series  of 
papers,  which  still  outrun  by  far  the  elementary  mathematics  of 
this  book.  If  we  cannot  follow^  him  in  all  he  does,  we  may  find 
useful  lessons  in  his  way  of  doing  it.  Cells  are  of  many  kinds;  they 
differ  in  size  and  shape,  in  visible  structure  and  chemical  com- 
position. Most  have  a  nucleus,  some  few  have  none;  most  need 
oxygen,  some  few  do  not;  some  metabolise  in  one  way,  some  in 
another.  What  small  residuum  of  properties  remains  common  to 
them  all?  A  living  cell  is  a  little  fluid  (or  semi-fluid)  system,  in 
which  work  is  being  done,  physical  forces  are  in  operation  and 
chemical  changes  are  going  on.  It  is  in  such  intimate  relation  with 
the  world  outside — its  own  milieu  interne  with  the  great  ynilieu 
externe — that  substances  are  continually  entering  the  cell,  some  to 
remain  there  and  contribute  to  its  growth,  some  to  pass  out  again 
with  loss  of  energy  and  metabolic  change.  The  picture  seems 
simplicity  itself,  but  it  is  less  simple  than  it  looks.  For  on  either 
side  of  the  boundary- wall,  both  in  the  adjacent  medium  and  in  the 
living  protoplasm  within,  there  will  be  no  uniformity,  but  only 
degrees  of  activity,  and  gradients  of  concentration.  Substances 
which  are  being  absorbed  and  consumed  will  diminish  from  periphery 
to  centre;  those  which  are  diffusing  outwards  have  their  greatest 
concentration  near  the  centre,  decrease  towards  the  periphery,  and 
diminish  further  with  increasing  distance  in  the  near  neighbourhood  of 
the  system.  Size,  shape,  diffusibility,  permeability,  chemical  properties 
of  this  and  that,  may  affect  the  gradients,  but  in  the  living  cell  the 
interchanges  are  always  going  on,  and  the  gradients  are  always  there  *. 

*  Outward  diffusion  makes  one  of  the  many  contrasts  between  cell-growth  and 
crystal-growth.  But  the  diffusion^gradients  round  a  growing  crystal  are  far  more 
complicated  than  was  once  supposed.  Cf.  W.  F.  Berg,  Crystal  growth  from 
solutions,  Proc.  R.8.  (A),  clxiv,  pp.  79-95,  1938. 


334  ON  THE  INTERNAL  FORM  [ch. 

If  the  cell  be  homogeneous,  taking  in  and  giving  out  at  a  constant 
rate  in  a  uniform  way,  its  shape  will  be  spherical,  the  concentration- 
field  of  force,  or  concentration-field,  will  likewise  have  a  spherical 
symmetry,  and  the  resultant  force  will  be  zero.  But  if  the  symmetry 
be  ever  so  little  disturbed,  and  the  shape  be  ever  so  little  deformed, 
then  there  will  be  forces  at  work  tending  to  increase  the  deformation, 
and  others  tending  to  equalise  the  surface-tension  and  restore  the 
spherical  symmetry,  and  it  can  be  shewn  that  such  agencies  are 
within  the  range  of  the  chemistry  of  the-  cell.  Since  surface- 
tension  becomes  more  and  more  potent  as  the  size  of  the  drop 
diminishes,  it  follows  that  (under  fluid  conditions)  the  smallest 
solitary  cells  are  least  likely  to  depart  from  a  spherical  shape,  and 
that  cell-division  is  only  likely  to  occur  in  cells  above  a  certain 
critical  order  of  magnitude;  and  using  such  physical  constants 
as  are  available,  Rashevsky  finds  that  this  critical  magnitude 
tallies  fairly  well  with  the  average  size  of  a  living  cell.  The  more 
important  lesson  to  learn,  however,  is  this,  that,  merely  by  virtue 
of  its  metabolism,  every  cell  contains  within  itself  factors  which  may 
lead  to  its  division  after  it  reaches  a  certain  critical  size. 

There  are  simple  corollaries  to  this  simple  setting  of  the  case. 
Since  unequal  concentration-gradients  are  the  chief  cause  which 
renders  non-spherical  shapes  of  cell  possible,  and  these  last  only  so 
long  as  the  cell  lives  and  metabolises,  it  follows  that,  as  soon  as  the 
gradients  disappear,  whether  in  death  or  in  a  "resting-stage",  the 
cell  reverts  to  a  spherical  shape  and  symmetry.  Again,  not  only  is 
there  a  critical  size  above  which  cell-division  becomes  possible,, 
and  more  and  more  probable,  but  there  must  also  be  a  size  beyond 
which  the  cell  is  not  likely  to  grow.  For  the  "specific  surface" 
decreases,  the  metabolic  exchanges  diminish,  the  gradients  become 
less  steep,  and  the  rate  of  growth  decreases  too ;  there  must  come 
a  stage  where  anabolism  just  balances  katabolism,  and  growth 
ceases  though  life  goes  on.  When  streaming  currents  are  visible 
within  the  cell,  they  seem  to  complicate  the  problem;  but  after  all, 
they  are  part  of  the  result,  and  proof  of  the  existence,  of  the  gradients 
•we  have  described.  In  any  further  account  of  Rashevsky 's  theories 
the  mathematical  difficulties  very  soon  begin.  But  it  is  well  to 
realise  that  pure  theory  often  carries  the  mathematical  physicist  a 
long  way ;  and  that  higher  and  higher  powers  of  the  microscope,  and 


IV]  AND  STRUCTURE  OF  THE  CELL  335 

greater  and  greater  histological  skill  are  not  the  one  and  only  way 
to  study  the  physical  forces  acting  within  the  cell  *. 

As  regards  the  phenomena  of  fertihsation,  of  the  union  of  the 
spermatozoon  with  the  "pronucleus"  of  the  egg,  we  might  study 
these  also  in  illustration,  up  to  a  certain  point,  of  the  forces  which 
are  more  or  less  manifestly  at  work.  But  we  shall  merely  take,  as 
a  single  illustration,  the  paths  of  the  male  and  female  pronuclei,  as 
they  travel  to  their  ultimate  meeting-place. 

The  spermatozoon,  when  within  a  very  short  distance  of  the  egg- 
cell,  is  attracted  by  it,  the  same  attraction  being  further  manifested 
in  a  small  conical  uprising  of  the  surface  of  the  eggt.  The  nature 
of  the  attractive  force  has  been  much  disputed.  Loeb  found  the 
spermatozoon  to  be  equally  attracted  by  other  substances,  even  by 
a  bead  of  glass.  It  has  been  held  also  that  the  attraction  is 
chemotropic,  some  substance  being  secreted  by  the  egg  which  drew 
the  sperm  towards  it:  just  as  Pfeifer,  having  shewn  that  maUc  acid 
has  an  attraction  for  fern-antheridia,  supposed  this  substance  to 
play  its  attractive  part  within  the  mucus  of  the  archegonia.  Again, 
the  chemical  secretion  may  be  neither  attractive  nor  directive,  but 
yet  play  a  useful  part  in  activating  the  spermatozoa.  However 
that  may  be,  Gray  has  shewn  reason  to  believe  that  an  electromotive 
force  is  developed  in  the  contact  between  active  spermatozoon  and 
inactive  ovum;  and  that  it  is  the  electrical  change  so  set  up,  and 
almost  instantaneously  propagated,  which  precludes  the  entry  of 
another  spermatozoon  J.  Whatever  the  force  may  be,  it  is  one 
which  acts  normally  to  the  surface  of  the  ovum,  and  after  entry  the 

*  Cf.  N.  Rashevsky,  Mathematical  Biophysics,  Chicago,  1938;  and  many  earlier 
papers.  Eg.  Physico-matheraatical  aspects  of  cellular  multiplication  and  de- 
velopment, Cold  Spring  Harbor  Symposia,  ii,  1934;  The  mechanism  of  division  of 
small  liquid  systems  which  are  the  seat  of  physico-chemical  reactions,  Physics,  in, 
pp.  374-379,  1934;    papers  in  Protoplasma,  xiv-xx,  1931-33,  etc. 

t  With  the  classical  account  by  H.  Fol,  C.R.  lxxxiii,  p.  667,  1876;  Mem.  Soc. 
Phys.  Geneve,  xxvi,  p.  89,  1879,  cf.  Robert  Chambers,  The  mechanism  of  the  entrance 
of  sperm  into  the  star-fish  egg,  Journ.  Gen,  Physiol,  v,  pp.  821-829,  1923.  Here 
a  delicate  filament  is  said  to  run  out  from  the  fertilisation -cone  and  drag  the 
spermatozoon  in;  but  this  is  disputed  and  denied  by  E.  Just,  Biol.  Bull.  LVii, 
pp.  311-325,  1929. 

I  But,  under  artificial  conditions,  "polyspermy"  may  take  place,  eg.  under 
the  action  of  dilute  poisons,  or  of  an  abnormally  high  temperature,  these  being 
doubtless  also  conditions  under  which  the  surface-tension  is  diminished. 


336  ON  THE  INTERNAL  FORM  [ch. 

spermatozoon  points  straight  towards  the  centre  of  the  egg.  From 
the  fact  that  other  spermatozoa,  subsequent  to  the  first,  fail  to 
effect  an  entry,  we  may  safely  conclude  that  an  immediate  con- 
sequence of  the  entry  of  the  spermatozoon  is  an  increase  in  the 
surface-tension  of  the  egg:  this  being  but  one  of  the  complex 
reactions  exhibited  by  the  surface,  or  cortex  of  the  cell*.  Some- 
where or  other,  within  the  egg,  near  or  far  away,  lies  its  own  nuclear 
body,  the  so-called  female  pronucleus,  and  we  find  that  after  a 
while  this  has  fused  with  the  "male  pronucleus"  or  head  of  the 
spermatozoon,  and  that,  the  body  resulting  from  their  fusion  has 
come  to  occupy  the  centre  of  the  egg.  This  must  be  due  (as  Whitman 
pointed  out  many  years  ago)  to  a  force  of  attraction  acting  between 
the  two  bodies,  and  another  force  acting  upon  one  or  other  or  both 
in  the  direction  of  the  centre  of  the  cell.  Did  we  know  the  magnitude 
of  these  several  forces,  it  would  be  an  easy  task  to  calculate  the 
precise  path  which  the  two  pronuclei  would  follow,  leading  to  con- 
jugation and  to  the  central  position.  As  we  do  not  know  the 
magnitude,  but  only  the  direction,  of  these  forces,  we  can  only  make 
a  general  statement:  (1)  the  paths  of  both  moving  bodies  will  he 
wholly  within  a  plane  triangle  drawn  between  the  two  bodies  and 
the  centre  of  the  cell;  (2)  unless  the  two  bodies  happen  to  he,  to 
begin  with,  precisely  on  a  diameter  of  the  cell,  their  paths  until  they 
meet  one  another  will  be  curved  paths,  the  convexity  of  the  curve 
being  towards  the  straight  line  joining  the  two  bodies;  (3)  the  two 
bodies  will  meet  a  httle  before  they  reach  the  centre;  and,  having 
met  and  fused,  will  travel  on  to  reach  the  centre  in  a  straight  hne. 
The  actual  study  and  observation  of  the  path  followed  is  not  very 
easy,  owing  to  the  fact  that  what  we  usually  see  is  not  the  path 
itself,  but  only  a  projection  of  the  path  upon  the  plane  of  the 
microscope ;  but  the  curved  path  is  particularly  well  seen  in  the  frog's 
egg,  where  the  path  of  the  spermatozoon  is  marked  by  a  little  streak 
of  brown  pigment,  and  the  fact  of  the  meeting  of  the  pronuclei  before 
reaching  the  centre  has  been  repeatedly  seen  by  many  observers  f. 

*  See  Mrs  Andrews'  beautiful  observations  on  "Some  spinning  activities  of 
protoplasm  in  starfish  and  echinoid  eggs,"  J  own.  Morphol.  xii,  pp.  307-389,  1897. 

t  W.  Pfeffer,  Locomotorische  Richtungsbewegungen  durch  chemische  Reize, 
Unters.  a.  d.  Botan.  Inst.  Tubingen,  i,  1884;  Physiology  of  Plants,  m,  p.  345,  Oxford, 
1906;  W.  J.  Dakin  and  M.  G.  C,  Fordham,  Journ.  Exp.  Biol.  t.  pp.  183-200,  1924. 
Cf.  J.  Loeb,  Dynamics  of  Living  Matter,  1906,  p.  153. 


IV]  AND  STRUCTURE  OF  THE  CELL  337 

The  problem  recalls  the  famous  problem  of  three  bodies,  which  has 
so  occupied  the  astronomers;  and  it  is  obvious  that  the  foregoing 
brief  description  is  very  far  from  including  all  possible  cases. 
Many  of  these  are  particularly  described  in  the  works  of  Fol,  Roux, 
Whitman  and  others*. 

The  intracellular  phenomena  of  which  we  have  now  spoken  have 
assumed  great  importance  in  biological  literature  and  discussion 
during  the  last  fifty  years;  but  it  is  open  to  us  to  doubt  whether 
they  will  be  found  in  the  end  to  possess  more  than  a  secondary, 
even  a  remote,  biological  significance.  Most,  if  not  all  of  them, 
would  seem  to  follow  immediately  and  inevitably  from  certain 
simple  assumptions  as  to  the  physical  constitution  of  the  cell,  and 
from  an  extremely  simple  distribution  of  polarised  forces  within  it. 
We  have  already  seen  that  how  a  thing  grows,  and  what  it  grows 
into,  is  a  dynamic  and  not  a  merely  material  problem;  so  far  as 
the  material  substance  is  concerned,  it  is  so  only  by  reason  of  the 
chemical,  electrical  or  other  forces  which  are  associated  with  it. 
But  there  is  another  consideration  which  would  lead  us  to  suspect 
that  many  features  in  the  structure  and  configuration  of  the  cell 
are  of  secondary  biological  importance;  and  that  is,  the  great 
variation  to  which  these  phenomena  are  subject  in  similar  or  closely 
related  organisms,  and  the  apparent  impossibihty  of  correlating 
them  with  the  pecuHarities  of  the  organism  as  a  whole.  In  a 
broad  and  general  way  the  phenomena  are  always  the  same.  Certain 
structures  swell  and  contract,  twine  and  untwine,  split  and  unite, 
advance  and  retire ;  certain  chemical  changes  also  repeat  themselves. 
But  Nature  rings  the  changes  on  all  the  details.  "Comparative 
study  has  shewn  that  almost  every  detail  of  the  processes  (of 
mitosis)  described  above  is  subject  to  variation  in  different  forms 
of  cells  |."  A  multitude  of  cells  divide  to  the  accompaniment  of 
caryokinetic  phenomena;  but  others  do  so  without  any  visible 
caryokinesis  at  all.     Sometimes  the  polarised  field  of  force  is  within, 

*  H.  Fol,  Becherches  sur  la  fecondation,  1879;  W.  Roux,  Beitrage  zur  Erit- 
wickelungsmechanik  des  Embryos,  Arch.  f.  Mikr.  Anat.  xix,  1887;  C.  0.  Whitman, 
Ookinesis,  Journ.  Morph.  i,  1887;  E.  Giglio-Tos,  Entwicklungsmechanische 
Studien,  I,  Arch.  f.  Entw.  Mech.  li,  p.  94,  1922,  See  also  Frank  R.  Lillie,  Problems 
of  Fertilisation,  Chicago,  1919. 

t  Wilson,  The  Cell.  p.  77;  cf.  3rd  ed.  (1925),  p.  120. 


338        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

sometimes  it  is  adjacent  to,  and  at  other  times  it  lies  remote  from, 
the  nucleus.  The  distribution  of  potential  is  very  often  symmetrical 
and  bipolar,  as  in  the  case  described;  but  a  less  symmetrical 
distribution  often  occurs,  with  the  result  that  we  have,  for  a  time 
at  least,  numerous  centres  of  force,  instead  of  the  two  main  correlated 
poles :  this  is  the  simple  explanation  of  the  numerous  stellate  figures, 


Haploid  number  of  chr.'raosomes 
Fig.  103.    Summation  diagram  shewing  the  %  number  of  instances  (among  2,415 
phanerogams  and  1,070  metazoa),  in  which  the  chromosomes  do  not  exceed 
a  given  number.    Data  from  M.  J.  D.  White. 

or  "Strahlungen,"  which  have  been  described  in  certain  eggs,  such  as 
those  of  Chaetopterus.  The  number  of  chromosomes  may  be  constant 
within  a  group,  as  in  the  tailed  Amphibia,  with  12;  or  very  variable, 
as  in  sedges,  and  in  grasshoppers  * ;  in  one  and  the  same  species 
of  worm  (Ascaris  megdlocephala),  one  group  or  two  groups  of 
chromosomes  may  be  present.  And  remarkably  constant,  in 
general,  as  the  number  in  any  one  species  undoubtedly  is,  yet  we 
must  not  forget  that,  in  plants  and  animals  alike,  the  whole  range 
of  observed  numbers  is  but  a  small  one  (Fig.  103);    for  (as  regards 

*  There  are  varieties  of  Artemia  salina  which  hardly  differ  in  outward  characters, 
but  differ  widely  in  the  number  of  their  chromosomes. 


IV]  OF  CHROMOSOMES  339 

the  germ-nuclei)  few  have  less  than  six  chromosomes,  and  few  have 
more  than  twenty*.  In  closely  related  animals,  such  as  various 
species  of  Copepods,  and  even  in  the  same  species  of  worm  or  insect, 
the  form  of  the  chromosomes  and  their  arrangement  in  relation*  to 
the  nuclear  spindle  have  been  found  to  differ  in  ways  alluded  to 
above ;  while  only  here  and  there,  as  among  the  chrysanthemums, 
do  related  species  or  varieties  shew  their  own  characteristic  chromo- 
some numbers.  In  contrast  to  the  narrow  range  of  the  chromo- 
some numbers,  we  may  reflect  on  the  all  but  infinite  possibilities  of 
chemical  variabihty.  Miescher  shewed  that  a  molecule  containing 
40  C-atoms  would  admit  (arithmetically  though  not  necessarily 
chemically)  of  a  million  possible  isomers;  and  changes  in  position 
of  the  N-atoms  of  a  protein,  for  instance,  might  vastly  increase 
that  prodigious  number.  In  short,  we  cannot  help  perceiving 
that  many  nuclear  phenomena  are  not  specifically  related  to  the 
particular  organism  in  which  they  have  been  observed,  and  that 
some  are  not  even  specially  and  indisputably  connected  with  the 
organism  as  such.  They  include  such  manifestations  of  the  physical 
forces,  in  their  various  permutations  and  combinations,  as  may  also 
be  witnessed,  under  appropriate  conditions,  in  non-living  things. 
When  we  attempt  to  separate  our  purely  morphological  or  "purely 

*  The  commonest  numbers  of  (haploid)  chromosomes,  both  in  plants  and 
animals,  are  8,  12  and  16.  The  median  number  is  12  in  both,  and  the  lower 
quartile  is  8,  likewise  in  both;  but  the  upper  quartile  is  24  or  thereby  in  animals, 
and  in  the  neighbourhood  of  16  in  plants.  If  we  may  judge  by  the  long  lists  given 
by  E.  B.  Wilson  (The  Cell,  3rd  ed.  pp.  855-865),  by  M.  Ishikawa  in  Botan.  Mag. 
Tokyo,  XXX,  1916,  by  M.  J.  D.  White  in  his  book  on  Chromosomes,  or  by  Tischler 
in  Tabulae  Biologicae  (1927),  fully  60  per  cent,  of  the  observed  cases  lie  between  6 
and  16.  As  Wilson  says  (p.  866)  "the  number  of  chromosomes  is  per  se  a  matter 
of  secondary  importance";  and  (p.  868)  "We  must  admit  the  present  inadequacy 
of  attempts  to  reduce  the  chromosome  numbers  to  any  single  or  consistent 
arithmetical  rules."  Clifford  Dobell  had  said  the  same  thing:  "Nobody  nowadays 
will  be  prepared  to  argue  that  chromosome  numbers,  as  such,  have  any  quantitative 
or  qualitative  relation  to  the  characters  exhibited  by  their  owners.  Complexity 
of  bodily  structure  is  certainly  not  correlated  in  any  way  with  multiplicity  of 
chromosomes  " ;  La  Cellule,  xxxv,  p.  188,  1924.  On  the  other  hand,  Tischler  stoutly 
maintains  that  chromosome-numbers  give  useful  evidence  of  phylogenetic  affinity 
{Biol.  Centralbl.  XLvni,  pp.  321-345,  1928);  and  there  axe  a  few  well-known  cases, 
such  as  the  chrysanthemums,  where,  undoubtedly,  the  numbers  are  constant  and 
specific.  Again  in  certain  cases,  the  number  of  the  chromosomes  may- differ  in 
dififerent  races  (diploid  and  tetraploid)  of  the  same  plant;  and  the  difference  is 
accompanied  by  differences  in  cell-size,  in  rate  of  growth,  and  even  in  the  shape 
of  the  fruit  (of.  Sinnott  and  Blakeslee,  Xat.  Acad,  of  Sci.  1938,  p.  476). 


340        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

embryological "  studies  from  physiological  and  physical  investiga- 
tions, we  tend  ipso  facto  to  regard  each  particular  structure  and 
configuration  as  an  attribute,  or  a  particular  "character,"  of  this  or 
that  particular  organism.  From  this  assumption  we  are  easily  led  to 
the  framing  of  theories  as  to  the  ancestral  history,  the  classificatory 
position,  the  natural  affinities  of  the  several  organisms :  in  fact,  to 
apply  our  embryological  knowledge  to  the  study  of  phylogeny. 
When  we  find,  as  we  are  not  long  of  finding,  that  our  phylogenetic 
hypotheses  become  complex  and  unwieldy,  we  are  nevertheless 
reluctant  to  admit  that  the  whole  method,  with  its  fundamental 
postulates,  is  at  fault;  and  yet  nothing  short  of  this  would  seem 
to  be  the  case,  in  regard  to  the  earher  phases  at  least  of  embryonic 
development.  All  the  evidence  at  hand  goes,  as  it  seems  to  me,  to 
shew  that  embryological  data,  prior  to  and  even  long  after  the 
epoch  of  segmentation,  are  essentially  a  subject  for  physiological  and 
physical  investigation  and  have  but  the  shghtest  fink,  if  any,  with 
the  problems  of  zoological  classification.  Comparative  embryology 
has  its  own  facts  to  classify,  and  its  own  methods  and  principles  of 
classification.  We  may  classify  eggs  according  to  the  presence  or 
absence,  the  paucity  or  abundance,  of  their  associated  food-yolk, 
the  chromosomes  according  to  their  form  and  their  number,  the 
segmentation  according  to  its  various  "types" — radial,  bilateral, 
spiral,  and  so  forth.  But  we  have  httle  right  to  expect,  and  in 
point  of  fact  we  shall  very  seldom  and  (as  it  were)  only  accidentally 
find,  that  these  embryological  categories  coincide  with  the  lines  of 
"natural"  or  "phylogenetic"  classification  which  have  been  arrived 
at  by  the  systematic  zoologist. 

The  efforts  to  explain  "heredity"  by  help  of  "genes"  and  chromo- 
somes, which  have  grown  up  in  the  hands  of  Morgan  and  others  since 
this  book  was  first  written,  stand  by  themselves  in  a  category  which 
is  all  their  own  and  constitutes  a  science  which  is  justified  of 
itself.  To  weigh  or  criticise  these  explanations  would  lie  outside 
my  purpose,  even  were  I  fitted  to  attempt  the  task.  When  these 
great  discoveries  began  to  be  made,  Bateson  crossed  the  ocean 
to  see  and  hear  for  himself  what  Morgan  and  his  pupils  had  to 
shew  and  to  tell.  He  came  home  convinced,  and  humbly  marvelling. 
And  I  leave  this  great  subject  on  one  side  not  because  I  doubt  for  a 
moment  the  facts  nor  dispute  the  hypotheses  nor  decry  the  im- 


IV]  OF  THE  CELL-THEORY  341 

portance  of  one  or  other;  but  because  we  are  so  much  in  the  dark 
as  to  the  mysterious  field  of  force  in  which  the  chromosomes  he, 
far  from  the  visible  horizon  of  physical  science,  that  the  matter  lies 
(for  the  present)  beyond  the  range  of  problems  which  this  book 
professes  to  discuss,  and  the  trend  of  reasoning  which  it  endeavours 
to  mamtain. 

The  cell*,  which  Goodsir  spoke  of  as  a  centre  of  force,  is  in  reality 
a  sphere  of  action  of  certain  more  or  less  localised  forces;  and  of 
these,  surface-tension  is  the  particular  force  which  is  especially 
responsible  for  giving  to  the  cell  its  outline  and  its  morphological 
individuahty.  The  partially  segmented  differs  from  the  totally 
segmented  egg,  the  unicellular  Infusorian  from  the  minute  multi- 

*  The  "  cell -theory "  began  early  and  grew  slowly.  In  a  curious  passage  which 
Mr  Clifford  Dobell  has  shewn  me  {Nov.  Org.  ii,  7,  ad  fin.).  Bacon  speaks  of  "cells" 
in  the  human  body:  of  a  "  coUocatio  spiritus  per  corpoream  molem,  eiusque  pori, 
meatus,  venae  et  cellulae,  et  rudimenta  sive  tentamenta  corporis  organici."  It  is 
"  surely  one  of  the  most  strangely  prophetic  utterances  which  even  Bacon  ever 
made."  Apart  from  this  the  story  begins  in  the  seventeenth  century,  with  Robert 
Hooke's  well-known  figure  of  the  "cells"  in  a  piece  of  cork  (1665),  with  Grew's 
"bladders"  or  "bubbles"  in  the  parenchyma  of  young  beans,  and  Malpighi's 
"utriculi"  or  "sacculi"  in  the  parenchyma  or  "utriculorum  substantia"  of 
various  plants.  Christian  Fr.  v.  Wolff  conceived,  about  the  same  time,  a  hypo- 
thetical "cell-theory,"  on  the  analogy  of  Leibniz's  Monads;  but  the  first  clear 
idea  of  a  cellular  parenchyma,  or  contextus  cellularis,  came  from  C.  Gottlieb 
Ludwig  (1742),  and  from  K.  Fr.  Wolff,  who  spoke  freely  of  cells  or  cellulae. 
Fontana,  author  of  a  curious  Traite  sur  le  venin  de  la  vipere  (1781),  described 
various  histological  elements,  caught  a  glimpse  of  the  nucleus,  and  experi- 
mented with  reagents,  using  syrup  of  violets  for  a  stain.  Early  in  the 
eighteenth  century  the  vessels  of  the  plant  played  an  important  role,  under  Kurt 
Sprengel  and  Treviranus;  but  it  was  not  till  1831  that  Hugo  v.  Mohl  recognised 
that  they  also  arose  from  "cells."  About  this  time  Robert  Brown  discovered, 
or  re-discovered,  the  nucleus  (1833),  which  Schleiden  called  the  cytohlast,  or  "cell- 
producer."  It  was  Schleiden's  idea,  and  a  far-seeing  one,  that  the  cell  lived  a  double 
life,  a  life  of  its  own  and  the  life  of  the  plant  to  which  it  belonged:  "jede  Zelle 
fuhrt  nun  ein  zweifaches  Leben :  ein  selbststandiges,  nur  ihrer  eigenen  Entwicklung 
angehorigen,  und  ein  anderes  mittelbares,  insofern  sie  integrierender  Theil  einer 
Pfianze  geworden  ist "  {Phylogenesis,  1838,  p.  1 ).  The  cell-theory,  so  long  a- building, 
may  be  said  to  have  been  launched,  and  christened,  with  Schwann's  Mikroskopische 
Untersmhungen  of  1839.  Within  the  next  five  years  Martin  Barry  shewed  how 
cell-division  starts  with  the  nucleus,  Henle  described  the  budding  of  certain  cells, 
and  Goodsir  declared  that  all  cells  originate  in  pre-existing  cells,  a  doctrine  at  once 
accepted  by  Remak,  and  made  famous  in  pathology  by  Virchow.  (Cf.  {int.  al.) 
J.  G.  McKendrick,  On  the  modern  cell-theory,  etc.,  Proc  Phil.  Soc.  Glasgow,  xix, 
pp.  1-55,  1887;  J.  Stephenson,  Robert  Brown. .  .and  the  cell-theory,  Proc.  Linn. 
Soc.  1931-2,  pp.  45-54;  M.  Mobius,  Hundert  Jahre  Zellenlehre,  Jen.  Ztschr.  lxxi, 
pp.  313-326,  1938.) 


342        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

cellular  Turbellarian,  in  the  intensity  and  the  range  of  those  surface- 
tensions  which  in  the  one  case  succeed  and  in  the  other  fail  to  form 
a  visible  separation  between  the  cells.  Adam  Sedgwick  used  to 
call  attention  to  the  fact  that  very  often,  even  in  eggs  that  appear 
to  be  totally  segmented,  it  is  yet  impossible  to  discover  an  actual 
separation  or  cleavage,  through  and  through,  between  the  cells  which 
on  the  surface  of  the  egg  are  so  clearly  delimited;  so  far  and  no 
farther  have  the  physical  forces  effectuated  a  visible  "cleavage." 
The  vacuolation  of  the  protoplasm  in  Actinophrys  or  Actinosphaerium 
is  due  to  localised  surface-tensions,  quite  irrespective  of  the  multi- 
nuclear  nature  of  the  latter  organism.  In  short,  the  boundary  walls 
due  to  surface-tension  may  be  present  or  may  be  absent,  with  or 
without  the  delimination  of  the  other  specific  fields  of  force  which 
are  usually  correlated  with  these  boundaries  and  with  the  inde- 
pendent individuality  of  the  cells.  What  we  may  safely  admit, 
however,  is  that  one  effect  of  these  circumscribed  fields  of  force  is 
usually  such  a  separation  or  segregation  of  the  protoplasmic 
constituents,  the  more  fluid  from  the  less  fluid  and  so  forth,  as  to 
give  a  field  where  surface-tension  may  do  its  work  and  bring  a 
visible  bouhdary  into  being.  When  the  formation  of  a  "surface" 
is  once  effected,  its  physical  condition,  or  phase,  will  be  bound  to 
differ  notably  from  that  of  the  interior  of  the  cell,  and  under 
appropriate  chemical  conditions  the  formation  of  an  actual  cell- wall, 
cellulose  or  other,  is  easily  inteUigible.  To  this  subject  we  shall 
return  again,  in  another  chapter. 

From  the  moment  that  we  enter  on  a  dynamical  conception  of 
the  cell,  we  perceive  that  the  old  debates  were  vain  as  to  what 
visible  portions  of  the  cell  were  active  or  passive,  living  or  non- 
living. For  the  manifestations  of  force  can  only  be  due  to  the 
interaction  of  the  various  parts,  to  the  transference  of  energy  from 
one  to  another.  Certain  properties  may  be  manifested,  certain 
functions  may  be  carried  on,  by  the  protoplasm  apart  from  the 
nucleus;  but  the  interaction  of  the  two  is  necessary,  that  other 
and  more  important  properties  or  functions  may  be  manifested. 
We  know,  for  instance,  that  portions  of  an  Infusorian  are  incapable 
of  regenerating  lost  parts  in  the  absence  of  a  nucleus,  while  nucleated 
pieces  soon  regain  the  specific  form  of  the  organism:  and  we  are 
told    that    reproduction    by    fission    cannot    be    initiated,    though 


IV]  OF  THE  CELL-THEORY  343 

apparently  all  its  later  steps  can  be  carried  on,  independently  of 
nuclear  action.  Nor,  as  Verworn  pointed  out,  can  the  nucleus 
possibly  be  regarded  as  the  "sole  vehicle  of  inheritance,"  since  only 
in  the  conjunction  of  cell  and  nucleus  do  we  find  the  essentials  of 
cell-hfe.  "Kern  und  Protoplasma  sind  nur  vereint  lebensfahig,"  as 
Nussbaum  said.  Indeed  we  may,  with  E.  B.  Wilson,  go  further, 
and  say  that  "the  terms  'nucleus'  and  'cell-body'  should  probably 
be  regarded  as  only  topographical  expressions  denoting  two 
differentiated  areas  in  a  common  structural  basis." 

Endless  discussion  has  taken  place  regarding  the  centrosome, 
some  holding  that  it  is  a  specific  and  essential  structure,  a  permanent 
corpuscle  derived  from  a  similar  pre-existing  corpuscle,  a  "fertilising 
element"  in  the  spermatozoon,  a  special  "organ  of  cell-division," 
a  material  "dynamic  centre"  of  the  cell  (as  Van  Beneden  and 
Boveri  call  it);  while  on  the  other  hand,  it  is  pointed  out  that 
many  cells  live  and  multiply  without  any  visible  centrosomes,  that 
a  centrosome  may  disappear  and  be  created  anew,  and  even  that 
under  artificial  conditions  abnormal  chemical  stimuh  may  lead  to 
the  formation  of  new  centrosomes.  We  may  safely  take  it  that  the 
centrosome,  or  the  "attraction  sphere,"  is  essentially  a  "centre  of 
force,"  and  that  this  dynamic  centre  may  or  may  not  be  constituted 
by  (but  will  be  very  apt  to  produce)  a  concrete  and  visible  con- 
centration of  matter. 

It  is  far  from  correct  to  say,  as  is  often  done,  that  the  cell- wall, 
or  cell-membrane,  belongs  "to  the  passive  products  of  protoplasm 
rather  than  to  the  hving  cell  itself";  or  to  say  that  in  the  animal 
cell,  the  cell-wall,  because  it  is  "slightly  developed,"  is  relatively 
unimportant  compared  with  the  important  role  which  it  assumes 
in  plants.  On  the  contrary,  it  is  quite  certain  that,  whether  visibly 
diiferentiated  into  a  semi-permeable  membrane  or  merely  con- 
stituted by  a  liquid  film,  the  surface  of  the  cell  is  the  seat  of 
important  forces,  capillary  and  electrical,  which  play  an  essential 
part  in  the  dynamics  of  the  cell.  Even  in  the  thickened,  largely 
solidified  cellulose  wall  of  the  plant-cell,  apart  from  the  mechanical 
resistances  which  it  affords,  the  osmotic  forces  developed  in  con- 
nection with  it  are  of  essential  importance. 

But  if  the  cell  acts,  after  this  fashion,  as  a  whole,  each  part 
interacting  of  necessity  with  the  rest,  the  same  is  certainly  true  of 


344   FORM  AND  STRUCTURE  OF  THE  CELL   [ch. 

the  entire  multicellular  organism :  as  Schwann  said  of  old,  in  very- 
precise  and  adequate  words,  "the  whole  organism  subsists  only  by 
means  of  the  recijpTocal  action  of  the  single  elementary  parts*."  As 
Wilson  says  again,  "the  physiological  autonomy  of  the  individual 
cell  falls  into  the  background . . .  and  the  apparently  composite 
character  which  the  multicellular  organism  may  exhibit  is  owing  to 
a  secondary  distribution  of  its  energies  among  local  centres  of 
action!."  I*  is  here  that  the  homology  breaks  down  which  is  so 
often  drawn,  and  overdrawn,  between  the  unicellular  organism  and 
the  individual  cell  of  the  metazoonj. 

Whitman,  Adam  Sedgwick  §,  and  others  have  lost  no  opportunity 
of  warning  us  against  a  too  hteral  acceptation  of  the  cell-theory, 
against  the  view  that  the  multicellular  organism  is  a  colony  (or,  as 
Haeckel  called  it,  in  the  case  of  the  plant,  a  "republic")  of  inde- 
pendent units  of  Ufe||.  As  Goethe  said  long  ago,  "Das  lebendige 
ist  zwar  in  Elemente  zerlegt,  aber  man  kann  es  aus  diesen  nicht 
wieder  zusammenstellen  und  beleben " ;  the  dictum  of  the  Cellular- 
pathologie  being  just  the  opposite,  "Jedes  Thier  erscheint  als  cine 
Summe  vitaler  Einheiten,  von  denen  jede  den  vollen  Charakter  des 
Lebens  an  sich  trdgt." 

Hofmeister  and  Sachs  have  taught  us  that  in  the  plant  the  growth 

♦  Theory  of  Cells,  p.  191. 

t  The  Cell  in  Development,  etc.,  p.  59;   cf.  3rd  ed.  (1925),  p.  102.- 

X  E.g.  Brticke,  Elementarorganismen,  p.  387:  "Wir  miissen  in  der  Zelle  einen 
kleinen  Thierleib  sehen,  und  diirfen  die  Analogien,  welche  zwischen  ihr  und  den 
kleinsten  Thierformen  existiren,  niemals  aus  den  Augen  lassen." 

§  C.  0.  Whitman,  The  inadequacy  of  the  cell-theory,  Journ.  Morphol.  viii, 
pp.  639-658,  1893;  A.  Sedgwick,  On  the  inadequacy  of  the  cellular  theory  of 
development,  Q.J. M.S.  xxxvii,  pp.  87-101,  1895;  xxxviii,  pp.  331-337,  1896. 
Cf.  G.  C.  Bourne,  ibid,  xxxviii,  pp.  137-174,  1896;  Clifford  Dobell,  The  principles 
of  Protistology,  Arch.  f.  Protistenk.  xxiii,  p.  270,  1911. 

II  Cf.  0.  Hertwig,  Die  Zelle  und  die  Gewebe,  1893,  p.  1:  "Die  Zellen,  in  welche 
der  Anatom  die  pflanzlichen  und  thierischen  Organismen  zerlegt,  sind  die  Trager 
der  Lebensfunktionen ;  sie  sind,  wie  Virchow  sich  ausgedruckt  hat,  die  'Lebensein- 
heiten.'  Von  diesem  Gesichtspunkt  aus  betrachtet,  erscheint  der  Gesammtlebens- 
prozess  eines  zusammengesetzten  Organismus  nichts  Anderes  zu  sein  als  das  hochst 
yerwickelte  Resultat  der  einzelnen  Lebensprozesse  seiner  zahlreichen,  verschieden 
functionirenden  Zellen."  But  in  1920  Doncaster  (Cytology,  p.  1)  declared  that  "the 
old  idea  of  discrete  and  independent  cells  is  almost  abandoned,"  and  that  the 
word  cell  was  coming  to  be  used  "rather  as  a  convenient  descriptive  term  than 
as  denoting  a  fundamental  concept  of  biology";  and  James  Gray  {Experimental 
Cytology,  p.  2)  said,  in  1931,  that  "we  must  be  careful  to  avoid  any  tacit  assumption 
that  the  cell  is  a  natural,  or  even  legitimate,  unit  of  life  and  function." 


IV]  OF  THE  CELL-THEORY  345 

of  the  mass,  the  growth  of  the  organ,  is  the  primary  fact,  that 
"cell  formation  is  a  phenomenon  very  general  in  organic  life,  but 
still  only  of  secondary  significance."  "Comparative  embryology,'* 
says  Whitman,  "reminds  us  at  every  turn  that  the  organism 
dominates  cell-formation,  using  for  the  same  purpose  one,  several, 
or  many  cells,  massing  its  material  and  directing  its  movements 
and  shaping  its  organs,  as  if  cells  did  not  exist*."  So  Rauber 
declared  that,  in  the  whole  world  of  organisms,  "das  Ganze  liefert 
die  Theile,  nicht  die  Theile  das  Ganze:  letzteres  setzt  die  Theile 
zusammen,  nicht  diese  jenesf."  And  on  the  botanical  side  De  Bary 
has  summed  up  the  matter  in  an  aphorism,  "Die  Pflanze  bildet 
Zellen,  nicht  die  Zelle  bildet  Pflanzen." 

Discussed  almost  wholly  from  the  concrete,  or  morphological 
point  of  view,  the  question  has  for  the  most  part  been  made  to  turn 
on  whether  actual  protoplasmic  continuity  can  be  demonstrated 
between  one  cell  and  another,  whether  the  organism  be  an  actual 
reticulum,  or  syncytium  J.  But  from  the  dynamical  point  of  view 
the  question  is  much  simpler.  We  then  deal  not  with  material 
continuity,  not  with  little  bridges  of  connecting  protoplasm,  but 
with  a  continuity  of  forces,  a  comprehensive  field  of  force,  which 
runs  through  and*  through  the  entire  organism  and  is  by  no  means 
restricted  in  its  passage  to  a  protoplasmic  continuum.  And  such 
a  continuous  field  of  force,  somehow  shaping  the  whole  organism, 
independently  of  the  number,  magnitude  and  form  of  the  individual 
cells,  which  enter  hke  a  froth  into  its  fabric,  seems  to  me  certainly 
and  obviously  to  exist.  As  Whitman  says,  "the  fact  that  physio- 
logical unity  is  not  broken  by  cell-boundaries  is  confirmed  in  so 
many  ways  that  it  must  be  accepted  as  one  of  the  fundamental 
truths  of  biology  §." 

*  Journ.  Morph.  viii,  p.  653,  1893. 

t  Neue  Grundlegungen  zur  Kenntniss  der  Zelle,  Morph.  Jahrb.  viii,  pp.  272, 
313,  333,  1883. 

X  Cf.  e.g.  Ch.  van  Bambeke,  A  propos  de  la  delimitation  cellulaire,  Bull.  Soc. 
beige  de  Microsc.  xxiii,  pp.  72-87,  1897. 

§  Journ.  Morph.  ii,  p.  49,  1889. 


CHAPTER  V 

THE   FORMS   OF  CELLS 

Protoplasm,  as  we  have  already  said,  is  a  fluid*  or  a  semi-fluid 
substance,  and  we  need  not  try  to  describe  the  particular  properties 
of  the  colloid  or  jelly-like 'substances  to  which  it  is  alHed,  or  rather 
the  characteristics  of  the  "colloidal  state"  in  which  it  and  they 
exist;  we  should  find  it  no  easy  matter f.  Nor  need  we  appeal  to 
precise  theoretical  definitions  of  fluidity,  lest  we  come  into  a 
debatable  land.  It  is  in  the  most  general  sense  that  protoplasm 
is  "fluid."  As  Graham  said  (of  colloid  matter  in  general),  "its 
softness  partakes  of  fluidity,  and  enables  the  colloid  to  become  a 
vehicle  for  liquid  diffusion,  like  water  itself  J."  When  we  can  deal 
with  protoplasm  in  sufficient  quantity  we  see  it /oi^§;  particles 
move  freely  through  it,  air-bubbles  and  liquid  droplets  shew  round 
or  spherical  within  it;  and  we  shall  have  much  to  say  about  other 
phenomena  manifested  by  its  own  surface,  which  are  those  especially 
characteristic  of  liquids.  It  may  encompass  and  contain  solid 
bodies,  and  it  may  "secrete"  solid  substances  within  or  around 
itself;  and  it  often  happens  in  the  complex  Hving  organism  that 
these  sohd  substances,  such  as  shell  or  nail  or  horn  or  feather, 
remain  when  the  protoplasm  which  formed  them  is  dead  and  gone. 
But  the  protoplasm  itself  is  fluid  or  semi-fluid,  and  permits  of  free 
(though  not  necessarily  rapid)  diffusion  and  easy  convection  of 
particles  within  itself,  which  simple  fact  is  of  elementary  importance 

*  Cf.  W.  Kuhne,  Ueber  das  Protoplasma,  1864. 

t  Sand,  or  a  heap  of  millet-seed,  may  in  a  sense  be  deemed  a  "fluid,"  and  such 
the  learned  Father  Boscovich  held  them  to  be  {Theoria,  p.  427),  but  at  best  they 
are  fluids  without  a  surface.  Galileo  had  drawn  the  same  comparison;  but  went  on 
to  contrast  the  continuity,  or  infinite  subdivision,  of  a  fluid  with  the  finite,  dis- 
continuous subdivision  of  a  fine  powder.  Cf.  Boyer,  Concepts  of  the  Calculus,  1939, 
p.  291.  . 

J  Phil  Trans,  clt,  p  183,  1861;  Researches,  ed.  Angus  Smith,  1877,  p  553. 
We  no  longer  speak,  however,  of  "colloids'  in  a  specific  sense,  as  Graham  did; 
for  any  substance  can  be  brought  into  the  "colloidal  state"  by  appropriate  means 
or  in  an  appropriate  medmm. 

§  The  copious  protoplasm  of  a  Myxomycete  has  been  passed  unharmed  through 
filter-paper  with  a  pore-size  of  about  1  /x,  or  0001  mm. 


CH.  V]  OF  THE  COLLOID  STATE  347 

in  connection  with  form,  throwing  light  on  what  seem  to  be  common 
characteristics  and  pecuHarities  of  the  forms  of  Hving  things. 

Much  has  been  done,  and  more  said,  about  the  nature  of  protoplasm  since 
this  book  was  written.  Calling  cytoplasm  the  cell-protoplasm  after  deduction 
of  chloroplasts  and  other  gross  inclusions,  we  find  it  to  contain  fats,  proteins, 
lecithin  and  some  other  substances  combined  with  much  water  (up  to 
97  per  cent.)  to  form  a  sort  of  watery  gel.  The  microscopic  structures 
attributed  to  it,  alveolar,  granular  or  fibrillar,  are  inconstant  or  invalid; 
but  it  does  appear  to  possess  an  invisible  or  submicroscopic  structure, 
distinguishing  it  from  an  ordinary  colloid  gel,  and  forming  a  quasi-solid 
framework  or  reticulum.  This  framework  is  based  on  proteid  macromolecules, 
in  the  form  of  polypeptide  chains,  of  great  length  and  carrying  in  side-chains 
other  organic  constituents  of  the  cytoplasm*.  The  polymerised  units 
represent  the  micellae  f  which  the  genius  of  Nageli  predicted  or  postulated 
more  than  sixty  years  ago;  and  we  may  speak  of  a  "micellar  framework" 
as  representing  in  our  cytoplasm  the  dispersed  phase  of  an  ordinary  colloid. 
In  short,  as  the  cytoplasm  is  neither  true  fluid  not  true  solid,  neither  is  it  true 
colloid  in  the  ordinary  sense.  Its  micellar  structure  gives  it  a  certain  rigidity 
or  tendency  to  retain  its  shape,  a  certain  plasticity  and  tensile  strength,  a 
certain  ductility  or  capacity  to  be  drawn  out  in  threads;  but  yet  leaves  it 
with  a  permeability  (or  semi-permeability),  a  capacity  to  swell  by  imbibition, 
above  all  an  ability  to  stream  and  flow,  which  justify  our  calling  it  "fluid 
or  semi-fluid,"  and  account  for  its  exhibition  of  surface-tension  and  other 
capillary  phenomena. 

The  older  naturalists,  in  discussing  the  differences  between  organic  and 
inorganic  bodies,  laid  stress  upon  the  circumstance  that  the  latter  grow  by 
"agglutination,"  and  the  former  by  what  they  termed  "intussusception." 
The  contrast  is  true;  but  it  applies  rather  to  solid  or  crystalline  bodies  as 
compared  with  colloids  of  all  kinds,  whether  living  or  dead.  But  it  so  happens 
that  the  great  majority  of  colloids  are  of  organic  origin;  and  out  of  them  our 
bodies,  and  our  food,  and  the  very  clothes  we  wear,  are  almost  wholly  made. 

A  crystal  "grows"  by  deposition  of  new  molecules,  one  by  one 
and  layer  by  layer,  each  one  superimposed  on  the  solid  substratum 

*  See  {int.  al.)  A.  Frey-Wyssling,  Subrnikroskopische  Morphologie  des  Protoplasmas, 
Berlin,  1938;  cf.  Nature,  June  10,  1939,  p.  965;  also  A.  R.  Moore,  in  Scientia, 
LXii,  July  1,  1937.  On  the  nature  of  viscid  fluid  threads,  cf.  Larmor,  Nature, 
July  11,  1936,  p.  74. 

f  Micella,  or  micula,  diminutive  of  mica,  a  crumb,  grain  or  morsel — ynica  panis, 
salis,  turis,  etc.  Nageli  used  the  word  to  mean  an  aggregation  of  molecules,  as 
the  molecule  is  an  aggregation  of  atoms;  the  one,  however,  is  a  physical  and  the 
other  a  chemical  concept.  Roughly  speaking,  we  may  think  of  micellae  as  varying 
from  about  1  to  200 /x/x;  they  play  a  corresponding  part  in  the  "disperse  phase" 
of  a  colloid  to  that  played  by  the  molecules  in  an  ordinary  solution.  The  macro- 
molecules  of  modern  chemistry  are  sometimes  distinguished  from  these  as  still 
larger  aggregates.  See  Carl  Nageli,  Das  Mikroskop  (2nd  ed.),  1877;  Theorie  der 
Gahrung,  1879. 


348  THE  FORMS  OF  CELLS  [ch. 

already  formed.  Each  particle  would  seem  to  be  influenced  only 
by  the  particles  in  its  immediate  neighbourhood,  and  to  be  in 
a  state  of  freedom  and  independence  from  the  influence,  either 
direct  or  indirect,  of  its  remoter  neighbours.  So  Lavoisier  was 
the  first  to  say.  And  as  Kelvin  and  others  later  on  explained 
the  formation  and  the  resulting  forms  of  crystals,  so  wo  beheve 
that  each  added  particle  takes  up  its  position  in  relation  to  its 
immediate  neighbours  already  arranged,  in  the  holes  and  corners 
that  their  arrangement  leaves,  and  in  closest  contact  with  the 
greatest  number*;  hence  we  may  repeat  or  imitate  this  process  of 
arrangement,  with  great  or  apparently  even  with  precise  accuracy 
(in  the  case  of  the  simpler  crystalhne  systems),  by  piling  up  spherical 
pills  or  grains  of  shot.  In  so  doing,  we  must  have  regard  to  the 
fact  that  each  particle  must  drop  into  the  place  where  it  can  go 
most  easily,  or  where  no  easier  place  offers.  In  more  technical 
language,  each  particle  is  free  to  take  up,  and  does  take  up,  its 
position  of  least  potential  energy  relative  to  those  already  there: 
in  other  words,  for  each  particle  motion  is  induced  until  the  energy 
of  the  system  is  so  distributed  that  no  tendency  or  resultant  force 
remains  to  move  it  more.  This  has  been  shewn  to  lead  to  the 
production  of  plane  surfaces  f  (in  all  cases  where,  by  the  hmitation 
of  material,  surfaces  must  occur);  where  we  have  planes,  there 
straight  edges  and  solid  angles  must  obviously  occur  also,  and,  if 
equihbrium  is  to  follow,  must  occur  symmetrically.  Our  pihng  up 
of  shot  to  make  mimic  crystals  gives  us  visible  demonstration  that 
the  result  is  actually  to  obtain,  as  in  the  natural  crystal,  plane 
surfaces  and  sharp  angles  symmetrically  disposed. 

*  Cf.  Kelvin,  On  the  molecular  tactics  of  a  crystal,  The  Boyle  Lecture,  Oxford, 
1893;  Baltimore  Lectures,  1904,  pp.  612-642.  Here  Kelvin  was  mainly  following 
Bravais's  (and  Frankenheim's)  theory  of  "space-lattices,"  but  he  had  been  largely 
anticipated  by  the  crystallographers.  For  an  account  of  the  development  of  the 
subject  in  modern  crystallography,  by  Sohncke,  von  Fedorow,  8chonfliess,  Barlow 
and  others,  see  (e.g.)  Tutton's  Crystallography,  and  the  many  papers  by  W.  E.  Bragg 
and  others. 

t  In  a  homogeneous  crystalline  arrangement,  symmetry  compels  a  locus  of  one 
property  to  be  a  plane  or  set  of  planes ;  the  locus  in  this  case  being  that  of  least 
surface  potential  energy.  Crystals  "seem  to  be,  as  it  were,  the  Elemental  Figures, 
or  the  A  B  C  of  Nature's  workmg,  the  reason  of  whose  curious  Geometrical  Forms 
(if  I  may  so  call  them)  is  very  easily  explicable"  (Robert  Hooke,  Posthumous  Works^ 
1745,  p.  280). 


V]  OF  INTUSSUSCEPTION  349 

But  the  living  cell  grows  in  a  totally  different  way,  very  much 
as  a  piece  of  glue  swells  up  in  water,  by  "imbibition,"  or  by  inter- 
penetration  into  and  throughout  its  entire  substance.  The  semi- 
fluid colloid  mass  takes  up  water,  partly  to  combine  chemically 
with  its  individual  molecules*;  partly  by  physical  diffusion  into 
the  interstices  between  molecules  or  micellae,  and  partly,  as  it  would 
seem,  in  other  ways;  so  that  the  entire  phenomenon  is  a  complex 
and  even  an  obscure  onef.  But,  so  far  as  we  are  concerned,  the 
net  result  is  very  simple.  For  the  equilibrium,  or  tendency  to 
equilibrium,  of  fluid  pressure  in  all  parts  of  its  interior  while  the 
process  of  imbibition  is  going  on,  the  constant  rearrangement  of  its 
fluid  mass,  the  contrast  in  short  with  the  crystalhne  method  of 
growth  where  each  particle  comes  to  rest  to  move  (relatively  to  the 
whole)  no  more,  lead  the  mass  of  jelly  to  swell  up  very  much  as  a 
bladder  into  which  we  blow  air,  and  so,  by  a  graded  and  harmonious 
distribution  of  forces,  to  assume  everywhere  a  rounded  and  more 
or  less  bubble-hke  external  form  J.  So,  when  the  same  school  of 
older  naturahsts  called  attention  to  a  new  distinction  or  contrast  of 
form  between  organic  and  inorganic  objects,  in  that  the  contours 
of  the  former  tended  to  roundness  and  curvature,  and  those  of  the 
latter  to  be  bounded  by  straight  lines,  planes  and  sharp  angles,  we 
see  that  this  contrast  was  not  a  new  and  different  one,  but  only 
another  aspect  of  their  former  statement,  and  an  immediate  con- 
sequence of  the  difference  between  the  processes  of  agglutination 
and  intussusception  §. 

So  far  then  as  growth  goes  on  undisturbed  by  pressure  or  other 
external  force,  the  fluidity  of  the  protoplasm,  its  mobility  internal 

*  This  is  what  Graham  called  the  water  of  gelatination,  on  the  analogy  of  water 
of  crystallisation;   Chem.  and  Phys.  Researches,  p.  597. 

t  On  this  important  phenomenon,  see  J.  R.  Katz,  Oesetze  der  Quellung,  Dresden, 
1916.  Swelling  is  due  to  "concentrated  solution,"  and  is  accompanied  by  increase 
of  volume  and  liberation  of  energy,  as  when  the  Egyptians  split  granite  by  the 
swelling  of  wood. 

X  Here,  in  a  non- crystalline  or  random  arrangement  of  particles,  symmetry 
ensures  that  the  potential  energy  shall  be  the  same  per  unit  area  of  all  surfaces; 
and  it  follows  from  geometrical  considerations  that  the  total  surface  energy  will 
be  least  if  the  surface  be  spherical. 

§  Intussusception  has  its  shades  of  meaning;  it  is  excluded  from  the  idea  of  a 
crystalline  body,  but  not  limited  to  the  ordinary  conception  of  a  colloid  one.  When 
new  micellar  strands  become  interwoven  in  the  micro-structure  of  a  cellulose  cell- 
wall,  that  is  a  special  kind  of  "intussusception." 


350  THE  FORMS  OF  CELLS  [ch. 

and  external*,  and  the  way  in  which  particles  move  freely  hither 
and  thither  within,  all  manifestly  tend  to  the  production  of  swelhng, 
rounded  surfaces,  and  to  their  great  predominance  over  plane  sur- 
faces in  the  contours  of  the  organism.  These  rounded  contours 
will  tend  to  be  preserved  for  a  while,  in  the  case  of  naked  protoplasm 
by  its  viscosity,  and  in  presence  of  a  cell-wall  by  its  very  lack  of 
fluidity.  In  a  general  way,  the  presence  of  curved  boundary 
surfaces  will  be  especially  obvious  in  the  unicellular  organisms,  and 
generally  in  the  external  form  of  all  organisms,  and  wherever 
mutual  pressure  between  adjacent  cells,  or  other  adjacent  parts, 
has  not  come  into  play  to  flatten  the  rounded  surfaces  into  planes. 

The  swelling  of  any  object,  organic  or  inorganic,  living  or  dead,  is  bound  to 
be  influenced  by  any  lack  of  structural  symmetry  or  homogeneity  f.  We 
may  take  it  that  all  elongated  structures,  such  as  hairs,  fibres  of  silk  or  cotton, 
fibrillae  of  tendon  and  connective  tissue,  have  by  virtue  of  their  elongation 
an  invisible  as  well  as  a  visible  polarity.  Moreover,  the  ultimate  fibrils  are 
apt  to  be  invested  by  a  protein  different  from  the  "collagen"  within,  and 
liable  to  swell  more  or  to  swell  less.  In  ordinary  tendons  there  is  a  "reticular 
sheath,"  which  swells  less,  and  is  apt  to  burst  under  pressure  from  within; 
it  breaks  into  short  lengths,  and  when  the  strain  is  relieved  these  roll  back, 
and  form  the  familiar  annuli.  Another  instance  is  the  tendency  to  swell  of 
the  "macro- molecules"  of  many  polymerised  organic  bodies,  proteins  among 
them. 

But  the  rounded  contours  which  are  assumed  and  exhibited  by 
a  piece  of  hard  glue  when  we  throw  it  into  water  and  see  it  expand 
as  it  sucks  the  water  up,  are  not  near  so  regular  nor  so  beautiful  as 
are  those  which  appear  when  we  blow  a  bubble,  or  form  a  drop,  or 
even  pour  water  into  an  elastic  bag.  For  these  curving  contours 
depend  upon  the  properties  of  the  bag  itself,  of  the  film  or  membrane 
which  contains  the  mobile  gas,  or  which  contains  or  bounds  the 
mobile  liquid  mass.  And  hereby,  in  the  case  of  the  fluid  or  semifluid 
mass,  we  are  introduced  to  the  subject  of  surface-tension:  of  which 
indeed  we  have  spoken  in  the  preceding  chapter,  but  which  we  must 
now  examine  with  greater  care. 

*  The  protoplasm  of  a  sea-urchin's  egg  has  a  viscosity  only  about  four  times, 
and  that  of  various  plants  not  more  than  ten  to  twenty  times,  that  of  water  itself. 
See,  for  a  general  discussion,  L.  V.  Heilbrunn,  Colloid  Symposium  Monograph^  1928. 

t  D.  Jordan  Lloyd  and  R.  H.  Marriott,  The  swelling  of  structural  proteins, 
Proc.  U.S.  (B),  No.  810,  pp.  439-44.3,  1935. 


V]  OF  SURFACE  TENSION  351 

Among  the  forces  which  determine  the  forms  of  cells,  whether 
they  be  sohtary  or  arranged  in  contact  with  one  another,  this  force 
of  surface-tension  is  certainly  of  great,  and  is  probably  of  paramount, 
importance.  But  while  we  shall  try  to  separate  out  the  phenomena 
which  are  directly  due  to  it,  we  must  not  forget  that,  in  each 
particular  case,  the  actual  conformation  which  we  study  may  be, 
and  usually  is,  the  more  or  less  complex  resultant  of  surface-tension 
acting  together  with  gravity,  mechanical  pressure,  osmosis,  or  other 
physical  forces.  The  peculiar  beauty  of  a  soap-bubble,  solitary  or 
in  collocation,  depends  on  the  absence  (to  all  intents  and  purposes) 
of  these  ahen  forces  from  the  field ;  hence  Plateau  spoke  of  the  films 
which  were  the  subject  of  his  experiments  as  "lames  fluides  sans 
pesanteur.''  The  resulting  form  is  in  such  a  case  so  pure,  and  simple 
that  we  come  to  look  on  it  as  wellnigh  a  mathematical  abstraction. 

Surface-tension,  then,  is  that  force  by  which  we  explain  the  form 
of  a  drop  or  of  a  bubble,  of  the  surfaces  external  and  internal  of 
a  "froth"  or  collocation  of  bubbles,  and  of  many  other  things  of 
like  nature  and  in  hke  circumstances*.  It  is  a  property  of  Hquids 
(in  the  sense  at  least  with  which  our  subject  is  concerned),  and  it 
is  manifested  at  or  very  near  the  surface,  where  the  liquid  comes 
into  contact  with  another  hquid,  a  solid  or  a  gas.  We  note  here 
that  the  term  surface  is  to  be  interpreted  in  a  wide  sense;  for 
wherever  we  have  solid  particles  embedded  in  a  fluid,  wherever  we 
have  a  non-homogeneous  fluid  or  semi-fluid,  or  a  "two-phase  colloid " 
such  as  a  particle  of  protoplasm,  wherever  we  have  the  presence  of 
"impurities"  as  in  a  mass  of  molten  metal,  there  we  have  always 
to  bear  in  mind  the  existence  of  surfaces  and  of  surface-phenomena, 
not  only  on  the  exterior  of  the  mass  but  also  throughout  its  inter- 
stices, wherever  like  and  unlike  meet. 

*  The  idea  of  a  "surface-tension"  in  liquids  was  first  enunciated  by  Segner,  and 
ascribed  by  him  to  forces  of  attraction  whose  range  of  action  was  so  small  "ut 
nullo  adhuc  sensu  percipi  potuerat"  {Defiguris  super ficier um  fiuidarum,  in  Comment. 
Soc.  Roy.  GoUingen,  1751,  p.  301).  Hooke,  in  the  Micrographia  (1665,  Obs.  viii, 
etc.),  had  called  attention  to  the  globular  or  spherical  form  of  the  little  morsels 
of  steel  struck  off  by  a  flint,  and  had  shewn  how  to  make  a  powder  of  such  spherical 
grains,  by  heating  fine  filings  to  melting  point,  "This  Phaenomenon"  he  said 
"proceeds  from  a  propriety  which  belongs  to  all  kinds  of  fluid  Bodies  more  or  less, 
and  is  caused  by  the  Incongruity  of  the  Atnbient  and  included  Fluid,  which  so 
acts  and  modulates  each  other,  that  they  acquire,  as  neer  as  is  possible,  a  spherical 
or  globular  form " 


352  THE  FORMS  OF  CELLS  [ch. 

A  liquid  in  the  mass  is  devoid  of  structure ;  it  is  homogeneous,  and 
without  direction  or  polarity.  But  the  very  concept  of  surface- 
tension  forbids  this  to  be  true  of  the  surface-layer  of  a  body  of  liquid, 
or  of  the  "interphase"  between  two  liquids,  or  of  any  film,  bubble, 
drop,  or  capillary  jet  or  stream.  In  all  these  cases,  and  more  empha- 
tically in  the  case  of  a  "monolayer,"  even  the  liquid  has  a  structure 
of  its  own ;  and  we  are  reminded  once  again  of  how  largely  the  living 
organism,  whether  high  or  low,  is  composed  of  colloid  matter  in 
precisely  such  forms  and  structural  conditions. 

Surface-tension  is  due  to  molecular  force  * :  to  force,  that  is  to  say, 
arising  from  the  action  of  one  molecule  upon  another;  and  since 
we  can  only  ascribe  a  small  "sphere  of  action"  to  each  several 
molecule,  this  force  is  manifested  only  within  a  narrow  range. 
Within  the  interior  of  the  liquid  mass  we  imagine  that  such  molecular 
interactions  negative  one  another ;  but  at  and  near  the  free  surface, 
within  a  layer  or  film  approximately  equal  to  the  range  of  the 
molecular  force — or  to  the  radius  of  the  aforesaid  "  sphere  of  action  " 
— there  is  a  lack  of  equilibrium  and  a  consequent  manifestation  of 
force. 

The  action  of  the  molecular  forces  has  been  variously  explained. 
But  one  simple  explanation  (or  mode  of  statement)  is  that  the 
molecules  of  the  surface-layer  are  being  constantly  attracted  into 
the  interior  by  such  as  are  just  a  little  more  deeply  situated;  the 
surface  shrinks  as  molecules  keep  quitting  it  for  the  interior,  and 
this  surface-shrinkage  exhibits  itself  as  a  surface-tension.  The  process 
continues  till  it  can  go  no  farther,  that  is  to  say  until  the  surface 
itself  becomes  a  "minimal  areaf."  This  is  a  sufficient  description 
of  the  phenomenon  in  cases  where  a  portion  of  liquid  is  subject  to 
no  other  than  its  own  molecular  forces,  and  (since  the  sphere  has, 

*  While  we  explain  certain  phenomena  of  the  organism  by  reference  to  atomic 
or  molecular  forces,  the  following  words  of  Du  Bois  Reymond's  seem  worth 
recalling:  ' Naturerkennen  ist  Zuriickfiihren  der  Veranderungen  in  der  Korperwelt 
auf  Bewegung  von  Atomen,  die  durch  deren  von  der  Zeit  unabhangige  Centralkrafte 
bewirkt  werden,  oder  Auflosung  der  Naturkrafte  in  Mechanik  der  Atome.  Es  ist 
eine  psychologische  Erfahrungstatsache  dass,  wo  solche  Auflosung  gelangt,  unser 
Causalbediirfniss  vorlaiifig  sioh  befriedigt  fiihlt"  (Ueber  die  Grenzen  des  Natnr- 
erkennens,  Leipzig,  1873). 

t  There  must  obviously  be  a  certain  kinetic  energy  in  the  molecules  within 
the  drop,  to  balance  the  forces  which  are  trying  to  contract  and  diminish  the 
surface. 


V]  OF  SURFACE  ENERGY  353 

of  all  solids,  the  least  surface  for  a  given  volume)  it  accounts  for 
the  spherical  form  of  the  raindrop*,  of  the  grain  of  shot,  or  of  the 
living  cell  in  innumerable  simple  organisms  f.  It  accounts  also,  as 
we  shall  presently  see,  for  many  much  more  complicated  forms, 
manifested  under  less  simple  conditions. 

Let  us  note  in  passing  that  surface-tension  is  a  comparatively 
small  force  and  is  easily  measurable:  for  instance  that  between 
water  and  air  is  equivalent  to  but  a  few  grains  per  linear  inch,  or 
a  few  grammes  per  metre.  But  this  small  tension,  when  it  exists 
in  a  curved  surface  of  great  curvature,  such  as  that  of  a  minute  drop, 
gives  nse  to  a  very  great  pressure,  directed  inwards  towards  the 
centre  of  curvature.  We  may  easily  calculate  this  pressure,  and  so 
satisfy  ourselves  that,  when  the  radius  of  curvature  approaches 
molecular  dimensions,  the  pressure  is  of  the  order  of  thousands  of 
atmospheres — a  conclusion  which  is  supported  by  other  physical 
considerations. 

The  contraction  of  a  liquid  surface,  and  the  other  phenomena  of 
surface-tension,  involve  the  doing  of  work,  and  the  power  to  do 
work  is  what  we  call  Energy.  The  whole  energy  of  the  system  is 
diffused  throughout  its  molecules,  as  is  obvious  in  such  a  simple 
case  as  we  have  just  considered;  but  of  the  whole  stock  of  energy 
only  the  part  residing  at  or  very  near  the  surface  normally  manifests 
itself  in  work,    and  hence  we  speak  (though  the  term  be  open  to 

*  Raindrops  must  be  spherical,  or  they  would  not  produce  a  rainbow;  and  the 
fact  that  the  upper  part  of  the  bow  is  the  brightest  and  sharpest  shews  that  the 
higher  raindrops  are  more  truly  spherical,  as  well  as  smaller  than  the  lower  ones. 
So  also  the  smallest  dewdrops  are  found  to  be  more  iridescent  than  the  large,  shewing 
that  they  also  are  the  more  truly  spherical;  cf.  T.  W.  Backhouse,  in  Monthly 
Meteorol.  Mag.  March,  1879.  Mercury  has  a  high  surface-tension,  and  its  globules 
are  very  nearly  round. 

t  That  the  offspring  of  a  spherical  cell  (whether  it  be  raindrop,  plant  or  animal) 
should  be  also  a  spherical  cell,  would  seem  to  need  no  other  explanation  than  that 
both  are  of  identical  substance,  and  each  subject  to  a  similar  equilibrium  of 
surface-forces;  but  the  biologists  have  been  apt  to  look  for  a  subtler  reason. 
Giglio-Tos,  speaking  of  a  sea-urchin's  dividing  egg,  asks  why  the  daughter-cells 
are  spherical  like  the  mother-cell,  and  finds  the  reason  in  "heredity":  "Wenn  also 
die  letztere  (d.  i.  die  Mutterzelle)  eine  spharische  Form  besass,  so  nehmen  auch  die 
Tochterzellen  dieselbe  ein;  ware  urspriinglich  eine  kubische  Form  vorhanden, 
so  wurden  also  auch  die  Tochterzellen  dieselbe  auch  aneignen.  Die  Ursache  warum 
die  Tochterzellen  die  spharische  Form  anzunehmen  trachten  liegt  darin,  dasa 
diese  die  Ur-  und  Grundform  alter  Zellen  ist,  sowohl  bet  Tieren  ivie  bei  den  Pflanzen'' 
[Arch.f.  Entw.  Mech.  Li,  p.  115,  1922). 


354  THE  FORMS  OF  CELLS  [ch. 

some  objection)  of  a  specific  surface-energy.  Surface-energy,  and 
the  way  it  is  increased  and  multiplied  by  the  multiphcation  of 
surfaces  due  to  the  subdivision  of  the  tissues  into  cells,  is  of  the 
highest  interest  tq  the  physiologist;  and  even  the  morphologist 
cannot  pass  it  by.  For  the  one  finds  surface-energy  present,  often 
perhaps  paramount,  in  every  cell  of  the  body ;  and  the  other  may  find, 
if  he  will  only  look  for  it,  the  form  of  every  solitary  cell,  hke  that  of 
any  other  drop  or  bubble,  related  to  if  not  controlled  by  capillarity. 
The  theory  of  "capillarity,"  or  "surface-energy,"  has  been  set  forth 
with  the  utmost  possible  lucidity  by  Tait  and  by  Clerk  Maxwell,  on 
whom  the  following  paragraphs  are  based:  they  having  based  their 
teaching  on  that  of  Gauss*,  who  rested  on  Laplace. 

Let  E  be  the  whole  potential  energy  of  a  mass  M  of  hquid;  let 
Cq  be  the  energy  per  unit  mass  of  the  interior  hquid  (we  may  call  it 
the  internal  energy);  and  let  e  be  the  energy  per  unit  mass  for  a 
layer  of  the  skin,  of  surface  S,  of  thickness  t,  and  density  p  (e  being 
what  we  call  the  surface-energy).  It  is  obvious  that  the  total  energy 
consists  of  the  internal  plus  the  surface-energy,  and  that  the  former 
is  distributed  through  the  whole  mass,  minus  its  surface  layers. 
That  is  to  say,  in  mathematical  language, 

E={M  -S.  l.tp)  e^  +  S  .  I^tpe. 

But  this  is  equivalent  to  writing : 

=  MeQ  +  S  .I.tp{e-  eo); 

and  this  is  as  much  as  to  say  that  the  total  energy  of  the  system 
may  be  taken  to  consist  of  two  portions,  one  uniform  throughout 
the  whole  mass,  and  another,  which  is  proportional  on  the  one  hand 
to  the  amount  of  surface,  and  on  the  other  hand  to  the  difference 
between  e  and  Cq,  that  is  to  say  to  the  difference  between  the  unit 
values  of  the  internal  and  the  surface  energy. 

It  was  Gauss  who  first  shewed  how,  from  the  mutual  attractions 
between  all  the  particles,  we  are  led  to  an  expression  for  what  we 

*  See  Gauss's  Principia  generalia  Theoriae  Figurae  Fluidorum  in  statu  equilibriif 
Gottingen,  1830.  The  historical  student  will  not  overlook  the  claims  to  priority 
of  Thomas  Young,  in  his  Essay  on  the  cohesion  of  fluids,  Phil.  Trans.  1805;  see 
the  account  given  in  his  Life  by  Dean  Peacock,  18.55,  pp.  199-210. 


V]  OF  SURFACE  ENERGY  355 

no\v'  call  the  potential  energy*  of  the  system;  and  we  know,  as  a 
fundamental  theorem  of  dynamics,  as  well  as  of  molecular  physics, 
that  the  potential  energy  of  the  system  tends  to  a  minimum,  and 
finds  in  that  minimum  its  stable  equihbrium. 

We  see  in  our  last  equation  that  the  term  Mcq  is  irreducible,  save 
by  a  reduction  of  the  mass  itself.  But  the  other  term  may  be 
diminished  (1)  by  a  reduction  in  the  area  of  surface,  S,  or  (2)  by 
a  tendency  towards  equahty  of  e  and  ^q  ,  that  is  to  say  by  a  diminu- 
tion of  the  specific  surface  energy,  e. 

These  then  are  the  two  methods  by  which  the  energy  of  the 
system  will  manifest  itself  in  work.  The  one,  which  is  much  the 
more  important  for  our  purposes,  leads  always  to  a  diminution  of 
surface,  to  the  so-called  "principle  of  minimal  areas";  the  other, 
which  leads  to  the  lowering  (under  certain  circumstances)  of  surface 
tension,  is  the  basis  of  the  theory  of  Adsorption,  to  which  we  shall 
have  some  occasion  to  refer  as  the  modus  operandi  in  the  develop- 
ment of  a  celi-wall,  and  in  a  variety  of  other  histological  phenomena. 
In  the  technical  phraseology  of  the  day,  the  '^capacity  factor"  is 
involved  in  the  one  case,  and  the  "intensity  factor"  in  the  otherf. 

Inasmuch  as  we  are  concerned  with  the  form  of  the  cell,  it  is  the 
former  which  becomes  our  main  postulate:  telhng  us  that  the 
energy-equations  of  the  surface  of  a  cell,  or  of  the  free  surfaces  of 
cells  in  partial  contact,  or  of  the  partition-surfaces  of  cells  in  contact 
with  one  another,  all  indicate  a  minimum  of  potential  energy  in  the 
system,  by  which  minimal  condition  the  system  is  brought,  ipso 
facto,  into  equihbrium.  And  we  shall  not  fail  to  observe,  with 
something  more  than  mere  historical  interest  and  curiosity,  how 


*  The  word  Energy  was  substituted  for  the  old  vis  viva  by  Thomas  Young  early 
in  the  nineteenth  century,  and  was  used  by  James  Thomson,  Lord  Kelvin's  brother, 
about  1852,  to  mean,  more  generally,  "capacity  for  doing  work."  The  term  potential, 
or  latent,  in  contrast  to  actual  energy,  in  other  words  the  distinction  between  "  energy 
of  activity  and  energy  of  configuration,"  was  proposed  by  Macquorn  Rankine,  and 
suggested  to  him  by  Aristotle's  use  of  8vva/j.L$  and  evepyeia;  see  Rankine's  paper 
On  the  general  law  of  the  transformation  of  energy,  Phil.  Soc.  Glasgow,  Jan. 
5,  1853,  cf.  ibid.  Jan.  23,  1867,  and  Phil.  Mag.  (4),  xxvii,  p.  404,  1864.  The  phrase 
potential  energy  was  at  once  adopted,  but  kinetic  was  substituted  for  actical  by 
Thomson  and  Tait. 

t  The  capacity  factor,  inasmuch  as  it  leads  to  diminution  of  surface,  is  responsible 
for  the  concrescence  of  droplets  into  drops,  of  microcrystals  into  larger  units,  for 
the  flocculation  of  colloids,  and  for  many  other  similar  "changes  of  state." 


356  THE  FORMS  OF  CELLS  [ch. 

deeply  and  intrinsically  there  enter  into  this  whole  class  of  problems 
the  method  of  maxima  and  minima  discovered  by  Fermat,  the  "loi 
universelle  de  repos"  of  Maupertuis,  "dont  tous  les  cas  d'equilibre 
dans  la  statique  ordinaire  ne  sont  que  des  cas  particuhers",  and  the 
lineae  curvae  maximi  miniynive  proprietatibus  gaudentes  of  Euler,  by 
which  principles  these  old  natural  philosophers  explained  correctly 
a  multitude  of  phenomena,  and  drew  the  lines  whereon  the  founda- 
tions of  great  part  of  modern  physics  are  well  and  truly  laid.  For 
that  physical  laws  deal  with  minima  is  very  generally  true,  and  is 
highly  characteristic  of  them.  The  hanging  chain  so  hangs  that  the 
height  of  its  centre  of  gravity  is  a  minimum ;  a  ray  of  hght  takes 
the  path,  however  devious,  by  which  the  time  of  its  journey  is  a 
minimum ;  two  chemical  substances  in  reaction  so  behave  that  their 
thermodynamic  potential  tends  to  a  minimum,  and  so  on.  The 
natural  philosophers  of  the  eighteenth  century  were  engrossed  in 
minimal  problems ;  and  the  differential  equations  which  solve  them 
nowadays  are  among  the  most  useful  and  most  characteristic  equa- 
tions in  mathematical  physics. 

"Voici,"  said  Maupertuis,  "dans  un  assez  petit  volume  a  quoi  je 
reduis  mes  ouvrages  mathematiques ! "  And  when  Lagrange,  fol- 
lowing Euler 's  lead*,  conceived  the  principle  of  least  action,  he 
regarded  it  not  as  a  metaphysical  principle  but  as  "un  resultat 
simple  et  general  des  lois  de  la  mecaniquef."  The  principle  of 
least  action  J  explains  nothing,  it  tells  us  nothing  of  causation, 
yet  it  illuminates  a  host  of  things.  Like  Maxwell's  equations  and 
other  such  flashes  of  genius  it  clarifies  our  knowledge,  adds  weight 
to  our  observations,  brings  order  into  our  stock-in-trade  of  facts. 
It  embodies  and  extends  that  "law  of  simplicity"  which  Borelli 
was  the  first  to  lay  down:  "Lex  perpetua  Naturae  est  ut  agat 
minimo  labore,  mediis  et  modis  simplicissimis,  facillimis,  certis  et 

*  Euler,  Traite  des  Isoperimetres,  Lausanne,  1744. 

t  Lagrange,  Mecanique  Analytique  (2),  ii,  p.  188;   ed.  in  4to,  1788. 

{  This  profound  conception,  not  less  metaphysical  in  the  outset  than  physical, 
began  in  the  seventeenth  century  with  Fermat,  who  shewed  (in  1629)  that  a  ray 
of  light  followed  the  quickest  path  available,  or,  as  Leibniz  put  it,  via  omnium 
facillima;  it  was  over  this  principle  that  Voltaire  quarrelled  with  Euler  and 
Maupertuis.  The  mathematician  will  think  also  of  Hamilton's  restatement  of  the 
principle,  and  of  its  extension  to  the  theory  of  probabilities  by  Boltzmann  and 
Willard  Gibbs.  Cf.  (int.  al.)  A.  Mayer,  Geschichte  des  Prinzips  der  kleinsten  Action, 
1877. 


vj  THE  MEANING  OF  SYMMETRY  357 

tutis:  evitando,  quam  maxime  fieri  potest,  incommoditates  et 
prolixitates."  The  principle  of  least  action  grew  up,  and  grew 
quickly,  out  of  cruder,  narrower  notions  of  "least  time"  or  "least 
space  or  distance."  Nowadays  it  is  developing  into  a  principle  of 
"least  action  in  space-time,"  which  shall  still  govern  and  predict 
the  motions  of  the  universe.  The  infinite  perfection  of  Nature  is 
expressed  and  reflected  in  these  concepts,  and  Aristotle's  great 
aphorism  that  "Nature  does  nothing  in  vain"  lies  at  the  bottom 
of  them  all. 

In  all  cases  where  the  principle  of  maxima  and  minima  comes 
into  play,  as  it  conspicuously  does  in  films  at  rest  under  surface- 
tension,  the  configurations  so  produced  are  characterised  by  obvious 
and  remarkable  symmetry'^.  Such  symmetry  is  highly  characteristic 
of  organic  forms,  and  is  rarely  absent  in  living  things — save  in  such 
few  cases  as  Amoeba,  where  the  rest  and  equilibrium  on  which 
symmetry  depends  are  likewise  lacking.  And  if  we  ask  what 
physical  equilibrium  has  to  do  with  formal  symmetry  and  structural 
regularity,  the  reason  is  not  far  to  seek,  nor  can  it  be  better  put 
than  in  these  words  of  Mach'sf:  "In  every  symmetrical  system 
every  deformation  that  tends  to  destroy  the  symmetry  is  com- 
plemented by  an  equal  and  opposite  deformation  that  tends  to 
restore  it.  In  each  deformation,  positive  and  negative  work  is  done. 
One  condition,  therefore,  though  not  an  absolutely  sufficient  one, 
that  a  niaximum  or  minimum  of  work  corresponds  to  the  form  of 
equiUbrium,  is  thus  supj)ned  by  symmetry.  Regularity  is  successive 
symmetry;  there  is  no  reason,  therefore,  to  be  astonished  that  the 
forms  of  equiUbrium  are  often  symmetrical  and  regular." 

A  crystal  is  the  perfection  of  symmetry  and  of  regularity; 
symmetry  is  displayed  in  its  external  form,  and  regularity  revealed 
in  its  internal  lattices.  Complex  and  obscure  as  the  attractions, 
rotations,  vibrations  and  what  not  within  the  crystal  may  be,  we 
rest  assured  that  the  configuration,  repeated  again  and  again,  of 

*  On  the  mathematical  side,  cf.  Jacob  Nteiner,  Einfache  Beweise  der  isoperi- 
metrischen  Hauptsatze,  Abh.  k.  Akad.  Wisa.  Berlin,  xxiii,  pp.  116-135,  1836  (1838). 
On  the  biological  side,  see  {int.  al.)  F.  M.  Jaeger,  Lectures  on  the  Principle  of  Symmetry, 
and  its  application  to  the  natural  sciences,  Amsterdam,  1917;  also  F.  T.  Lewis, 
Symmetry. .  .in  evolution,  Amer.  Nat.  lvii,  pp.  5-41,  1923. 

f  Science  of  Mechanics,  1902,  p.  395;  see  also  Mach's  article  Ueber  die  physika- 
lische  Bedeutung  der  Gesetze  der  Symmetrie,  Lotos,  xxi,  pp.  139-147,  1871. 


358  THE  FORMS  OF  CELLS  [ch. 

the  component  atoms  is  precisely  that  for  which  the  energy  is  a 
minimum;  and  we  recognise  that  this  minimal  distril^ution  is  of 
itself  tantamount  to  symmetry  and  to  stability. 

Moreover,  the  principle  of  least  action  is  but  a  setting  of  a  still 
more  universal  law — that  the  world  and  all  the  parts  thereof  tend 
ever  to  pass  from  less  to  more  probable  configurations;  in  which 
the  physicist  recognises  the  principle  of  Clausius,  or  second  law  of 
thermodynamics,  and  with  which  the  biologist  must  somehow 
reconcile  the  whole  "theory  of  evolution." 

As  we  proceed  in  our  enquiry,  and  especially  when  we  approach 
the  subject  of  tissues,  or  agglomerations  of  cells,  we  shall  have  from 
time  to  time  to  call  in  the  help  of  elementary  mathematics.  But 
already,  with  very  Httle  mathematical  help,  we  find  ourselves  in  a 
position  to  deal  with  some  simple  examples  of  organic  forms. 

When  we  melt  a  stick  of  sealing-wax  in  the  flame,  surface-tension 
(which  was  ineffectively  present  in  the  solid  but  finds  play  in  the 
now  fluid  mass)  rounds  off  its  sharp  edges  into  curves,  so  striving 
tbwards  a  surface  of  minimal  area ;  and  in  like  manner,  by  merely 
melting  the  tip  of  a  thin  rod  of  glass,  Hooke  made  the  little  spherical 
beads  which  served  him  for  a  microscope*.  When  any  drop  of 
protoplasm,  either  over  all  its  surface  or  at  some  free  end,  as  at  the 
extremity  of  the  pseudopodium  of  an  amoeba,  is  seen  hkewise  to 
"round  itself  off,"  that  is  not  an  effect  of  "vital  contractility,"  but, 
as  Hofmeister  shewed  so  long  ago  as  1867,  a  simple  consequence  of 
surface-tension;  and  almost  immediately  afterwards  Engelmannf 
argued  on  the  same  lines,  that  the  forces  which  cause  the  contraction 
of  protoplasm  in  general  may  "be  just  the  same  as  those  which  tend 
to  make  every  non-spherical  drop  of  fluid  become  spherical."  We 
are  not  concerned  here  with  the  many  theories  and  speculations 
which  would  connect  the  phenomena  of  surface-tension  with  con- 
tractility, muscular  movement,  or  other  special  'physiological  func- 

*  Similarly,  Sir  David  Brewster  and  others  made  powerful  lenses  by  simply 
dropping  small  drops  of  Canada  balsam,  castor  oil,  or  other  strongly  refractive 
liquids,  on  to  a  glass  plate:  On  New  Philosophical  Instruments  (Description  of  a 
new  fluid  microscope),  Edinburgh,  1813,  p.  413.  See  also  Hooke's  Micrographia, 
1665;  and  Adam's  Essay  on  the  Microscope,  1798,  p.  8:  "No  person  has  carried 
the  use  of  these  globules  so  far  as  Father  Torre  of  Naples,  etc."  Leeuwenhoek. 
on  the  other  hand,  ground  his  lenses  with  exquisite  skill. 

t  Beitrage  zur  Physiologie  des  Protoplasma,  Pfluger's  Archlv,  ii,  p.  307,  1869. 


V]  OF  SURFACE  ACTION  359 

tions,  but  we  find  ample  room  to  trace  the  operation  of  the  same 
cause  in  producing,  under  conditions  of  rest  and  equiUbrium,  certain 
definite  and  inevitable  forms. 

It  is  of  great  importance  to  observe  that  the  living  cell  is  one 
of  those  cases  where  the  phenomena  of  surface-tension  are  by  no 
means  limited  to  the  outer  surface;  for  within  the  heterogeneous 
emulsion  of  the  cell,  between  the  protoplasm  and  its  nuclear  and 
other  contents,  and  in  the  "alveolar  network"  of  the  cytoplasm 
itself  (so  far  as  that  alveolar  structure  is  actually  present  in  life), 
we  have  a  multitude  of  interior  surfaces;  and,  especially  among 
plants,  we  may  have  large  internal  "interfacial  contacts"  between 
the  protoplasm  and  its  included  granules,  or  its  vacuoles  filled  with 
the  "cell-sap."  Here  we  have  a  great  field  for  surface-action;  and 
so  long  ago  as  1865,  Nageli  and  Schwendener  shewed  that  the 
streaming  currents  of  plant  cells  might  be  plausibly  explained  by 
this  phenomenon.  Even  ten  years  earlier,  Weber  had  remarked 
upon  the  resemblance  between  the  protoplasmic  streamings  and 
the  currents  to  be  observed  in  certain  inanimate  drops  for  which 
no  cause  but  capillarity  could  be  assigned*.  What  sort  of  chemical 
changes  lead  up  to,  or  go  hand  in  hand  with,  the  variations  of 
surface-tension  in  a  hving  cell,  is  a  vastly  important  question.  It 
is  hardly  one  for  us  to  deal  with ;  but  this  at  least  is  clear,  that  the 
phenomenon  is  more  complicated  than  its  first  investigators,  such 
as  BUtschli  and  Quincke,  ever  took  it  to  be.  For  the  lowered 
surface-tension  which  leads,  say,  to  the  throwing  out  of  a  pseudo- 
podium,  is  accompanied  first  by  local  acidity,  then  by  local 
adsorption  of  proteins,  lastly  and  consequently  by  gelation;  and 
this  last  is  tantamount  to  the  formation  of  "ectoplasm" — a  step 
in  the  direction  of  encystmentf. 

The  elementary  case  of  Amoeba  is  none  the  less  a  complicated  one. 
The  "amoeboid"  form  is  the  very  negation  of  rest  or  of  equihbrium; 

*  Poggendorff's  Annalen,  xciv,  pp.  447-459,  1855.  Cf.  Strethill  Wright,  Phil. 
Mag.  Feb.  1860;   Journ.Anat.  and  Physiol,  i,  p.  337,  1867. 

t  Cf.  C,  J.  Pantin,  Journ.  Mar.  Biol.  Assoc,  xiii,  p.  24,  1923;  Journ.  Exp.  Biol. 
1923  and  1926;  S.  0.  Mast,  Jo^lrn.  Morph.  xli,  p.  347,  1926;  and  0.  W.  Tiegs, 
Surface  tension  and  the  theory  of  protoplasmic  movement,  Protoplasma,  iv, 
pp.  88-139,  1928.  See  also  (int.  al.)  N.  K.  Adam,  Physics  and  Chemistry  of  Surfaces, 
1930;  also  Discussion  on  colloid  science  applied  to  biology  (passim),  Trans.  Faraday 
Soc.  XXVI,  pp.  663  seq.,  1930. 


360  THE  F0RM8  OF  CELLS  [ch. 

the  creature  is  always  moving,  from  one  protean  configuration  to 
another;  its  surface-tension  is  never  constant,  but  continually 
varies  from  here  to  there.  Where  the  surface  tension  ^is  greater, 
that  portion  of  the  surface  will  contract  into  spherical  or  spheroidal 
forms;  where  it  is  less,  the  surface  will  correspondingly  extend. 
While  generally  speaking  the  surface-energy  has  a  minimal  value, 
it  is  not  necessarily  constant.  It  may  be  diminished  by  a  rise  of 
temperature;  it  may  be  altered  by  contact  with  adjacent  sub- 
stances*, by  the  transport  of  constituent  materials  from  the  interior 
to  the  surface,  or  again  by  actual  chemical  and  fermentative  change ; 
for  within  the  cell,  the  surface-energies  developed  about  its  hetero- 
geneous contents  will  continually  vary  as  these  contents  are  affected 
by  chemical  metabolism.  As  the  colloid  materials  are  broken  down 
and  as  the  particles  in  suspension  are  diminished  in  size  the  "free 
surface-energy"  will  be  increased,  but  the  osmotic  energy  will  be 
diminished!.  Thus  arise  the  various  fluctuations  of  surface-tension, 
and  the  various  phenomena  of  amoeboid  form  and  motion,  which 
Biitschli  and  others  have  reproduced  or  imitated  by  means  of  the 
fine  emulsions  which  constitute  their  "artificial  amoebae." 

A  multitude  of  experiments  shew  how  extraordinarily  dehcate  is 
the  adjustment  of  the  surface-tension  forces,  and  how  sensitive  they 
are  to  the  least  change  of  temperature  or  chemical  state.     Thus, 

*  Haycraft  and  Carlier  pointed  out  long  ago  {Proc.  R.S.E.  xv,  pp.  220-224, 
1888)  that  the  amoeboid  movements  of  a  white  blood-corpuscle  are  only  manifested 
when  the  corpuscle  is  in  contact  with  some  solid  substance:  while  floating  freely 
in  the  plasma  or  serum  of  the  blood,  these  corpuscles  are  spherical,  that  is  to  say 
they  are  at  rest  and  in  equilibrium.  The  same  fact  was  recorded  anew  by 
Ledingham  (On  phagocytosis  from  an  adsorptive  point  of  view,  Journ.  Hygiene, 
XII,  p.  324,  1912).  On  the  emission  of  pseudopodia  as  brought  about  by  changes 
in  surface  tension,  see  also  {int.  al.)  J.  A.  Ryder,  Dynamics  in  Evolution,  1894; 
Jensen,  L'eber  den  Geotropismus  niederer  Orgahismen,  Pfliiger's  Archiv,  liii,  1893. 
Jensen  remarks  that  in  Orbitolites,  the  pseudopodia  issuing  through  the  pores  of 
the  shell  first  float  freely,  then  as  they  grow  longer  bend  over  till  they  touch  the 
ground,  whereupon  they  begin  to  display  amoeboid  and  streaming  motions. 
\'erworn  indicates  {Ally.  Physiol.  189o,  p.  429),  and  Davenport  says  {Exper. 
Morphology,  ii,  p.  376),  that  "this  persistent  clinging  to  the  substratum  is  a 
'  thigmotropic '  reaction,  and  one  which  belongs  clearly  to  the  category  of '  response '. " 
Cf.  Putter,  Thigmotaxis  bei  Protisten,  Arch.  f.  Physiol.  1900,  Suppl.  p.  247;  but 
it  is  not  clear  to  my  mind  that  to  account  for  this  simple  phenomenon  we  need 
invoke  other  factors  than  gravity  and  surface-action. 

t  Cf.  Pauli,  Allgemeine  physikalische  Chemie  d.  Zellen  u.  Gewebe,  in  Asher-Spiro's 
Ergebnisse  der  Physiologic,  1912;    Przibram,  Vitalitdt,  1913,  p.  6. 


V]  THE  FORM  OF  AMOEBA  361 

on  a  plate  which  we  have  warmed  at  one  side  a  drop  of  alcohol 
runs  towards  the  warm  area,  a  drop  of  oil  away  from  it;  and  a 
drop  of  water  on  the  glass  plate  exhibits  lively  movements  when 
we  bring  into  its  neighbourhood  a  heated  wire,  or  a  glass  rod  dipped 
in  ether*.  The  water-colour  painter  makes  good  use  of  the  surface- 
tension  effect  of  the  minutest  trace  of  ox-gall.  When  a  plasmodium 
of  Aethalium  creeps  towards  a  damp  spot  or  a  warm  spot,  or 
towards  substances  which  happen  to  be  nutritious,  and  creeps 
away  from  solutions  of  sugar  or  of  salt,  we  are  dealing  with  pheno- 
mena too  often  ascribed  to  'purposeful'  action  or  adaptation,  but 
every  one  of  which  can  be  paralleled  by  ordinary  phenomena  of 
surface-tensiont-  The  soap-bubble  itself  is  never  in  equilibrium: 
for  the  simple  reason  that  its  film,  Hke  the  protoplasm  of  Amoeba 
or  Aethalium,  is  exceedingly  heterogeneous.  Its  surface-energies 
vary  from  point  to  point,  and  chemical  changes  and  changes  of 
temperature  increase  and  magnify  the  variation.  The  surface  of 
the  bubble  is  in  continual  movement,  as  more  concentrated  portions 
of  the  soapy  fluid  make  their  way  outwards  from  the  deeper  layers ; 
it  thins  and  it  thickens,  its  colours  change,  currents  are  set  up  in 
it  and  little  bubbles  glide  over  it;  it  continues  in  this  state  of 
restless  movement  as  its  parts  strive  one  with  another  in  their 
interactioAs  towards  unattainable  equilibrium  J.  On  reaching  a 
certain  tenuity  the  bubble  bursts:  as  is  bound  to  happen  when 
the  attenuated  film  has  no  longer  the  properties  of  matter  in  mass. 

*  80  Bernstein  shewed  that  a  drop  of  mercury  in  nitric  acid  moves  towards,  or 
is  "attracted  by,"  a  crystal  of  potassium  bichromate;  Pfliiger's  Archiv,  lxxx, 
p.  628,  1900.      "^ 

t  The  surface-tension  theory  of  protoplasmic  movement  has  been  denied  by 
many.  Cf.  (e.g.)  H.  S.  Jennings,  Contributions  to  the  behaviour  of  the  lower 
organisms,  Carnegie  Instit.  1904,  pp.  130-230;  O.  P.  Dellinger,  Locomotion  of 
Amoebae,  etc.,  Journ.  Exp.  Zool.  iii,  pp.  337-3o7,  1906;  also  various  papers  by 
Max  Heidenhain,  in  Merkel  u.  Bonnet's  Anatomische  Hefte;   etc. 

X  These  motions  of  a  liquid  surface,  and  other  still  more  striking  movements, 
such  as  those  of  a  piece  of  camphor  floating  on  water,  were  at  one  time  ascribed 
by  certain  physicists  to  a  peculiar  force, ^sui  generis,  the  force  epipolique  of 
Dutrochet;  until  van  der  Mensbrugghe  shewed  that  differences  of  surface-tension 
were  enough  to  account  for  this  whole  series  of  phenomena  (Sur  la  tension  super- 
ficielle  des  liquides,  consideree  au  point  de  vue  de  certains  mouvements  observes 
a  leur  surface,  Mem.  Cour.  Acad,  de  Belgique,  xxxiv,  1869,  Phil.  Mag.  Sept.  1867; 
cf.  Plateau,  Statique  des  Liquides,  p.  283).  An  interesting  early  paper  is  by  Dr 
G.  Carradini  of  Pisa,  DelF  adesione  o  attrazione  di  superficie,  Mem.  di  Matem.  e 
di  Fisica  d.  Soc.  Ital.  d.  Sci.  (Modena),  xi,  p.  75,  xii,  p.  89,  1804-5. 


362  THE  FORMS  OF  CELLS  [ch. 

The  film  becomes  a  mere  bimolecular,  or  even  a  monomolecular, 
layer;  and  at  last  we  may  treat  it  as  a  simple  "surface  of  discon- 
tinuity." So  long  as  the  changes  due  to  imperfect  equihbrium  are 
taking  place  very  slowly,  we  speak  of  the  bubble  as  "at  rest";  it  is 
then,  as  Willard  Gibbs  remarks,  that  the  characters  of  a  film  are 
most  striking  and  most  sharply  defined*. 

So  also,  and  surely  not  less  than  the  soap-bubble,  is  every  cell- 
surface  a  complex  affair.  Face  and  interface  have  a  molecular 
orientation  of  their  own, -depending  both  on  the  partition-membrane 
and  on  the  phases  on  either  side.  It  is  a  variable  orientation, 
changing  at  short  intervals  of  space  and  time;  it  coincides  with 
inconstant  fields  of  force,  electrical  and  other;  it  initiates,  and 
controls  or  catalyses,  chemical  reactions  of  great  variety  and 
importance.  In  short  we  acknowledge  and  confess  that,  in  sim- 
pHfying  the  surface  phenomena  of  the  cell,  for  the  time  being  and 
for  our  purely  morphological  ends,  we  may  be  losing  sight,  or 
making  abstraction,  of  some  of  its  most  specific  physical  and 
physiological  characteristics. 

In  the  case  of  the  naked  protoplasmic  cell,  as  the  amoeboid  phase 
is  emphatically  a  phase  of  freedom  and  activity,  of  unstable  equi- 
librium, of  chemical  and  physiological  change,  so  on  the  other  hand 
does  the  spherical  form  indicate  a  phase  of  stabihty,  of  inactivity, 
of  rest.  In  the  one  phase  we  see  unequal  surface-tensions  manifested 
in  the  creeping  movements  of  the  amoeboid  body,  in  the  rounding- 
off  of  the  ends  of  its  pseudopodia,  in  the  flowing  out  of  its  substance 
over  a  particle  of  "food,"  and  in  the  current-motions  in  the  interior 
of  its  mass ;  till,  in  the  alternate  phase,  when  internal  homogeneity 
and  equilibrium  have  been  as  far  as  possible  attained  and  the 
potential  energy  of  the  system  is  at  a  minimum,  the  cell  assumes  a 
rounded  or  spherical  form,  passes  into  a  state  of  "rest,"  and  (for  a 
reason  which  we  shall  presently  consider)  becomes  at  the  same  time 
encysted  f. 

*  On  the  equilibrium  of  heterogeneous  substances,  Collected  Works,  i,  pp.  55-353; 
Trans.  Conn.  Acad.  1876-78. 

f  We  still  speak  of  the  naked  protoplasm  of  Amoeba;  but  short,  and  far  short, 
of  "encystment,"  there  is  always  a  certain  tendency  towards  adsorptive  action, 
leading  to  a  surface-layer,  or  "plasma- membrane,"  still  semi-fluid  but  less  fluid  than 
before,  and  different  from  the  protoplasm  within ;  it  was  one  of  the  first  and  chief 
things  revealed  by  the  new  technique  of  "micro-dissection."     Little  is  known  of 


v]  THE  FORM  OF  AMOEBA  363 

In  their  amoeboid  phase  the  various  Amoebae  are  just  so  many 
varying  distributions  of  surface-energy,  and  varying  amounts  of 
surface-potential*.  An  ordinary  floating  drop  is  a  figure  of  equi- 
librium under  conditions  of  which  we  shall  soon  have  something  to 
say;  and  if  both  it  and  the  fluid  in  which  it  floats  be  homogeneous 
it  will  be  a  round  drop,  a  "figure  of  revolution."  But  the  least 
chemical  heterogeneity  will  cause  the  surface-tension  to  vary  here 
and  there,  and  the  drop  to  change  its  form  accordingly.  The  httle 
swarm-spores  of  many  algae  lose  their  flagella  as  they  settle  down, 
and  become  mere  drops  of  protoplasm  for  the  time  being;  they 
"put  out  pseudopodia" — in  other  words  their  outline  changes;  and 
presently  this  amoeboid  outHne  grows  out  into  characteristic  lobes 
or  lappets,  a  sign  of  more  or  less  symmetrical  heterogeneity  in  the 
cell-substance. 

In  a  budding  yeast-cell  (Fig.  103  A),  we  see  a  definite  and  restricted 
change  of  surface-tension.  When  a  "bud"  appears,  whether  with 
or  without  actual  growth  by  osmosis  or  otherwise 
of  the  mass,  it  does  so  because  at  a  certain  part 
of  the  cell-surface  the  tension  has  diminished,  and 
the  area  of  that  portion  expands  accordingly ;  but 
in  turn  the  surface-tension  of  the  expanded  or  ex- 
truded portion  makes  itself  felt,  and  the  bud 
rounds  itself  off  into  a  more  or  less  spherical  form.  ^^' 

The  yeast-cell  with  its  bud  is  a  simple  example  of  an  important 
principle.  Our  whole  treatment  of  cell-form  in  relation  to  surface- 
tension  depends  on  the  fact  (which  Errera  was  the  first  to  give  clear 
expression  to)  that  the  incipient  cell-wall  retains  with  but  little 
impairment  the  properties  of  a  liquid  filmf,  and  that  the  growing 
cell,  in  spite  of  the  wall  by  which  it  has  begun  to  be  surrounded, 

the  physical  nature  of  this  so-called  membrane.  It  behaved  more  or  less  like  a  fluid 
lipoid  envelope,  immiscible  with  its  surroundings.  It  is  easily  injured  and  easily 
repaired,  and  the  well-being  of  the  internal  protoplasm  is  said  to  depend  on  the 
maintenance  of  its  integrity.  Robert  Chambers,  Physical  Properties  of  Protoplasm, 
1926;  The  living  cell  as  revealed  by  microdissection,  Harvey  Lectures,  Ser.  xxii, 
1926-27;   Journ.  Gen.  Physiol,  vm,  p.  369,  1926;   etc. 

*  See  (int.  al.)  Mary  J.  Hogue,  The  effect  of  media  of  different  densities  on  the 
shape  of  Amoebae,  Journ.  Exp.  Zool.  xxii,  pp.  565-572,  1917.  Scheel  had  said 
in  1889  that  A.  radiosa  is  only  an  early  stage  of  ^.  proteus  {Festschr.  z.  70.  Geburtstag 
0.  V.  Kuptfer). 

t  Cf.  infra,  p.  561. 


364  THE  FORMS  OF  CELLS  [ch. 

behaves  very  much  Hke  a  fluid  drop.  So,  to  a  first  approximation, 
even  the  yeast-cell  shews,  by  its  ovoid  and  non-spherical  form,  that 
it  has  acquired  its  shape  under  some  influence  other  than  the  uniform 
and  symmetrical  surface-tension  which  makes  a  soap-bubble  into  a 
sphere.  This  oval  or  any  other  asymmetrical  form,  once  acquired, 
may  be  retained  by  virtue  of  the  solidification  and  consequent 
rigidity  of  the  membrane-like  wall  of  the  cell;  and,  unless  rigidity 
ensue,  it  is*  plain  that  such  a  conformation  as  that  of  the  yeast-cell 
with  its  attached  bud  could  not  be  long  retained  as  a  figure  of  even 
partial  equilibrium.  But  as  a  matter  of  fact,  the  cell  in  this  case 
is  not  in  equilibrium  at  all ;  it  is  in  process  of  budding,  and  is  slowly 
altering  its  shape  by  rounding  off  its  bud.  In  like  manner  the 
developing  egg,  through  all  its  successive  phases  of  form,  is  never 
in  complete  equilibrium:  but  is  constantly  responding  to  slowly 
changing  conditions,  by  phases  of  partial,  transitory,  unstable  and 
conditional  equihbrium. 

There  are  innumerable  solitary  plant-cells,  and  unicellular 
organisms  in  general,  which,  hke  the  yeast-cell,  do  not  correspond 
to  any  of  the  simple  forms  which  may  be  generated  under  the 
influence  of  simple  and  homogeneous  surface-tension ;  and  in  many 
cases  these  forms,  which  we  should  expect  to  be  unstable  and 
transitory,  have  become  fixed  and  stable  by  reason  of  some  com- 
paratively sudden  solidification  of  the  envelope.  This  is  the  case, 
for  instance,  in  the  more  comphcated  forms  of  diatoms  or  of  desmids, 
where  we  are  dealing,  in  a  less  striking  but  even  more  curious  way 
than  in  the  budding  yeast-cell,  not  with  one  simple  act  of  formation, 
but  with  a  complicated  result  of  successive  stages  of  localised  growth, 
interrupted  by  phases  of  partial  consolidation.  The  original  cell 
has  acquired  a  certain  form,  and  then,  under  altering  conditions 
and  new  distributions  of  energy,  has  thickened  here  or  weakened 
there,  and  has  grown  out,  or  tended  (as  it  were)  tc  branch,  at  par- 
ticular points.  We  can  often  trace  in  each  particular  stage  of 
growth,  or  at  each  particular  temporary  growing  point,  the  laws  of 
surface  tension  manifesting  themselves  in  what  is  for  the  time  being 
a  fluid  surface ;  nay  more,  even  in  the  adult  and  completed  structure 
we  have  little  difficulty  in  tracing  and  recognising  (for  instance  in 
the  outline  of  such  a  desmid  as  Euastrum)  the  rounded  lobes  which 
have  successively  grown  or  flowed  out  from  the  original  rounded  and 


V]  OF  LIQUID  FILMS  365 

flattened  cell.  What  we  see  in  a  many  chambered  foraminifer,  such 
as  Glohigerina  or  Rotalia,  is  the  same  thing,  save  that  the  stages  are 
more  separate  and  distinct,  and  the  whole  is  carried  out  to  greater 
completeness  and  perfection.  The  little  organism  as  a  whole  is  not 
a  figure  of  equihbrium  nor  of  minimal  area;  but  each  new  bud  or 
separate  chamber  is  such  a  figure,  conditioned  by  the  forces  of 
surface-tension,  and  superposed  upon  the  complex  aggregate  of 
similar  bubbles  after  these  latter  have  become  consoHdated  one  by 
one  into  a  rigid  system. 

Let  us  now  make  some  enquiry  into  the  forms  which  a  fluid 
surface  can  assume  under  the  mere  influence  of  surface-tension. 
In  doing  so  we  are  limited  to  conditions  under  which  other  forces 
are  relatively  unimportant,  that  is  to  say  where  the  surface  energy 
is  a  considerable  fraction  of  the  whole  energy  of  the  system;  and 
in  general  this  will  be  the  case  when  we  are  dealing  with  portions 
of  liquid  so  small  that  their  dimensions  come  within  or  near  to  what 
we  have  called  the  molecular  range,  or,  more  generally,  in  which 
the  "specific  surface"  is  large.  In  other  words  it  is  the  small  or 
minute  organisms,  or  small  cellular  elements  of  larger  organisms, 
whose  forms  will  be  governed  by  surface-tension;  while  the  forms 
of  the  larger  organisms  are  due  to  other  and  non-molecular  forces. 
A  large  surface  of  water  sets  itself  level  because  here  gravity  is 
predominant;  but  the  surface  of  water  in  a  narrow  tube  is  curved, 
for  the  reason  that  we  are  here  deahng  with  particles  which  he  within 
the  range  of  each  other's  molecular  forces.  The  like  is  the  case  with 
the  cell-surfaces  and  cell-partitions  which  we  are  about  to  study,  and 
the  effect  of  gravity  will  be  especially  counteracted  and  concealed 
when  the  object  is  immersed  in  a  hquid  of  nearly  its  own  density. 

We  have  already  learned,  as  a  fundamental  law  of  "capillarity," 
that  a  liquid  film  in  equilibrium  assumes  a  form  which  gives  it  a 
minimal  area  under  the  conditions  to  which  it  is  subject.  These 
conditions  include  (1)  the  form  of  the  boundary,  if  such  exist,  and 
(2)  the  pressure,  if  any,  to  which  the  film  is  subject:  which  pressure 
is  closely  related  to  the  volume  of  air,  or  of  liquid,  that  the  film 
(if  it  be  a  closed  one)  may  have  to  contain.  In  the  simplest  of  cases, 
as  when  we  take  up  a  soap-film  on  a  plane  wire  ring,  the  film  is 
exposed  to  equal  atmospheric  pressure  on  both  side^,  and  it  ob- 


36G  THE  FORMS  OF  CELLS  [ch. 

viously  has  its  minimal  area  in  the  form  of  a  plane.  So  long  as  our 
wire  ring  lies  in  one  plane  (however  irregular  in  outline),  the  film 
stretched  across  it  will  still  be  in  a  plane;  but  if  we  bend  the  ring 
so  that  it  lies  no  longer  in  a  plane,  then  our  film  will  become  curved 
into  a  surface  which  may  be  extremely  comphcated,  but  is  still  the 
smallest  possible  surface  which  can  be  drawn  continuously  across 
the  uneven  boundary. 

The  question  of  pressure  involves  not  only  external  pressures 
acting  on  the  film,  but  also  that  which  the  film  itself  is  capable  of 
exerting.  For  we  have  seen  that  the  film  is  always  contracting  to 
the  utmost;  and  when  the  film  is  curved,  this  leads  to  a  pressure 
directed  inwards — perpendicular,  that  is  to  say,  to  the  surface  of 
the  film.  In  the  case  of  the  soap-bubble,  the  uniform  contraction 
of  whose  surface  has  led  to  its  spherical  form,  this  pressure  is 
balanced  by  the  pressure  of  the  air  within;  and  if  an  outlet  be 
given  for  this  air,  then  the  bubble  contracts  with  perceptible  force 
until  it  stretches  across  the  mouth  of  the  tube,  for  instance  across 
the  mouth  of  the  pipe  through  which  we  have  blown  the  bubble. 
A  precisely  similar  pressure,  directed  inwards,  is  exercised  by  the 
surface  layer  of  a  drop  of  water  or  a  globule  of  mercury,  or  by  the 
surface  pellicle  on  a  portion  or  "drop"  of  protoplasm.  Only  we 
must  always  remember  that  in  the  soap-bubble,  or  the  bubble  which 
a  glass-blower  blows,  there  is  a  twofold  pressure  as  compared  with 
that  which  the  surface-film  exercises  on  the  drop  of  liquid  of  which 
it  is  a  part ;  for  the  bubble  consists  (unless  it  be  so  thin  as  to  consist 
of  a  mere  layer  of  molecules*)  of  a  Kquid  layer,  with  a  free  surface 
within  and  another  without,  and  each  of  these  two  surfaces  exercises 
its  own  independent  and  coequal  tension  and  its  corresponding 
pressure!. 

If  we  stretch  a  tape  upon  a  flat  table,  whatever  be  the  tension 
of  the  tape  it  obviously  exercises  no  pressure  upon  the  table  below. 
But  if  we  stretch  it  over  a  curved  surface,  a  cy finder  for  instance, 
it  does  exercise  a  downward  pressure;  and  the  more  curved  the 
surface  the  greater  is  this  pressure,  that  is  to  say  the  greater  is  this 
share  of  the  entire  force  of  tension  which  is  resolved  in  the  down- 

*  Or,  more  strictly  speaking,  unless  its  thickness  be  less  than  twice  the  range 
of  the  molecular  forces. 

t  It  follows  that  the  tension  of  a  bubble,  depending  only  on  the  surface-conditions, 
is  independent  of  the  thickness  of  the  film. 


v]  OF  LIQUID  FILMS  367 

ward  direction.  In  mathematical  language,  the  pressm-e  (p)  varies 
directly  as  the  tension  (T),  and  inversely  as  the  radius  of  curvature 
(R) :   that  is  to  say,  p  =  T/R,  per  unit  of  surface. 

If  instead  of  a  cylinder,  whose  curvature  lies  only  in  one  direction, 
we  take  a  case  of  curvature  in  two  dimensions  (as  for.  instance  a 
sphere),  then  the  effects  of  these  two  curvatures  must  be  added 
together  to  give  the  resulting  pressure  p:  which  becomes  equal  to 
T/R  +  TIR\  or  1       1 

R'^R' 


i.=  H+-* 


And  if  in  addition  to  the  pressure  p,  which  is  due  to  surface-tension, 
we  have  to  take  into  account  other  pressures,  p\  p",  etc.,  due  to 
gravity  or  other  forces,  then  we  may  say  that  the  total  pressure 


p=/+/'+2'(l+l) 


We  may  have  to  take  account  of  the  extraneous  pressures  in 
some  cases,  as  when  we  come  to  speak  of  the  shape  of  a  bird's  egg ; 
but  in  this  first  part  of  our  subject  we  are  able  for  the  most  part 
to  neglect  them. 

Our  equation  is  an  equation  of  equihbrium.  The  resistance  to 
compression — the  pressure  outwards — of  our  fluid  mass  is  a  constant 
quantity  (P);  the  pressure  inwards,  T  (IjR  +  l/R'),  is  also  con- 
stant; and  if  the  surface  (unlike  that  of  the  mobile  amoeba)  be 
homogeneous,  so  that  T  is  everywhere  equal,  it  follows  that 
1/i?  +  l/R'  =  C  (a  constant),  throughout  the  whole  surface  in  question. 

Now  equihbrium  is  reached  after  the  surface-contraction  has 
done  its  utmost,  that  is  to  say  when  it  has  reduced  the  surface  to 
the  least  possible  area.  So  we  arrive  at  the  conclusion,  from  the 
physical  side,  that  a  surface  such  that  IjR  +  IjR'  =  C,  in  other 
\<^ords  a  surface  which  has  the  same  7nean  curvature  at  all  points, 
is  equivalent  to  a  surface  of  minimal  area  for  the  volume  enclosed  f ; 

*  This  simple  but  immensely  important  formula  is  due  to  Laplace  (Mecanique 
Celeste,  Bk  x,  suppl.  Theorie  de  Vaction  capillaire,  1806). 

t  A  surface  may  be  "minimal"  in  respect  of  the  area  occupied,  or  of  the  volume 
enclosed:  the  former  being  such  as  the  surface  which  a  soap-film  forms  when  it 
fills  up  a  ring,  whether  plane  or  no.  The  geometers  are  apt  to  restrict  the  term 
"minimal  surface"  to  such  as  these,  or,  more  generally,  to  all  cases  where  the  mean 
curvature  i»  nil;  the  others,  being  only  minimal  with  respect  to  the  volume  con- 
tained, they  call  "surfaces  of  constant  mean  curvature." 


368  THE  FORMS  OF  CELLS  [ch. 

and  to  the  same  conclusion  we  may  also  come  by  ways  purely 
mathematical.  The  plane  and  the  sphere  are  two  obvious  examples 
of  such  surfaces,  for  in  both  the  radius  of  curvature  is  everywhere 
constant. 

From  the  fact  that  we  may  extend  a  soap-film  across  any  ring 
of  wire,  however  fantastically  the  wire  be  bent,  we  see  that  there 
is  no  end  to  the  number  of  surfaces  of  minimal  area  which  may  be 
constructed  or  imagined*.  While  some  of  these  are  very  com- 
phcated  indeed,  others,  such  as  a  spiral  or  helicoid  screw,  are 
relatively  simple.  But  if  we  Umit  ourselves  to  surfaces  of  revolution 
(that  is  to  say,  to  surfaces  symmetrical  about  an  axis),  w^e  find,  as 
Plateau  was  the  first  to  shew,  that  those  which  meet  the  case  are 
few  in  number.  They  are  six  in  all,  nam-ely  the  plane,  the  sphere, 
the  cyhnder,  the  catenoid,  the  unduloid,  and  a  curious  surface  which 
Plateau  called  the  nodoid. 


A  B  CD 

Fig.  104.   Roulettes  of  the  conic  sections. 

These  several  surfaces  are  all  closely  related,  and  the  passage 
from  one  to  another  is  generally  easy.  Their  mathematical  inter- 
relation is  expressed  by  the  fact  (first  shewn  by  Delaunayf,  in  1841) 
that  the  plane  curves  by  whose  revolution  they  are  generated  are 
themselves  generated  as  "roulettes"  of  the  conic  sections. 

Let  us  imagine  a  straight  line,  or  axis,  on  which  a  circle,  ellipse  or 
other  conic  section  rolls ;  the  focus  of  the  conic  section  will  describe 
a  line  in  some  relation  to  the  fixed  axis,  and  this  fine  (or  roulette), 
when  we  rotate  it  around  the  axis,  will  describe  in  space  one  or 
another  of  the  six  surfaces  of  revolution  of  which  we  are  speaking. 

If  we  imagine  an  elhpse  so  to  roll  on  a  base-hne,  either  of  its  foci 
will  describe  a  sinuous  or  wavy  fine  (Fig.   104,  B)   at  a  distance 

*  To  fit  a  minimal  surface  to  the  boundary  of  any  given  closed  curve  in  space  is 
a  problem  formulated  by  Lagrange,  and  commonly  known  as  the  "problem  of 
Plateau,"  who  solved  it  with  his  soap-films. 

f  Sur  la  surface  de  revolution  dont  la  courbure  moyenne  est  constante,  Journ. 
de  M.  Liouville,  vi,  p.  309,  1841.  Cf.  {int.  al.)  J.  Clerk  Maxwell,  On  the  theory  of 
rolUng  curves.  Trans.  R.S.E.  xvi,  pp.  519-540,  1849;  J.  K.  Wittemore,  Minimal 
surfaces  of  rotation,  Ann.  Math.  (2),  xix,  1917,  Amer.  Journ.  Math,  xl,  p.  69, 
1918;   Crino  Loria,  Courbes  planes  speciales,  theorie  ef  histoire,  Milan,  574  pp.,  1930. 


V]  OF  MINIMAL  SURFACES 

alternately  maximal  and  minimal  from  the  axis;  this  wavy  Hne, 
by  rotation  about  the  axis,  becomes  the  meridional  Hne  of  the 
surface  which  we  call  the  unduloid,  and  the  more  unequal  the  two 
axes  are  of  our  elhpse,  the  more  pronounced  will  be  the  undulating 
sinuosity  of  the  roulette.  If  the  two  axes  be  equal,  then  our  elhpse 
becomes  a  circle ;  the  path  described  by  its  roUing  centre  is  a  straight 
line  parallel  to  the  axis  (A),  and  the  sohd  of  revolution  generated 
therefrom  will  be  a  cylinder:  in  other  words,  the  cylinder  is  a 
"Hmiting  case"  of  the  unduloid.  If  one  axis  of  our  ellipse  vanish, 
while  the  other  remains  of  finite  length,  then  the  elhpse  is  reduced 
to  a  straight  line  with  its  foci  at  the  two  ends,  and  its  roulette  will 
appear  as  a  succession  of  semicircles  touching  one  another  upon  the 
axis  (C);  the  solid  of  revolution  will  be  a  series  of  equal  spheres. 
If  as  before  one  axis  of  the  ellipse  vanish,  but  the  other  be  infinitely 
long,  then  the  roulette  described  by  the  focus  of  this  ellipse  will  be 
a  circular  arc  at  an  infinite  distance;  i.e.  it  will  be  a  straight  line 
normal  to  the  axis,  and  the  surface  of  revolution  traced  by  this 
straight  hne  turning  about  the  axis  will  be  a  plane.  If  we  imagine 
one  focus  of  our  ellipse  to  remain  at  a  given  distance  from  the  axis, 
but  the  other  to  become  infinitely  remote,  that  is  tantamount  to 
saying  that  the  elhpse  becomes  transformed  into  a  parabola;  and 
by  the  rolling  of  this  curve  along  the  axis  there  is  described  a 
catenary  (D),  whose  solid  of  revolution  is  the  catenoid. 

Lastly,  but  this  is  more  difficult  to  imagine,  we  have  the  case  of 
the  hyperbola.  We  cannot  well  imagine  the  hyperbola  rolling  upon 
a  fixed  straight  line  so  that  its  focus  shall  describe  a  continuous 
curve.  But  let  us  suppose  that  the  fixed  hne  is,  to  begin  with, 
asymptotic  to  one  branch  of  the  hyperbola,  and  that  the  rolUng 
proceeds  until  the  hne  is  now  asymptotic  to  the  other  branch,  that 
is  to  say  touching  it  at  an  infinite  distance;  there  will  then  be 
mathematical  continuity  if  we  recommence  rolling  with  this  second 
branch,  and  so  in  turn  with  the  other,  when  each  has  run  its  course. 
We  shall  see,  on  reflection,  that  the  line  traced  by  one  and  the 
same  focus  will  be  an  "elastic  curve,"  describing  a  succession  of 
kinks  or  knots  (E),  and  the  solid  of  revolution  described  by  this 
meridional  line  about  the  axis  is  the  so-called  nodoid. 
I 

The  physical  transition  of  one  of  these  surfaces  into  another  can 


370  THE  FORMS  OF  CELLS  [ch. 

be  experimentally  illustrated  by  means  of  soap-bubbles,  or  better 
still,  after  another  method  of  Plateau's,  by  means  of  a  large  globule 
of  oil,  supported  when  necessary  by  wire  rings,  and  lying  in  a  fluid 
of  specific  gravity  equal  to  its  own. 

To  prepare  a  mixture  of  alcohol  and  water  of  a  density  precisely 
equal  to  that  of  the  oil-globule  is  a  troublesome  matter,  and  a 
method  devised  by  Mr  C.  R.  Darhng  is  a  great  improvement  on 
Plateau's*.  Mr  Darhng  used  the  oily  hquid  orthotoluidene,  which 
does  not  mix  with  water,  has  a  beautiful  and  conspicuous  red 
colour,  and  has  precisely  the  same  density  as  water  when  both 
are  kept  at'  a  temperature  of  24°  C.  We  have  therefore  only  to 
run  the  liquid  into  water  at  this  temperature  in  order  to  produce 
beautifully  spherical  drops  of  any  required  size :  and  by  adding  a 
little  salt  to  the  lower  layers  of  water,  the  drop  may  be  made  to 
rest  or  float  upon  the  denser  liquid. 


Fig.  105. 

We  have  seen  that,  the  soap-bubble,  spherical  to  begin  with,  is 
transformed  into  a  plane  when  we  release  its  internal  pressure  and 
let  the  film  shrink  back  upon  the  orifice  of  the  pipe.  If  we  blow 
a  bubble  and  then  catch  it  up  on  a  second  pipe,  so  that  it  stretches 
between,  we  may  draw  the  two  pipes  apart,  with  the  result  that 
the  spheroidal  surface  will  be  gradually  flattened  in  a  longitudinal 
direction,  and  the  hubble  will  be  transformed  into  a  cyhnder.  But 
if  we  draw  the  pipes  yet  farther  apart,  the  cyhnder  narrows  in  the 
middle  into  a  sort  of  hour-glass  form,  the  increasing  curvature  of 
its  transverse  section  being  balanced  by  a  gradually  increasing 
negative  curvature  in  the  longitudinal  section;  the  cyhnder  has,  in 
turn,  been  converted  into  an  unduloid.  When  we  hold  a  soft  glass 
tube  in  the  flame  and  "draw  it  out,"  we  are  in  the  same  identical 
fashion  converting  a  cylinder  into  an  unduloid  (Fig.  105,  A)\  when 
on  the  other  hand  we  stop  the  end  and  blow,  we  again  convert  the 
cylinder  into  an  unduloid  (B),  but  into  one  which  is  now  positively, 
while   the   former   was   negatively,   curved.     The   two  figures  are 

*  See  Liquid  Drops  and  Globules,  1914,  p.  11.  Robert  Boyle  used  turpentine 
in  much  the  same  way;   for  other  methods  see  Plateau,  op.  cit.  p.  154. 


v]  OF  MINIMAL  SURFACES  371 

essentially  the  same,  save  that  the  two  halves  of  the  one  change 
places  in  the  other. 

That  spheres,  cylinders  and  unduloids  are  of  the  commonest 
occurrence  among  the  forms  of  small  unicellular  organisms  or  of 
individual  cells  in  the  simpler  aggregates,  and  that  in  the  processes 
of  growth,  reproduction  and  development  transitions  are  frequent 
from  one  of  these  forms  to  another,  is  obvious  to  the  naturalist*, 
and  we  shall  deal  presently  with  a  few  of  these  phenomena. 
But  before  we  go  further  in  this  enquiry  we  must  consider,  to 
some  small  extent  at  least,  the  curvatures  of  the  six  different  sur- 
faces, so  far  as  to  determine  what  modification  is  required,  in  each 
case,  of  the  general  equation  which  apphes  to  them  all.  We  shall 
find  that  with  this  question  is  closely  connected  the  question  of 
the  pressures  exercised  by  or  impinging  on  the  film,  and  also  the 
very  important  question  of  the  limiting  conditions  which,  from  the 
nature  of  the  case,  set  bounds  to  the  extension  of  certain  of  the 
figures.  The  whole  subject  is  mathematical,  and  we  shall  only  deal 
with  it  in  the  most  elementary  way. 

We  have  seen  that,  in  our  general  formula,  the  expression 
IjR  +  IjR'  =  C,  a  constant;  and  that  this  is,  in  all  cases,  the 
condition  of  our  surface  being  one  of  minimal  area.  That  is  to  say, 
it  is  always  true  for  one  and  all  of  the  six  surfaces  which  we  have 
to  consider;  but  the  constant  C  may  have  any  value,  positive, 
negative  or  nil. 

In  the  case  of  the  plane,  where  R  and  R'  are  both  infinite, 
IjR  +  IjR'  =  0.  The  expression  therefore  vanishes,  and  our  dy- 
namical equation  of  equihbrium  becomes  P  =  p.  In  short,  we  can 
only  have  a  plane  film,  or  we  shall  only  find  a  plane  surface  in  our 
cell,  when  on  either  side  thereof  we  have  equal  pressures  or  no 
pressure  at  all;  a  simple  case  is  the  plane^partition  between  two 
equal  and  similar  cells,  as  in  a  filament  of  Spirogyra. 

In  the  sphere  the  radii  are  all  equal,  R=  R';  they  are  also  positive, 
and  T  (l/R  +  l/R'),  or  2T/R,  is  a-  positive  quantity,  involving  a 
constant  positive  pressure  P,  on  the  other  side  of  the  equation. 

In  the  cylinder  one  radius  of  curvature  has  the  finite  and  positive 
value  R;   but  the  other  is  infinite.     Our  formula  becomes  TjR,  to 

*  They  tend  to  reappear,  no  less  obviously,  in  those  precipitated  structures  which 
simulate  organic  form  in  the  experiments  of  Leduc,  Herrera  and  Lillie. 


372  THE  FORMS  OF  CELLS  [ch. 

which  corresponds  a  positive  pressure  P,  suppHed  by  the  surface- 
tension  as  in  the  case  of  the  sphere,  but  evidently  of  just  half  the 
magnitude. 

In  plane,  sphere  and  cyUnder  the  two  principal  curvatures  are 
constant,  separately  and  together;  but  in  the  unduloid  the  curva- 
tures change  from  one  point  to  another.  At  the  middle  of  one  of 
the  swollen  "beads"  or  bubbles,  the  curvatures  are  both  positive; 
the  expression  (l/R  +  1/^')  is  therefore  positive,  and  it  is  also  finite. 
The  film  exercises  (like  the  cyUnder)  a  positive  pressure  inwards, 
to  be  compensated  by  an  equivalent  outward  pressure  from  within. 
Between  two  adjacent  beads,  at  the  middle  of  one  of  the  narrow 
necks,  there  is  obviously  a  much  stronger  curvature  in  the  trans- 
verse direction;  but  the  total  pressure  is  unchanged,  and  we  now 
see  that  a  negative  curvature  along  the  unduloid  balances  the 
increased  curvature  in  the  transverse  direction.  The  sum  of  the  two 
must  remain  positive  as  well  as  constant;  therefore  the  convex  or 
positive  curvature  must  always  be  greater  than  the  concave  or 
negative  curvature  at  the  same  point,  and  this  is  plainly  the  case 
in  our  figure  of  the  unduloid. 

The  catenoid,  in  this  respect  a  limiting  case  of  the  unduloid,  has 
its  curvature  in  one  direction  equal  and  opposite  to  its  curvature 
in  the  other,  this  property  holding  good  for  all  points  of  the  surface ; 
R  =  —  R';  and  the  expression  becomes 

{l/R  -+-  l/R')  =  (L/R  -  l/R)  =  0. 

That  is  to  say,  the  mean  curvature  is  zero,  and  the  catenoid, 
like  the  plane  itself,  has  no  curvature,  and  exerts  no  pressure. 
None  of  the  other  surfaces  save  these  two  share  this  remarkable 
property;  and  it  follows  that  we  may  have  at  times  the  plane  and 
the  catenoid  co-existing  as  parts  of  one  and  the  same  boundary 
system,  just  as  the  cyHnder  or  the  unduloid  may  be  capped  by 
portions  of  spheres.  It  follows  also  that  if  we  stretch  a  soap-film 
between  two  rings,  and  so  form  an  annular  surface  open  at  both  ends, 
that  surface  is  a  catenoid :  the  simplest  case  being  when  the  rings  are 
parallel  and  normal  to  the  axis  of  the  figure*. 

*  A  topsail  bellied  out  by  the  wind  is  not  a  catenoid  surface,  but  in  vertical 
section  it  is  everywhere  a  catenary  curve;  and  Diirer  shews  beautiful  catenary 
curves  in  the  wrinkles  under  an  Old  Man's  eyes.     A  simple  experiment  is  to  invert 


v]  OF  FIGURES  OF  EQUILIBRIUM  373 

The  nodoid  is,  like  the  iinduloid,  a  continuous  curve  which  keeps 
altering  its  curvature  as  it  alters  its  distance  from  the  axis;  but  in 
this  case  the  resultant  pressure  inwards  is  negative  instead  of 
positive.  But  this  curve  is  a  complicated  one,  and  its  full  mathe- 
matical treatment  is  too  hard  for  us. 

In  one  of  Plateau's  experiments,  a  bubble  of  oil  (protected  from 
gravity  by  a  fluid  of  equal  density  to  its  own)  is  balanced  between 
annuh;  and  by  adjusting  the  distance  apart  of  these,  it  may  be 
brought  to  assume  the  form  of  Fig.  106,  that  is  to  say,  of  a  cyhndet 
with  spherical  ends;  there  is  then  everywhere  a  pressure  inwards 
on  the  fluid  contents  of  the  bubble  a  pressure  due  to  the  convexity 


Fig.  106. 


Fig.  107. 


of  the  surface  film.  This  cylinder  may  be  converted  into  an  undu- 
loid,  either  by  drawing  the  rings  farther  apart  or  by  abstracting 
some  of  the  oil,  imtil  at  length  rupture  ensues,  and  the  cylinder 
breaks  up  into  two  spherical  drops.  Or  again,  if  the  surrounding 
liquid  be  made  ever  so  little  heavier  or  hghter  than  that  which 
constitutes  the  drop,  then  gravity  comes  into  play,  the  conditions 
of  equihbrium  are  modified  accordingly,  and  the  cylinder  becomes 
part  of  an  unduloid,  with  its  dilated  portion  above  or  below  as  the 
case  may  be  (Fig.  107). 

In  all  cases  the  unduloid,  like  the  original  cyhnder,  is  capped 
by  spherical  ends,  the  sign  and  the  consequence  of  a  positive 
pressure  produced  by  the  curved  walls  of  the  unduloid.  But  if 
our  initial  cyhnder,  instead  of  being  tall,  be  a  flat  or  dumpy  one 

a  small  funnel  in  a  large  one,  wet  them  with  soap-solution,  and  draw  them  apart; 
the  film  which  develops  between  them  is  a  catenoid  surface,  set  perpendicularly 
to  the  two  funnels.  On  this  and  other  geometrical  illustrations  of  the  fact  that 
a  soap-film  sets  itself  at  right  angles  to  a  solid  boundary,  see  an  elegant  paper  by 
Mary  E.  Sinclair,  in  Annals  of  Mathematics,  viii,  1907. 


374 


THE  FORMS  OF  CELLS 


[CH. 


(with  certain  definite  relations  of  height  to  breadth),  then  new 
phenomena  may  occur.  For  now,  if  oil  be  cautiously  withdrawn 
from  the  mass  by  help  of  a  small  syringe,  the  cylinder  may  be  made 
to  flatten  down  so  that  its  upper  and  lower  surfaces  become  plane: 
which  is  of  itself  a  sufficient  indication  that  the  pressure  inwards 
is  now  nil.  But  at  the  very  moment  when  the  upper  and  lower 
surfaces  become  plane,  it  will  be  found  that  the  sides  curve  inwards, 
in  the  fashion  shewn  in  Fig.  108  B.     This  figure  is  a  catenoid,  which, 


B 


Fig.  108. 


as  we  have  seen,  is,  like  the  plane  itself,  a  surface  exercising  no 
pressure,  and  which  therefore  may  coexist  with  the  plane  as  part 
of  one  and  the  same  system. 

We  may  continue  to  withdraw  more  oil  from  our  bubble,  drop 
by  drop,  and  now  the  upper  and  lower  surfaces  dimple  down  into 
concave  portions  of  spheres,  as  the 
result  of  the  negative  internal 
pressure ;  and  thereupon  the  peri- 
pheral catenoid  surface  alters  its 
form  (perhaps,  on  this  small  scale, 
imperceptibly),  and  becomes  a 
portion  of  a  nodoid.  It  represents, 
in  fact,  that  portion  of  the  nodoid 
which  in  Fig.  109  lies  between  such  points  as  0,  P.  While  it  is  easy  to 
draw  the  outline,  or  meridional  section,  of  the  nodoid,  it  is  obvious 
that  the  solid  of  revolution  to  be  derived  from  it  can  never  be 
reahsed  in  its  entirety :  for  one  part  of  the  solid  figure  would  cut,  or 
entangle  with,  another.  All  that  we  can  ever  do,  accordingly,  is  to 
reahse  isolated  portions  of  the  nodoid*. 

*  This  curve  resembles  the  looped  Elastic  Curve  (see  Thomson  and  Tait,  ii, 
p.  148,  fig.  7),  but  has  its  axis  on  the  other  side  of  the  curve.  The  nodoid  was 
represented  upside-down  in  the  first  edition  of  this  book,  a  mistake  into  which  others 
have  fallen,  including  no  less  a  person  than  Clerk  Maxwell,  in  his  article  "Capillarity  " 
in  the  Encycl.  Brit.  9th  ed. 


Fig.  109. 


V}  OF  FIGURES  OF  EQUILIBRIUM  375 

In  all  these  cases  the  ring  or  annulus  is  not  merely  a  means  of 
mechanical  restraint,  controlling  the  form  of  the  drop  or  bubble ;  it 
also  marks  the  boundary,  or  "locus  of  discontinuity,"  between  one 
surface  and  another. 

If,  in  a  sequel  to  the  preceding  experiment  of  Plateau's,  we  use 
solid  discs  instead  of  annuli,  we  may  exert  pressure  on  our  oil- 
globule  as  we  exerted  traction  before.  We  begin  again  by  adjusting 
the  pressure  of  these  discs  so  that  the  oil  assumes  the  form  of  a 
cylinder:  our  discs,  that  is  to  say,  are  adjusted  to  exercise  a 
mechanical  pressure  just  equal  to  what  in  the  former  case  was 
supplied  by  the  surface-tension  of  the  spherical  caps  or  ends  of  the 
bubble.  If  we  now  increase  the  pressure  slightly,  the  peripheral 
walls  become  convexly  curved,  exercising  a  precisely  corresponding 
pressure;  the  form  assumed  by  the  sides  of  our  figure  is  now  that 
of  a  portion  of  an  unduloid.  If  we  increase  the  pressure,  the 
peripheral  surface  of  oil  will  bulge  out  more  and  more,  and  will 
presently  constitute  a  portion  of  a  sphere.  But  we  may  continue 
the  process  yet  further,  and  find  within  certain  limits  the  system 
remaining  perfectly  stable.  What  is  this  new  curved  surface  which 
has  arisen  out  of  the  sphere,  as  the  latter  was  produced  from  the 
unduloid?  It  is  no  other  than  a  portion  of  a  nodoid,  that  part 
which  in  Fig.  109  lies  between  M  and  N.  But  this  surface,  which  is 
concave  in  both  directions  towards  the  surface  of  the  oil  within, 
is  exerting  a  pressure  upon  the  latter,  just  as  did  the  sphere  out  of 
which  a  moment  ago  it  was  transformed;  and  we  had  just  stated, 
in  considering  the  previous  experiment,  that  the  pressure  inwards 
exerted  by  the  nodoid  was  a  negative  one.  The  explanation  of  this 
seeming  discrepancy  lies  in  the  simple  fact  that,  if  we  follow  the 
outline  of  our  nodoid  curve  in  Fig.  109,  from  OP,  the  surface  con- 
cerned in  the  former  case,  to  MN,  that  concerned  in  the  present, 
we  shall  see  that  in  the  two  experiments  the  surface  of  the  Uquid 
is  not  the  same,  but  lies  on  the  positive  side  of  the  curve  in  the  one 
case,  and  on  the  negative  side  in  the  other. 

These  capillary  surfaces  of  Plateau's  form  a  beautiful  example 
of  the  "materialisation"  of  mathematical  law.  Theory  leads  to 
certain  equations  which  determine  the  position  of  points  in  a 
system,  and  these  points  we  may  then  plot  as  curves  on  a  coordinate 
diagram;    but  a  drop  or  a  bubble  may  realise  in  an  instant  the 


376  THE  FORMS  OF  CELLS  [ch. 

whole  result  of  our  calculations,  and  materialise  our  whole  ap- 
paratus of  curves.  Such  a  case  is  what  Bacon  calls  a  "collective 
instance,"  bearing  witness  to  the  fact  that  one  common  law  is 
obeyed  by  every  point  or  particle  of  the  system.  Where  the  under- 
lying equations  are  unknown  to  us,  as  happens  in  so  many  natural 
configurations,  we  may  still  rest  assured  that  kindred  mathematical 
laws  are  being  automatically  followed,  and  rigorously  obeyed,  and 
sometimes  half-revealed. 

Of  all  the  surfaces  which  we  have  been  describing,  the  sphere  is 
the  only  one  which  can  enclose  space  of  itself;  the  others  can  only 
help  to  do  so,  in  combination  with  one  another  or  with  the  sphere. 
Moreover,  the  sphere  is  also,  of  all  possible  figures,  that  which 
encloses  the  greatest  volume  with  the  least  area  of  surface  * ;  it  is 
strictly  and  absolutely  the  surface  of  minimal  area,  and  it  is,  ipso 
facto,  the  form  which  will  be  assumed  by  a  unicellular  organism 
(just  as  by  a  raindrop),  if  it  be  practically  homogeneous  and  if,  Hke 
Orhulina  floating  in  the  ocean,  its  surroundings  be  likewise  homo- 
geneous and  its  field  of  force  symmetrical  f.  It  is  only  relatively 
speaking  that  the  rest  of  these  configurations  are  surfaces  minimae 
areae',  for  they  are  so  under  conditions  which  involve  various 
pressures  or  restraints.  Such  restraints  are  imposed  by  the  pipe  or 
annulus  which  supports  and  confines  our  oil-globule  or  soap-bubble ; 
and  in  the  case  of  the  organic  cell,  similar  restraints  are  supplied 
by  solidifications  partial  or  complete,  or  other  modifications  local 
or  general,  of  the  cell-surface  or  cell-wall. 

One  thing  we  must  not  fail  to  bear  in  mind.  In  the  case  of  the 
soap-bubble  we  look  for  stabihty  or  instability,  equihbrium  or  non- 
equilibrium,  in  its  several  configurations.  But  the  living  cell  is 
seldom  in  equihbrium.  It  is  continually  using  or  expending  energy; 
and  this  ceaseless  flow  of  energy  gives  rise  to  a  "steady  state," 
taking  the  place  of  and  simulating  equilibrium.     In  like  manner  the 

*  On  the  circle  and  sphere  as  giving  the  smallest  boundary  for  a  given  content, 
see  (e.g.)  Jacob  Steiner,  Einfache  Beweisen  der  isoperimetrischen  Hauptsatze, 
Berlin.  Abhandlungen,  1836,  pp.  123-132. 

t  The  essential  conditions  of  homogeneity  and  symmetry  are  none  too  common, 
and  a  spherical  organism  is  only  to  be  looked  for  among  simple  things.  The 
floating  (or  pelagic)  eggs  of  fishes,  the  spores  of  red  seaweeds,  the  oospheres  of 
Fucus  or  Oedogonium,  the  plasma-masses  escaping  from  the  cells  of  Vaucheriaf 
are  among  the  instances  which  come  to  mind. 


OF  FIGURES  OF  EQUILIBRIUM 


377 


hardly  changing  outHne  of  a  jet  or  waterfall  is  but  in  pseudo- 
equiHbrium;  it  is  in  a  steady  state,  dynamically  speaking.  Many 
puzzling  and  apparent  paradoxes  of  physiology,  such  (to  take  a 
single  instance)  as  the  maintenance  of  a  constant  osmotic  pressure 
on  either  side  of  a  cell-membrane,  are  accounted  for  by  the  fact 
that  energy  is  being  spent  and  work  done,  and  a  steady  state  or 
pseudo-equilibrium  maintained  thereby. 

Before  we  pass  to  biological  illustrations  of  our  surface-tension 
figures  we  have  still  another  matter  to  deal  with.  We  have  seen 
from  our  description  of  two  of  Plateau's  classical  experiments,  that 
at  some  particular  point  one  type  of  surface  gives  place  to  another ; 
and  again  we  know  that,  when  we  draw  out  our  soap-bubble  into 
a  cyhnder,  and  then  beyond,  there  comes  a  certain  point  at  which 
the  bubble  breaks  in  two,  and  leaves  us  with  two  bubbles  of  w^hich 
each  is  a  sphere  or  a  portion  of  a  sphere.  In  short  there  are  certain 
limits  to  the  dimensions  of  our  figures,  within  which  limits  equi- 
librium is  stable,  but  at  which  it  becomes  unstable,  and  beyond  which 
it  breaks  down.  Moreover,  in  our  composite  surfaces,  when  the 
cyhnder  for  instance  is  capped  by  two  spherical  cups  or  lenticular 
discs,  there  are  well-defined  ratios  which  regulate  their  respective 
curvatures  and  their  respective  dimensions.  These  two  matters  we 
may  deal  with  together. 

Let  us  imagine  a  Hquid  drop  which  in 
appropriate  conditions  has  been  made  to 
assume  the  form  of  a  cylinder;  we  have 
already  seen  that  its  ends  will  be  capped 
by  portions  of  spheres.  Since  one  and 
the  same  liquid  film  covers  the  sides  and 
ends  of  the  drop  (or  since  one  and  the 
same  delicate  membrane  encloses  the 
sides  and  ends  of  the  cell),  w^e  assume 
the  surface-tension  (T)  to  be  everywhere 
identical;  and  it  follows,  since  the 
internal  fluid-pressure  is  also  every- 
where identical,  that  the  expression  (IjR  +  1/i?')  for  the  cylinder 
is  equal  to  the  corresponding  expression,  which  we  may  cafl 
(1/r  +  1/r'),   in   the    case    of   the    terminal    spheres.     But   in    the 


378  THE  FORMS  OF  CELLS  [ch. 

cylinder  l/R'  =-  0,  and  in  the  sphere  1/r  =  1/r'.  Therefore  our 
relation  of  equality  becomes  l/R  =  2/r,  or  r  =  2R;  which  means 
that  the  sphere  in  question  has  just  twice  the  radius  of  the  cylinder 
of  which  it  forms  a  cap. 

And  if  Ob,  the  radius  of  the  sphere,  be  equal  to  twice  the  radius 
(Oa)  of  the  cyhnder,  it  follows  that  the  angle  aOb  is  an  angle  of  60°, 
and  bOc  is  also  an  angle  of  60° ;  that  is  to  say,  the  arc  be  is  equal  to 
Jtt.  In  other  words,  the  spherical  disc  which  (under  the  given 
conditions)  caps  our  cylinder  is  not  a  portion  taken  at  haphazard, 
but  is  neither  more  nor  less  than  that  portion  of  a  sphere  which  is 
subtended  by  a  cone  of  60°.  Moreover,  it  is  plain  that  the  height 
of  the  spherical  cap,  de,  =  Ob  —  ab  =  R  (2  —  ^/3)  =  0'27R,  where 
R  is  the  radius  of  our  cylinder,  or  one-half  the  radius  of  our  spherical 
cap:  in  other  words  the  normal  height  of  the  spherical  cap  over 
the  end  of  the  cylindrical  cell  is  just  a  very  little  more  than  one- 
eighth  of  the  diameter  of  the  cylinder,  or  of  the  radius  of  the  sphere. 
And  these  are  the  proportions  which  we  recognise,  more  or  less, 
under  normal  circumstances,  in  such  a  case  as  the  cylindrical  cell 
of  Spirogyra,  when  one  end  is  free  and  capped  by  a  portion  of  a 
sphere*. 

Among  the  many  theoretical  discoveries  which  we  owe  to  Plateau, 
one  to  which  we  have  just  referred  is  of  peculiar  importance: 
namely  that,  with  the  exception  of  the  sphere  and  the  plane,  the 
surfaces  with  which  we  have  been  dealing  are  only  in  complete 
equilibrium  within  certain  dimensional  limits,  or  in  other  words, 
have  a  certain  definite  limit  of  stabihty;  only  the  plane  and  the 
sphere,  or  any  portion  of  a  sphere,  are  perfectly  stable,  because 
they  are  perfectly  symmetrical,  figures. 

Perhaps  it  were  better  to  say  that  their  symmetry  is  such  that 
any  small  disturbance  will  probably  readjust  itself,  and  leave  the 
plane  or  spherical  surface  as  it  was  before,  while  in  the  other 
configurations  the  chances  are  that  a  disturbance  once  set  up  will 
travel  in  one  direction  or  another,  increasing  as  it  goes.  For 
equihbrium  and  probabihty  (as  Boltzman  told  us)  are  nearly  alHed: 

*  The  conditions  of  stability  of  the  cyhnder,  and  also  of  the  catenoid,  are 
explained  with  the  utmost  simplicity  by  Clerk  Maxwell,  in  his  article,  already 
quoted,  on  "Capillarity.''  On  the  catenoids,  see  A.  Terquem.  C.R.  xcii,  pp.  407-9, 
1881. 


V] 


OF  FIGURES  OF  EQUILIBRIUM 


379 


so  nearly  that  that  state  of  a  system  which  is  most  likely  to  occur, 
or  most  likely  to  endure,  is  precisely  that  which  we  call  the  state  of 
equilibrium. 

For  experimental  demonstration,  the  case  of  the  cylinder  is  the 
simplest.  If  we  construct  a  liquid  cylinder,  either  by  drawing  out 
a  bubble  or  by  supporting  a  globule  of  oil  between  two  rings,  the 
experiment  proceeds  easily  until  the  length  of  the  cylinder  becomes 
just  about  three  times  as  great  as  its  diameter.  But  soon  afterwards 
instability  begins,  and  the  cylinder  alters  its  form;    it  narrows  at 


Fig.  111. 


the  waist,  so  passing  into  an  unduloid,  and  the  deformation  pro- 
gresses quickly  until  our  cylinder  breaks  in  two,  and  its  two  halves 
become  portions  of  spheres.  This  physical  change  of  one  surface  into 
another  corresponds  to  what  the  mathematicians  call  a  "discon- 
tinuous solution"  of  a  problem  of  minima.  The  theoretical  limit  of 
stability,  according  to  Plateau,  is  when  the  length  of  the  cylinder  is 
equal  to  its  circumference,  that  is  to  say,  when  L  =  Inr,  or  when  the 
ratio  of  length  to  diameter  is  represented  by  -n. 

The  fact  is  that  any  small  disturbance  takes  the  form  of  a  wave, 
and  travels  along  the  cylinder.  Short  waves  do  not  affect  the 
stability  of  the  system ;  but  waves  whose  length  exceeds  that  of  the 
circumference  tend  to  grow  in  amplitude:  until,  contracting  here, 
expanding  there,  the  cylinder  turns  into  a  pronounced  unduloid, 
and  soon  breaks  into  two  parts  or  more.     Thus  the  cylinder  is  a 


380  THE  FORMS  OF  CELLS  [ch. 

stable  figure  until  it  becomes  longer  than  its  own  circumference, 
and  then  the  risk  of  rupture  may  be  said  to  begin.  But  Rayleigh 
shewed  that  still  longer  waves,  leading  to  still  greater  instability, 
are  needed  to  break  down  material  resistance*.  For,  as  Plateau 
knew  well,  his  was  a  theoretical  result,  to  be  departed  from  under 
material  conditions;  it  is  affected  largely  by  viscosity,  and,  as  in 
the  case  of  a  flowing  cylinder  or  jet,  by  inertia.  When  inertia  plays 
a  leading  part,  viscosity  being  small,  the  node  of  maximum  in- 
stability corresponds  to  nearly  half  as  much  again  as  in  the  simple 
or  theoretical  case:  and  this  result  is  very  near  to  what  Plateau 
himself  had  deduced  from  Savart's  experiments  on  jets  of  water "j". 
When  the  fluid  is  external  >(as  when  the  cyhnder  is  of  air)  the  wave- 
length of  maximal  instability  is  longer  still.  Lastly,  when  viscosity 
is  very  large,  and  becomes  paramount,  then  the  wave-length  between 
regions  of  maximal  instability  may  become  very  long  indeed:  so 
that  (as  Rayleigh  put  it)  ''long  (viscid)  threads  do  not  tend  to  divide 
themselves  into  drops  at  mutual  distances  comparable  with  the 
diameter  of  the  cylinder,  but  rather  to  give  way  by  attenuation  at 
few  and  distant  places."  It  is  this  that  renders  possible  the  making 
of  long  glass  tubes,  or  the  spinning  of  threads  of  "viscose"  and  like 
materials;  but  while  these  latter  preserve  their  continuity,  the 
principle  of  Plateau  tends  to  give  them  something  of  a  wavy, 
unduloid  surface,  to  the  great  enhancement  of  their  beauty.  We 
are  prepared,  then,  to  find  that  such  cylinders  and  unduloids  as 
occur  in  organic  nature  seldom  approach  in  regularity  to  those  which 
theory  prescribes  or  a  soap-film  may  be  made  to  shew;  but  rather 
exhibit  all  manner  of  gradations,  from  something  exquisitely  neat 
and  regular  to  a  coarse  and  distant  approximation  to  the  ideal 
thing  {. 

The  unduloid  has  certain  peculiar  properties  as  regards  its  limita- 
tions of  stabihty,  but  we  need  mention  two  facts  only:  (1)  that 
when  the  unduloid,  which  we  produce  with  our  soap-bubble  or  our 

*  Rayleigh,  On  the  instability  of  fluid  surfaces,  Sci.  Papers,  iii,  p.  594. 

t  Cf.  E.  Tylor,  Phil.  J^ag.  xvi,  pp.  504-518,  1933. 

X  Cf.  F.  Savart,  Sur  la  constitution  des  veines  liquides  lancees  par  des  orifices, 
etc.,  Ann.  de  Chimie,  liii,  pp.  337-386,  1833.  Rayleigh,  On  the  instability  of 
a  cylinder  of  viscous  liquid,  etc.,  Phil.  Mag.  (5),  xxxiv,  1892,  or  Sci.  Papers,  i, 
p.  361.  See  also  Larmor,  On  the  nature  of  viscid  fluid  threads.  Nature,  July  11, 
1936,  p.  74. 


V]  OF  FIGURES  OF  EQUILIBRIUM  381 

oil-globule,  consists  of  the  figure  containing  a  complete  constriction, 
then  it  has  somewhat  wide  limits  of  stabiHty;  but  (2)  if  it  contain 
the  swollen  portion,  then  equilibrium  is  limited  to  the  case  of  the 
figure  consisting  of  one  complete  unduloid,  no  less  nor  more;  that 
is  to  say  when  the  ends  of  the  figure  are  constituted  by  the  narrowest 
portions,  and  its  middle  by  the  widest  portion  of  the  entire  curve. 
The  theoretical  proof  of  this  is  difficult ;  but  if  we  take  the  proof  for 
granted,  the  fact  itself  will  serve  to  throw  light  on  what  we  have 
learned  regarding  the  stabihty  of  the  cyhnder.  For,  when  we 
remember  that  the  meridional  section  of  our  unduloid  is  generated 
by  the  rolhng  of  an  ellipse  upon  a  straight  line  in  its  own  plane, 
we  easily  see  that  the  length  of  the  entire  unduloid  is  equal  to  the 
circumference  of  the  generating  ellipse.  As  the  unduloid  becomes 
less  and  less  sinuous  in  outUne  it  approaches,  and  in  time  reaches, 
the  form  of  the  cylinder,  as  a  "Hmiting  case";  and  jpari  passu,  the 
ellipse  which  generated  it  passes  into  a  circle,  as  its  foci  come  closer 
and  closer  together.  The  cyhnder  of  a  length  equal  to  the  circum- 
ference of  its  generating  circle  is  homologous  to  an  unduloid  whose 
length  is  equal  to  the  circumference  of  its  generating  elhpse;  and 
this  is  just  what  we  recognise  as  constituting  one  complete  segment 
of  the  unduloid. 

The  cylinder  turns  so  easily  into  an  unduloid,  and  the  unduloid 
is  capable  of  assuming  so  many  graded  differences  of  form,  that  we 
may  expect  to  find  it  abundantly  and  variously  represented  among 
the  simpler  living  things.  For  the  same  reason  it  is  the  very 
stand-by  of  the  glass-blower,  whose  flasks  and  bottles  are,  of 
necessity,  unduloids*.  The  blown-glass  bottle  is  a  true  unduloid, 
and  the  potter's  vase  a  close  approach  to  an  unduloid;  but  the 
alabaster  bottle,  turned  on  the  lathe,  is  another  story.  It  may  be 
an  imitation,  or  a  reminiscence,  of  the  potter's  or  the  glass-blower's 
work;   but  it  is  no  unduloid  nor  any  surface  of  minimal  area  at  all. 

The  catenoid,  as  we  have  seen,  is  a  surface  of  zero  pressure,  and  as 
such  is  unlikely  to  form  part  (unless  momentarily)  of  the  closed 
boundary  of  a  cell.  It  forms  a  limiting  case  between  unduloid  and 
nodoid,  and,  were  it  realised,  it  would  seldom  be  visibly  different  from 
the  other  two.     In  Trichodina  pediculus,  a  minute  infusorian  para- 

*  Unless,  that  is  to  say,  their  shape  be  cramped  and  their  mathematical  beauty 
annihilated,  by  compression  in  a  mould. 


382  THE  FORMS  OF  CELLS  [ch. 

site  of  the  freshwater  polype,   we  have  a  circular  disc   bounded 
(apparently)  by  two  parallel  rings  of  cilia,  with  a  pulley-like  groove 

between.  The  groove  looks  very  like 
that  catenoid  surface  which  we  have 
produced    from    two    parallel    and 
opposite  annuli;   and  the  fact  that 
the  lower  surface  of  the  little  creature 
is  practically  plane,  where  it  creeps 
^.^^^^^^^^^     over  the  smooth  body  of  the  Hydra, 
i\  '\\\  vvjN  ^         looks  like  confirming  the  catenoid 
Fig.  112.    Triclwdiruz  pediculus.      analogy.     But  the  upper  surface  of 

the  infusorian,  with  its  ciliated 
"gullet,"  gives  no  assurance  of  a  zero  pressure;  and  we  must 
take  it  that  the  equatorial  groove  of  Trichodina  resembles,  or 
approaches,  but  is  not  mathematically  identical  with,  a  catenoid 
surface. 

While  those  figures  of  equilibrium  which  are  also  surfaces  of 
revolution  are  only  six  in  number,  there  is  an  infinite  number  of 
other  figures  of  equilibrium,  that  is  to  say  of  surfaces  of  constant 
mean  curvature,  which  are  jiot  surfaces  of  revolution;  and  it  can 
be  shewn  mathematically  that  any  given  contour  can  be  occupied 
by  a  finite  portion  of  some  one  such  surface,  in  stable  equilibrium. 
The  experimental  verification  of  this  theorem  lies  in  the  simple  fact 
(already  noted)  that  however  we  bend  a  wire  into  a  closed  curve, 
plane  or  not  plane,  we  may  always  fill  the  entire  area  with  a  con- 
tinuous film.  No  more  interesting  problem  has  ever  been  pro- 
pounded to  mathematicians  as  the  outcome  of  experiment  than  the 
general  problem  so  to  describe  a  minimal  surface  passing  through 
a  closed  contour;  and  no  complete  solution,  no  general  method  of 
approach,  has  yet  been  discovered*. 

Of  the  regular  figures  of  equihbrium,  or  surfaces  of  constant  mean 
curvature,  apart  from  the  surfaces  of  revolution  which  we  have 
discussed,  the  helicoid  spiral  is  the  most  interesting  to  the  biologist. 

*  Partial  solutions,  closely  connected  with  recent  developments  of  mathematical 
analysis,  are  due  to  Riemann,  Weierstrass  and  Schartz.  Cf.  (int.  al.)  G.  Darboux, 
Theorie  des  surfaces,  1914,  pp.  490-601;  T.  Bonneson,  Problemes  des  i^operimetres 
et  des  iaipipJianes,  Paris,  1929;  Hilbert's  AnschauUche  Geometrie,  1932,  p.  237  seq.; 
a  good  account  also  in  G.  A.  Bliss's  Calculus  of  Variations,  Chicago,  1925.  See  also 
(int.  al.)  Tibor  Rado,  Mathem.  Ztschr.  xxxir,  1930;  -Jesse  Douglas,  Amer.  Math. 
Journ.  XXXIII,  1931,  Journ.  Math.  Phys.  xv,  1936. 


V]  OF  FIGURES  OF  EQUILIBRIUM  383 

This  is  a  helicoid  generated  by  a  straight  Hne  perpendicular  to  an 
axis,  about  which  it  turns  at  a  uniform  rate,  while  at  the  same  time 
it  slides,  also  uniformly,  along  this  same  axis.  At  any  point  in  this 
surface,  the  curvatures  are  equal  and  of  opposite  sign,  and  the  sum 
of  the  curvatures  is  accordingly  nil.  Among  what  are  called  "ruled 
surfaces,"  or  surfaces  capable  of  being  defined  by  a  system  of 
stretched  strings*,  the  plane  and  the  hehcoid  are  the  only  two  whose 
mean  curvature  is  null,  while  the  cyhnder  is  the  only  one  whose 
curvature  is  finite  and  constant.  As  this  simplest  of  hehcoids 
corresponds,  in  three  dimensions,  to  what  in  two  dimensions  is 
merely  a  plane  (the  latter  being  generated  by  the  rotation  of  a 
straight  line  about  an  axis  without  the  superadded  gliding  motion 
which  generates  the  hehcoid),  so  there  are  other  and  much  more 
complicated  helicoids  which  correspond  to  the  sphere,  the  unduloid 
and  the  rest  of  our  figures  of  revolution,  the  generating  planes  of 
these  latter  being  supposed  to  wind  spirally  about  an  axis.  In  the 
case  of  the  cyhnder  it  is  obvious  that  the  resulting  figure  is  indis- 
tinguishable from  the  cyhnder  itself.  In  the  case  of  the  unduloid 
we  obtain  a  grooved  spiral,  and  we  meet  with  something  very  like 
it  in  nature  (for  instance  in  Spirochaetes,  Bodo  gracilis,  etc.) ;  but  in 
point  of  fact,  the  screw  motion  given  to  an  unduloid  or  catenary 
curve  fails  to  give  a  minimal  screw  surface,  as  we  might  have 
expected  it  to  do. 

The  foregoing  considerations  deal  with  a  small  part  only  of  the 
theory  of  surface-tension,  or  capillarity:  with  that  part,  namely, 
which  relates  to  the  surfaces  capable  of  subsisting  in  equilibrium 
under  the  action  of  that  force,  either  of  itself  or  subject  to  certain 
simple  constraints.  And  as  yet  we  have  limited  ourselves  to  the 
case  of  a  single  surface,  or  of  a  single  drop  or  bubble,  leaving  to 
another  occasion  a  discussion  of  the  forms  assumed  when  such  drops 
or  vesicles  meet  and  combine  together.  Ip  short,  what  we  have 
said  may  help  us  to  understand  the  form  of  a  cell — considered,  as 
with  certain  hmitations  we  may  legitimately  consider  it,  as  a  Hquid 
drop  or  liquid  vesicle ;  the  conformation  of  a  tissue  or  cell-aggregate 
must  be  dealt  with  in  the  light  of  another  series  of  theoretical  con- 
siderations. In  both  cases,  we  can  do  no  more  than  touch  on  the 
fringe   of  a   large   and   difficult   subject.     There   are   many  forms 

*  Or  rather,  surfaces  such  that  through  every  point  there  runs  a  straight  line 
which  lies  wholly  in  the  surface. 


384  THE  FORMS  OF  CELLS  [ch. 

capable  of  realisation  under  surface-tension,  and  many  of  them 
doubtless  to  be  recognised  among  organisms,  which  we  cannot 
deal  with  in  this  elementary  account.  The  subject  is  a  very 
general  one;  it  is,  in  its  essence,  more  mathematical  than  physical; 
it  is  part  of  the  mathematics  of  surfaces,  and  only  comes  into  relation 
with  surface-tension  because  this  physical  phenomenon  illustrates 
and  exemplifies,  in  a  concrete  way,  the  simple  an4  symmetrical 
conditions  with  which  the  mathematical  theory  is  capable  of  deahng. 
And  before  we  pass  to  illustrate  the  physical  phenomena  by  biological 
examples,  we  must  repeat  that  the  simple  physical  conditions  which 
we  presuppose  will  never  be  wholly  realised  in  the  organic  cell. 
Its  substance  will  never  be  a  perfect  fluid,  and  hence  equilibrium 
will  be  slowly  reached;  its  surface  will  seldom  be  perfectly  homo- 
geneous, and  therefore  equilibrium  will  seldom  be  perfectly  attained ; 
it  will  very  often,  or  generally,  be  the  seat  of  other  forces,  symmetrical 
or  unsymmetrical ;  and  all  these  causes  will  more  or  less  perturb  the 
surface-tension  effects*.  But  we  shall  find  that,  on  the  whole,  these 
effects  of  surface-tension  though  modified  are  not  obliterated  nor 
even  masked;  and  accordingly  the  phenomena  to  which  I  have 
devoted  the  foregoing  pages  will  be  found  manifestly  recurring  and 
repeating  themselves  among  the  phenomena  of  the  organic  cell. 

In  a  spider's  web  we  find  exemplified  several  of  the  principles  of 
surface-tension  which  we  have  now  explained.  The  thread  is  spun 
out  of  a  glandular  secretion  which  issues  from  the  spider's  body  as 
a  semi-fluid  cyHnder,  the  force  of  expulsion  giving  it  its  length  and 
that'  of  surface-tension  giving  it  its  circular  section.  It  is  too  viscid, 
and  too  soon  hardened  on  exposure  to  the  air,  to  break  up  into  drops 
or  spherules ;  but  it  is  otherwise  with  another  sticky  secretion  which, 
coming  from   another  gland,   is  simultaneously  poured   over  the 

*  That  "every  particular  that  worketh  any  effect  is  a  thing  compounded  more 
or  less  of  diverse  single  natures,  more  manifest  and  more  obscure"  is  a  point  made 
and  dwelt  on  by  Bacon.  Of  the  same  principle  a  great  astronomer  speaks  as 
follows:  "It  is  one  of  the  fundamental  characteristics  of  natural  science  that  we 
never  get  beyond  an  approximation. .  .Nature  never  offers  us  simple  and  undivided 
phenomena  to  observe,  but  always  infinitely  complex  compounds  of  many  different 
phenomena.  Each  single  phenomenon  can  be  described  mathematically  in  terms 
of  the  accepted  fundamental  laws  of  Nature : . . .  but  we  can  never  be  sure  that  we 
have  carried  the  analysis  to  its  full  exhaustion,  and  have  isolated  one  single  simple 
phenomenon."     W.  de  Sitter,  in  Nature,  Jan.  21,  1928,  p.  99. 


V]  OF  SPIDEKS'  WEBS  385 

slacker  cross-threads  as  they  issue  to  form  the  spiral  portion  of  the 
web.  This  latter  secretion  is  more  fluid  than  the  first,  and  only 
dries  up  after  several  hours*.  By  capillarity  it  "wets"  the  thread, 
spreading  over  it  in  an  even  film  or  liquid  cylinder.  As  such  it 
has  its  limits  of  stabihty,  and  tends  to  disrupt  at  points  more 
distant  than  the  theoretical  wave-length,  owing  to  the  imperfect 
fluidity  of  the  viscous  film  and  still  more  to  the  frictional  drag  of 
the  inner  thread  with  which  it  is  in  contact.  Save  for  this  qualifi- 
cation the  cyhnder  disrupts  in  the  usual  manner,  passing  first  into 
the  wavy  outhne  of  an  unduloid,  whose  swollen  internodes  swell 
more  and  more  till  the  necks  between  them  break  asunder,  and  leave 
a  row  of  spherical  drops  or  beads  strung  like  dewdrops  at  regular 
intervals  along  the  thread.  If  we  try  to  varnish  a  thin  taut  wire 
we  produce  automatically  the  same  identical  result  t;  imless  our 
varnish  be  such  as  to  dry  almost  instantaneously  it  gathers  into 
beads,  and  do  what  we  will  we  fail  to  spread  it  smooth.  It  follows 
that,  according  to  the  drying  quahties  of  our  varnish,  the  process 
may  stop  at  any  point  short  of  the  formation  of  perfect  spherules; 
and  as  our  final  stage  we  may  only  obtain  half-formed  beads  or  the 
wavy  outlines  of  an  unduloid.  The  beads  may  be  helped  to  form 
by  jerking  the  stretched  thread,  and  so  disturbing  the  unstable 
equilibrium  of  the  viscid  cyhnder.  This  the  spider  has  been  said 
to  do,  but  Dr  G.  T.  Bennett  assures  me  that  she  does  nothing  of  the 
kind.  She  only  draws  her  thread  out  a  little,  and  leaves  it  a  trifle 
slack ;  if  the  gum  should  break  into  droplets,  well,  and  good,  but  it 
matters  httle.  The  web  with  its  sticky  threads  is  not  improved 
thereby.     Another  curious  phenomenon  here  presents  itself. 

In  Plateau's  experimental  separation  of  a  cyhnder  of  oil  into  two 
spherical  halves,  it  was  noticed  that,  when  contact  was  nearly 
broken,  that  is  to  say  when  the  narrow  neck  of  the  unduloid  had 
become  very  thin,  the  two  spherical  bullae,  instead  of  absorbing 
the  fluid  out  of  the  narrow  neck  into  themselves  as  they  had  done 
with  the  preceding  portion,  drew  out  this  small  remaining  part  of 

*  When  we  see  a  web  bespangled  with  dew  of  a  morning,  the  dewdrops  are  not 
drops  of  pure  water,  but  of  water  mixed  with  the  sticky,  gummy  fluid  of  the  cross- 
threads  ;  the  radii  seldom  if  ever  shew  dewdrops.  See  F,  Strehlke,  Beobachtungen 
an  Spinnengewebe,  Poggendorff's  Annalen,  XL,  p.  146,  1937. 

t  Felix  Plateau  recommends  the  use  of  a  weighted  thread  or  plumb-line,  to  be 
drawn  up  slowly  out  of  a  jar  of  water  or  oil;  Phil.  Mag.  xxxiv,  p.  246,  1867. 


386  THE  FORMS  OF  CELLS  [ch. 

the  liquid  into  a  thin  thread  as  they  completed  their  spherical  f^rm 
and  receded  from  one  another:  the  reason  being  that,  after  the 
thread  or  "neck"  has  reached  a  certain  tenuity,  internal  friction 
prevents  or  retards  a  rapid  exit  of  the  fluid  from  the  thread  to  the 
adjacent  spherule.  It  is  for  the  same  reason  that  we  are  able  to 
draw  a  glass  rod  or  tube,  which  we  have  heated  in  the  middle,  into 
a  long  and  uniform  cyhnder  or  thread  by  quickly  separating  the 
two  ends.  But  in  the  case  of  the  glass  rod  the  long  thin  thread 
quickly  cools  and  sohdifies,  while  in  the  ordinary  separation  of  a 
liquid  cylinder  the  corresponding  intermediate  cylinder  remains 
liquid;   and  therefore,  Hke  any  other  liquid  cylinder,  it  is  liable  to 


Fig.  113.     Dew-drops  on  a  spider's  web. 

break  up,  provided  that  its  dimensions  exceed  the  limit  of  stability. 
And  its  length  is  generally  such  that  it  breaks  at  two  points,  thus 
leaving  two  terminal  portions  continuous  and  confluent  with  the 
spheres,  and  one  median  portion  w^hich  resolves  itself  into  .a 
tiny  spherical  drop,  midway  between  the  original  and  larger  two. 
Occasionally,  the  same  process  of  formation  of  a  connecting  thread 
repeats  itself  a  second  time,  between  the  small  intermediate  spherule 
and  the  large  spheres;  and  in  this  case  we  obtain  two  additional 
spherules,  still  smaller  in  size,  and  lying  one  on  either  side  of  our 
first  little  one.  This  whole  phenomenon,  of  equal  and  regularly 
interspaced  beads,  often  with  little  beads  regularly  interspaced 
between  the  larger  ones,  and  now  and  then  with  a  third  order  of 
still  smaller  beads  regularly  intercalated,  may  be  easily  observed 
in  a  spider's  web,  such  as  that  of  Epeira,  very  often  with  beautiful 
regularity — sometimes  interrupted  and  disturbed  by  a  sKght  want 
of  homogeneity  in  the  secreted  fluid ;   and  the  same  phenomenon  is 


V]  OF  BEADS  OR  GLOBULES  387 

repeated  on  a  grosser  scale  when  the  web  is  bespangled  with  dew, 
and  its  threads  bestrung  with  pearls  innumerable.  To  the  older 
naturalists,  these  regularly  arranged  and  beautifully  formed  globules 
on  the  spider's  -web  were  a  frequent  source  of  wonderment.  Black- 
wall,  counting  some  twenty  globules  in  a  tenth  of  an  inch,  calculated 
that  a  large  garden-spider's  web  should  comprise  about  120,000 
globules;  the  net  was  spun  and  finished  in  about  forty  minutes, 
and  Blackwall  was  filled  with  admiration  of  the  skill  and  quickness 
with  which  the  spider  manufactured  these  little  beads.  And  no 
wonder,  for  according  to  the  above  estimate  they  had  to  be  made 
at  the  rate  of  about  50  per  second*. 

Here  we  see  exemphfied  what  Plateau  told  us  of  the  law  of  minimal 
areas  transforming  the  cylinder  into  the  unduloid  and  disrupting  it 


Fig.  114.     Root-hair  of  Trianea,  in  glycerine.     After  Berthold. 

into  spheres.  The  httle  dehcate  beads  which  stud,  the  long  thin 
pseudopodia.o.f  a  foraminifer,  such  as  Gromia,  or  which  appear  in 
like  manner  on  the  film  of  protoplasm  coating  the  long  radiating 
spicules  of  Globigerina,  represent  an  identical  phenomenon.  Indeed 
we  may  study  in  a  protoplasmic  filament  the  whole  process  of 
formation  of  such  beads:  if  we  squeeze  out  on  a  shde  the  viscid 
contents  of  a  mistletoe-berry,  the  long  sticky  threads  into  which  the 
substance  runs  shew  the  whole  phenomenon  particularly  well.  True, 
many  long  cylindrical  cells,  such  as  are  common  in  plants,  shew  no 
sign  of  beading  or  disruption ;  but  here  .the  cell- walls  are  never  fluid 
but  harden  as  they  grow,  and  the  protoplasm  within  is  kept  in  place 
and  shape  by  its  contact  with  the  cell-wall.  It  was  noticed  many 
years  ago  by  Hofmeisterf,  and  afterwards  explained  by  Berthold, 
that  if  we  dip  the  long  root-hairs  of  certain  water-plants,  such  as 
Hydrocharis  or  Trianea,  in  a  denser  fluid  (a  httle  sugar-solution  or 

*  J.  Blackwall,  Spiders  of  Great  Britain  (Ray  Society),  1859,  p.  10;  Trans.  Linn. 
Soc.  XVI,  p.  477,  1833.  On  the  strength  and  elasticity  of  the  spider's  web,  see 
J.  R.  Benton,  Amer.  Journ.  Science,  xxiv,  pp.  75-78,  1907. 

t  Lehrbuch  von  der  Pflanzenzelley  p.  71 ;  cf.  Nageli,  Pflanzenphysiologische  Unter- 
suchungen  [Spirogyra),  ni,  p.  10. 


388  THE  FORMS  OF  CELLS  [ch. 

dilute  glycerine),  the  cell-sap  tends  to  diffuse  outwards,  the  proto- 
plasm parts  company  with  its  surrounding  and  supporting  wall,  and 
then  hes  free  as  a  protoplasmic  cylinder  in  the  interior  of  the  cell. 
Thereupon  it  soon  shews  signs  of  instability,  and  commences  to 
disrupt;  it  tends  to  gather  into  spheres,  which  however,  as  in  our 
illustration,  may  be  prevented  by  their  narrow  quarters  from 
assuming  the  complete  spherical  form;  and  in  between  these 
spheres,  we  have  more  or  less  regularly  alternate  ones,  of  smaller 
size*.  We  could  not  wish  for  a  better  or  a  simpler  proof  of  the 
essential  fluidity  of  the  protoplasm  f.  Similar,  but  less  regular, 
beads  or  droplets  may  be  caused  to  appear,  under  stimulation  by  an 
alternating  current,  in. the  protoplasmic  threads  within  the  living 
cells  of  the  hairs  of  Tradescantia;  the  explanation  usually  given  is, 
that  the  viscosity  of  the  protoplasm  is  reduced,  or  its  fluidity 
increased ;  but  an  increase  of  the  surface-tension  would  seem  a  more 
likely  reason}. 

In  one  of  Robert  Chambers's  delicate  experiments,  a  filament  of 
protoplasm  is  drawn  off,  by  a  micro-needle,  from  the  fluid  surface 
of  a  starfish-egg.  If  drawn  too  far  it  breaks,  and  part  returns  within 
the  protoplasm  while  the  other  rounds  itself  off  on  the  needle's 
point.  If  drawn  out  less  far,  it  looks  hke  a  row  of  beads  or  chain 
of  droplets;  if  yet  more  relaxed,  the  droplets  begin  to  fuse  until 
the  whole  filament  is  withdrawn;  if  drawn  out  anew  the  process 
repeats  itself.  The  whole  story  is  a  perfect  description  of  the 
behaviour  of  a  fltiid  jet  or  cy finder,  of  varying  length  and 
thickness  §. 

We  may  take  note  here  of  a  remarkable  series  of  phenomena, 
which,  though  they  seem  at  first  sight  to  be  of  a  very  different  order, 

*  The  intermediate  spherules  appear  with  great  regularity  and  beauty  whenever 
a  liquid  jet  breaks  up  into  drops.  So  a  bursting  soap-bubble  scatters  a  shower 
of  droplets  all  around,  sometimes  all  alike,  but  often  with  a  beautiful  alternation 
of  great  and  small.  How  the  breaking  up  of  thread  or  jet  into  drops  may  be  helped, 
regularised,  and  sometimes  complicated,  by  external  vibrations  is  another  and  by 
no  means  unimportant  story. 

t  Though  doubtless  to  speak  of  the  viscid  thread  as  a  fluid  is  but  a  first  approxi- 
mation;  cf.  Larmor,  in  Nature,  July  11,  1936. 

X  Kiihne,  Untersuchungen  ilber  das  Protoplasma,  1864,  p.  75,  etc. 

§  Cf.  R.  Chambers  in  Colloid  Chemistry,  theoretical  and  applied,  ii,  cap.  24,  1928; 
also  Ann.  de  Physiol,  vi,  p.  234,  1930;  etc. 


V]  THE  SHAPE  OF  A  SPLASH  389 

are  closely  related  to  those  which  attend  and  which  bring  about  the 
breaking-up  of  a  liquid  cylinder  or  thread. 

In  Mr  Worthington's  beautiful  experiments  on  splashes*,  it  was 
found  that  the  fall  of  a  found  pebble  into  water  from  a  height  first 
formed  a  dip  or  hollow  in  the  surface,  and  then  caused  a  filmy 
"cup"  of  water  to  rise  up  all  round,  opening  out  trumpet-fashion 


Fig.  115.     Phases  of  a  splash.     From  Worthington. 


""''^WiW^JSSM, 


''S^^s?*^ 


Fig.  116.     A  wave  breaking  into  spray. 

or  closing  in  like  a  bubble,  according  to  the  height  from  which  the 
pebble  fell.  The  cup  or  "crater"  tends  to  be  fluted  in  alternate 
ridges  and  grooves,  its  edges  get  scolloped  into  corresponding  lobes 
and  notches,  and  the  projecting  lobes  or  prominences  tend  to  break 
off  or  break  up  into  drops  or  beads  (Fig.  115).  A  similar  appearance 
is  seen  on  a  great  scale  in  the  edge  of  a  breaking  wave :  for  the  smooth 

*  A  Study  of  SplasJies,  1908,  p.  38,  etc.;    also  various  papers  in  Proc.  R.S. 
1876-1882,  and  Phil  Trans.  (A),  1897  and  1900. 


390  THE  FORMS  OF  CELLS  [ch. 

edge  becomes  notched  or  sinuous,  and  the  surface  near  by  becomes 
ribbed  or  fluted,  owing  to  the  internal  flow  being  helped  here  and 
hindered  there  by  a  viscous  shear;  and  then  all  of  a  sudden  the 
uneven  edge  shoots  out  an  array  of  tiny  jets,  which  break  up  into 
the  countless  droplets  which  constitute  "spray"  (Fig.  116).  The 
naturalist  may  be  reminded  also  of  the  beautifully  symmetrical 
notching  of  the  calycles  of  many  hydroid  zoophytes,  which  little 
cups  had  begun  their  existence  as  Hquid  or  semi-hquid  films  before 
they  became  stiff  and  rigid.  The  next  phase  of  the  splash  (with 
which  we  are  less  directly  concerned)  is  that  the  crater  subsides, 
and  where  it  stood  a  tall  column  rises  up,  which  also  tends,  if  it  be 
tall  enough,  to  break  up  into  drops.  Lastly  the  column  sinks  down 
in  its  turn,  and  a  ripple  runs  out  from  where  it  stood. 

The  edge  of  our  little  cup  forms  a  hquid  ring  or  annulus,  com- 
parable on  the  one  hand  to  the  edge  of  an  advancing  wave,  and 
on  the  other  to  a  hquid  thread  or  cylinder  if  only  we  conceive  the 
thread  to  be  bent  round  into  a  ring;  and  accordingly,  just  as  the 
thread  segments  first  into  an  unduloid  and  then  into  separate 
spherical  drops,  so  likewise  will  the  edge  of  cup  or  annulus  tend  to 
do.  This  phase  of  notching,  or  beading,  of  the  edge  of  the  splash 
is  beautifully  seen  in  many  of  Worthington's  experiments*,  and  still 
more  beautifully  in  recent  work  (Frontispiece f).  In  the  second  place 
the  fact  that  the  crater  rises  up  means  that  liquid  is  flowing  in  from 
below;  the  segmentation  of  the  rim  means  that  channels  of  easier 
flow  are  being  created,  along  which  the  liquid  is  led  or  driven  into 
the  protuberances;  and  these  last  are  thereby  exaggerated  into  the 
jets  or  streams  which  become  conspicuous  at  the  edge  of  the  crater. 
In  short  any  film  or  film-like  fluid  or  semi-fluid  cup  will  be  unstable ; 
its  instability  will  tend  to  show  itself  in  a  fluting  of  the  surface  and 
a  notching  of  the  edge;  and  just  such  a  fluting  and  notching  are 
conspicuous  features  of  many  minute  organic  cup-like  structures. 
In  the  hydroids  (Fig.  117),  we  see  that  these  common  features  of  the 

*  Cf.  A  Study  of  Splashes,  pp.  17,  77.  The  same  phenomenon  is  often  well  seen 
in  the  splash  of  an  oar.  It  is  beautifully  and  continuously  evident  when  a  strong 
jet  of  water  from  a  tap  impinges  on  a  curved  surface  and  then  shoots  off  again. 

t  We  owe  this  picture  to  the  kindness  of  Mr  Harold  E.  Edgerton,  of  the 
Massachusetts  Institute  of  Technology.  It  shews  the  splash  caused  by  a  drop 
falling  into  a  thin  layer  of  milk;  a  second  drop  of  milk  is  seen  above,  following 
the  first.     The  exposure-time  was  1/50,000  of  a  second. 


The  latter  phase  of  a  splash:  the  crater  has  subsided,  a  columnar  jet  has  risen  up, 

and  the  jet  is  dividing  into  droplets.  From  Harold  E.  Edgerton, 

Massachusetts  Institute  of  Technology 


V] 


THE  SHAPE  O'F  A  SPLASH 


391 


cup  and  the  annulation  of  the  stem  are  phenomena  of  the  same  order. 
A  cord-hke  thickening  of  the  edge  of  the  cup  is  a  variant  of  the 
same  order  of  phenomena;  it  is  due  to  the  checking  at  the  rim  of 
the  flow  of  hquid  from  below,  and  a  similar  thickening  is  to  be  seen, 
not  only  in  some  hydroid  calycles  but  also  in  many  Vorticellae 
(cf.  Fig.  124)  and  other  cup-shaped  organisms.  And  these  are  by 
no  means  the  only  manifestations  of  surface-tension  in  a  splash 
which  shew  resemblances  and  analogies  to  organic  form*. 

The  phenomena  of  an  ordinary  liquid  splash  are  so  swiftly  tran- 
sitory that  their  study  is  only  rendered  possible  by  photography: 


01  [VA~/"'^^^VY 


Fig.  117.     Calycles  of  Campanularia  spp. 

but  this  excessive  rapidity  is  not  an  essential  part  of  the  pheno- 
menon. For  instance,  we  can  repeat  and  demonstrate  many  of  the 
simpler  phenomena,  in  a  permanent  or  quasi-permanent  form,  by 
splashing  water  on  to  a  surface  of  dry  sandf,  or  by  firing  a  bullet 
into  a  soft  metal  target.  There  is  nothing,  then,  to  prevent  a  slow 
and  lasting  manifestation,  in  a  viscous  medium  such  as  a  proto- 
plasmic organism,  of  phenomena  which  appear  and  disappear  with 

*  The.  same  phenomena  are  modified  in  various  ways,  and  the  drops  are  given 
off  much  more  freely,  when  the  splash  takes  place  in  an  electric  field — all  owing 
to  the  general  instability  of  an  electrified  liquid -surface;  and  a  study  of  this  aspect 
of  the  subject  might  suggest  yet  more  analogies  with  organic  form.  Cf.  J.  Zeleny, 
Phys.  Rev.  x,  1917;   J.  P.  Gott,  Proc.  Cambridge  Philos.  Soc.  xxxi,  1935;   etc. 

t  We  find  now  and  then  in  certain  brick-clays  of  glacial  origin,  hard,  quoit- 
shaped  rings,  each  with  an  equally  indurated,  round  or  flattened  ball  resting  on  it. 
These  may  be  precisely  imitated  by  splashing  large  drops  of  water  on  a  smooth 
surface  of  fine  dry  sand.  The  ring  corresponds,  apparently,  to  the  crater  of  the 
splash,  and  the  ball  (or  its  water  content)  to  the  pillar  rising  in  the  middle. 


392  THE  FORMS  OF  CELLS  [ch. 

evanescent  rapidity  in  a  more  mobile  liquid.  Nor  is  there  anything 
pecuhar  in  the  splash  itself;  it  is  simply  a  convenient  method  of 
setting  up  certain  motions  or  currents,  and  producing  certain  surface- 
forms,  in  a  Hquid  medium — or  even  in  such  an  imperfect  fluid  as  a  bed 
of  sand.  Accordingly,  we  have  a  large  range  of  possible  conditions 
under  which  the  organism  might  conceivably  display  configurations 
analogous  to,  or  identical  with,  those  which  Mr  Worthington  has 
shewn  us  how  to  exhibit  by  one  particular  experimental  method. 

To  one  who  has  watched  the  potter  at  his  wheel,  it  is  plain  that 
the  potter's  thumb,  like  the  glass-blower's  blast  of  air,  depends  for 
its  efficacy  upon  the  physical  properties  of  the  clay  or  "shp"  it 
works  on,  which  for  the  time  being  is  essentially  a  fluid.  The  cup 
and  the  saucer,  like  the  tube  and  the  bulb,  display  (in  their  simple 
and  primitive  forms)  beautiful  surfaces  of  equilibrium  as  manifested 
under  certain  hmiting  conditions.  They  are  neither  more  nor  less 
than  glorified  "splashes,"  formed  slowly,  under  conditions  of 
restraint  which  enhance  or  reveal  their  mathematical  symmetry. 
We  have  seen,  and  we  shall  see  again  before  we  are  done,  that  the 
art  of  the  glass-blower  is  full  of  lessons  for  the  naturahst  as  also 
for  the  physicist:  illustrating  as  it  does  the  development  of  a  host 
of  mathematical  configurations  and  organic  conformations  which 
depend  essentially  on  the  establishment  of  a  constant  and  uniform 
pressure  within  a  closed  elastic  shell  or  fluid  envelope  or  bubble. 
In  hke  manner  the  potter's  art  illustrates  the  somewhat  obscurer 
and  more  complex  problems  (scarcely  less  frequent  in  biology)  of  a 
figure  of  equilibrium  which  is  an  open  surface  of  revolution.  The 
two  series  of  problems  are  closely  akin;  for  the  glass-blower  can 
make  most  things  which  the  potter  makes,  by  cutting  off  portions 
of  his  hollow  ware;  besides,  when  this  fails  and  the  glass-blower, 
ceasing  to  blow,  begins  to  use  his  rod  to  trim  the  sides  or  turn  the 
edges  of  wineglass  or  of  beaker,  he  is  merely  borrowing  a  trick  from 
the  still  older  craft  of  the  potter. 

It  would  seem  venturesome  to  extend  our  comparison  with  these 
liquid  surface-tension  phenomena  from  the  cup  or  calycleof  the 
hydrozoon  to  the  little  hydroid  polyp  within:  and  yet  there  is 
something  to  be  learned  by  such  a  comparison.  The  cylindrical 
body  of  the  tiny  polyp,  the  jet-hke  row  of  tentacles,  the  beaded 


V]  THE  SPLASH  AND  THE  BUBBLE  393 

annulations  which  these  tentacles  exhibit,  the  web-like  film  which 
sometimes  (when  they  stand  a  httle  way  apart)  conjoins  their  bases, 
the  thin  annular  film  of  tissue  which  surrounds  the  little  organism's 
mouth,  and  the  manner  in  which  this  annular  "peristome"  con- 
tracts*, like  a  shrinking  soap-bubble,  to  close  the  aperture,  are 
every  one  of  them  features  to  which  we  may  find  a  singular  and 
striking  parallel  in  the  surface-tension  phenomena  of  the  splash "f". 

Some  seventy  years  ago  much  interest  was  aroused  by  Helmholtz's 
work  (and  also  Kirchhoff's)  on  "discontinuous  motions  of  a  fluidj"; 
that  is  to  say,  on  the  movements  of  one  body  of  fluid  within  another, 
and  the  resulting  phenomena  due  to  friction  at  the  surfaces  between. 
What  Kelvin§  called  Helmholtz's  "admirable  discovery  of  the  law 
of  vortex-motion  in 'a  perfect  fluid"  was  the  chief  result  of  this 
investigation;  and  was  followed  by  much  experimental  work,  in 
order  to  illustrate  and  to  extend  the  mathematical  conclusions. 

The  drop,  the  bubble  and  the  splash  are  parts  of  a  long  story; 
and  a  "falling  drop,"  or  a  drop  moving  through  surrounding  fluid, 
is  a  case  deserving  to  be  considered.  A  drop  of  water,  tinged  with 
fuchsin,  is  gently  released  (under  a  pressure  of  a  couple  of  milh- 
metres)  at  the  bottom  of  a  glass  of  water  || .  Its  momentum  enables 
it  to  rise  through  a  few  centimetres  of  the  surrounding  water,  and 
in  doing  so  it  communicates  motion  to  the  water  around.  In  front 
the  rising  drop  thrusts  its  way  through,  almost  like  a  sohd  body; 
behind  it  tends  to  drag  the  surrounding  water  after  it,  by  fluid 
friction^ ;  and  these  two  motions  together  give  rise  to  beautiful  vorti- 
coid  configurations,  the  Strdmungspilze  or  Tintenpilze  of  their  first 
discoverers  (Fig.  119).     Under  a  higher  and  more  continuous  pressure 


*  See  a  Study  of  Splashes,  p.  54. 

t  There  is  little  or  no  difference  between  a  splash  and  a  burst  bubble.  The  craters 
of  the  moon  have  been  compared  with,  and  explained  by,  both  of  these. 

X  Helmholtz,  in  Berlin.  Monatsber.  1868,  pp.  215-228;  Kirchhoflf,  in  CreUe'a 
Journal,  lxx,  pp.  289-298,  lxxi,  237-273,  1869-70. 

§  W.  Thomson,  in  Proc.  R.S.E.  vi,  p.  94,  1867. 

II  See  A.  Overbeck,  Ucber  discontinuirliche  Fliissigkeitsbewegungen,  Wiedemann'' s 
Annalen,  ii,  1877;  W.  Bezold,  Ueber  Stromungsfiguren  in  Fliissigkeiten,  ibid. 
XXIV,  pp.  569-593,  1885;   P.  Czermak,  ibid,  l,  p.  329,  1893;   etc. 

%  The  frictional  drag  on  the  hinder  part  of  the  drop  is  felt  alike  in  the  ship,  the 
bird  and  the  aeroplane,  and  tends  to  produce  retarding  vortices  in  them  all.  It  is 
always  minimised  in  one  way  or  another,  and  it  is  autoraaticaUy  minimised  in  the 
present  instance,  as  the  drop  thins  off  and  tapers  down. 


394 


THE  FORMS  OF  CELLS 


[CH. 


__Md3:::,,^^ 


Fig.  118.     a,  b.   More  phases  of  a  splash,  after  Worthington. 
c.    A  hydroid  polype,  after  Allman. 


o 


^t 


Fip.  119.     -Liiquid  jets.     From  A.  Overbeck. 


V]  OF  FALLING  DROPS  395 

the  drop  becomes  a  jet;  the,  form  of  the  vortex  is  modified  thereby, 
and  may  be  further  modified  by  sHght  differences  of  temperature 
(i.e.  of  density),  or  by  interrupting  the  rate  of  flow.  To  let  a  drop 
of  ink  fall  into  water  is  a  simple  and  most  beautiful  experiment*. 
The  effect  is  more  violent  than  in  the  former  case.    The  descending 


Fig.  120.     Falling  drops.     A,  ink  in  water,  after  J.  J.  Thomson  and  Newall. 
B,  fusel  oil  in  paraffin,  after  Tomlinson. 

drop  turns  into  a  complete  vortex-ring ;  it  expands  and  attenuates ; 
it  waves  about,  and  the  descending  loops  again  turn  into  incipient 
vortices  (Fig.  120). 

Lastly,  instead  of  letting  our  drop  rise  or  fall  freely,  we  may  use 
a  hanging  drop,  which,  while  it  sinks,  remains  suspended  to  the 
surface.     Thus  it  cannot  form  a    complete    annulus,  but  only  a 

*  J.  J.  Thomson  and  H.  F.  Newall,  On  the  formation  of  vortex-rings  by  drops, 
Proc.  R.S.  XXXIX,  pp.  417-436,  1885.  Emil  Hatschek,  On  forms  assumed  by  a 
gelatinising  liquid  in  various  coagulating  solutions,  ibid.  (A)  xciv,  pp.  303-316,  1918. 


396  THE  FORMS  OF  CELLS  [ch. 

partial  vortex  suspended  by  a  thread  or  column — just  as  in  Over- 
beck's  jet-experiments;  and  the  figure  so  produced,  in  either  case, 
is  closely  analogous  to  that  of  a  medusa  or  jellyfish,  with  its  bell 
or  "umbrella,"  and  its  clapper  or  "manubrium"  as  well.  Some 
years  ago  Emil  Hatschek  made  such  vortex-drops  as  these  of  hquid 
gelatine  dropped  into  a  hardening  fluid.  These  "  artificial  medusae  " 
sometimes  show  a  symmetrical  pattern  of  radial  "ribs",  due  to 
shrinkage,  and  this  to  dehydration  by  the  coagulating  fluid.     An 


Fig.  121.     Various  medusoids:    1,  Syncoryne;    2,  Cordylophora; 
3,  Cladonema  (after  Allmaii). 

extremely  curious  result  of  Hatschek's  experiments  is  to  shew  how 
sensitive  these  vorticoid  drops  are  to  physical  conditions.  For  using 
the  same  gelatine  all  the  while,  and  merely  varying  the  density  of 
the  fluid  in  the  third  decimal  place,  we  obtain  a  whole  range  of 
configurations,  from  the  ordinary  hanging  drop  to  the  same  with  a 
ribbed  pattern,  and  then  to  medusoid  vortices  of  various  graded  forms. 
The  living  medusa  has  a  geometrical  symmetry  so  marked  and  regular 
as  to  suggest  a  physical  or  mechanical  element  in  the  little  creature's 
growth  and  construction.  It  has,  to  begin  with,  its  vortex-hke  bell 
or  umbrella,  with  its  cylindrical  handle  or  manubrium.     The  bell  is 


OF  VORTICOID  OR  MEDUSOID  DROPS 


397 


traversed  by  radial  canals,  four  or  in  multiples  of  four;  its  edge  is 
beset  with  tentacles,  smooth  or  often  beaded,  at  regular  intervals 
and  of  graded  sizes ;  and  certain  sensory  structures,  including  sohd 
concretions  or  "otohths,"  are  also  symmetrically  interspaced.  No 
sooner  made,  than  it  begins  to  pulsate ;  the  little  bell  begins  to  "  ring." 


m 


& 


Fig.  1216.     "Medusoid  drops",  of  gelatin.     After  Hatschek, 

Buds,  miniature  replicas  of  the  parent-organism,  are  very  apt  to  appear 
on  the  tentacles,  or  on  the  manubrium  or  sometimes  on  the  edge  of  the 
bell;  we  seem  to  see  one  vortex  producing  others  before  our  eyes. 
The  development  of  a  medusoid  deserves  to  be  studied  without 
prejudice,  from  this  point  of  view.  Certain  it  is  that  the  tiny 
medusoids  of  Obelia,  for  instance,  are ,  budded  off  with  a  rapidity 
and  a  complete  perfection  which  suggest  an  automatic  and  all  but 
instantaneous  act  of  conformation,  rather  than  a  gradual  process  of 
growth. 

Moreover,  not  only  do  we  recognise  in  a  vorti- 
coid  drop  a  "schema"  or  analogue  of  medusoid 
form,  but  we  seem  able  to  discover  -various  actual 
phases  of  the  splash  ot  drop  in  the  all  but  in- 
numerable living  types  of  jellyfish;  in  Cladoneyna 
we  seem  to  see  an  early  stage  of  a  breaking  drop, 
and  in  Cordylophom  a  beautiful  picture  of  incipient 
vortices.  It  is  hard  indeed  to  say  how  much  or  little  all  these 
analogies  imply.  But  they  indicate,  at  the  very  least,  how  certain 
simple  organic  forms  might  be  naturally  assumed  by  one  fluid  mass 
within  another,  when  gravity,  surface  tension  and  fluid  friction  play 


Fig.  122.  Meckbsach- 
loris,  a  ciliate 
infusoiia. 


398  THE  FORMS  OF  CELLS  [ch. 

their  part,  under  balanced  conditions  of  temperature,  density  and 
chemical  composition. 

A  little  green  infusorian  from  the  Baltic  Sea  is,  as  near  as  may 
be,  a  medusa  in  miniature*.  It  is  curious  indeed  to  find  the  same 
medusoid,  or  as  we  may  now  call  it  vorticoid,  configuration  occurring 
in  a  form  so  much  lower  in  the  scale,  and  so  much  less  in  order  of 
magnitude,  than  the  ordinary  medusae. 

According  to  Plateau,  the  viscidity  of  the  liquid,  while  it 
retards  the  breaking  up  of  the  cylinder  and  increases  the  length 
of  the  segments  beyond  that  which  theory  demands,  has  never- 
theless less  influence  in  this  direction  than  we  might  have  expected. 
On  the  other  hand  any  external  support  or  adhesion,  or  mere 
contact  with  a  sohd  body,  will  be  equivalent  to  a  reduction  of 
surface-tension  and  so  will  very  greatly  increase  the  stability  of 
our  cylinder.  It  is  for  this  reason  that  the  mercury  in  our  thermo- 
meters seldom  separates  into  drops:  though  it  sometimes  does  so, 
much  to  our  inconvenience.  And  again  it  is  for  this  reason  that 
the  protoplasm  in  a  long  tubular  or  cyhndrical  cell  need  not  divide 
into  separate  cells  and  internodes  until  the  length  of  these  far 
exceeds  the  theoretical  limits. 

An  interesting  case  is  that  of  a  viscous  drop  immersed  in  another 
viscous  fluid,  and  drawn  out  into  a  thread  by  a  shearing  motion  of 
the  latter.  The  thread  seems  stable  at  first,  but  when  left  to  rest 
it  breaks  up  into  drops  of  a  very  definite  and  uniform,  size,  the  size 
of  the  drops,  or  wave-length  of  the  unduloid  of  which  they  are  made, 
depending  on  the  relative  viscosities  of  the  two  threads  f- 

Plateau's  results,  though  discovered  by  way  of  experiment  and 
though  (as  we  have  said)  they  illustrate  the  "  materiahsation "  of 
mathematical  law,  are  nevertheless  essentially  theoretical  results 
approached  rather  than  realised  in  material  systems.  That  a  hquid 
cylinder  begins  to  be  unstable  when  its  length  exceeds  ^-nr  is  all 
but  mathematically  true  of  an  all  but  immaterial  soap-bubble ;  but 
very  far  from  true,  as  Plateau  himself  was  well  aware,  in  a  flowing 
jet,  retarded  by  viscosity  and  by  inertia.  The  principle  is  true  and 
universal;    but  our  living  cylinders  do  not  follow  the  abstract  laws 

*  Medusachloris  phiale,  of  A.  Pascher,  Biol.  Centralbl.  xxxvii,  pp.  421-429,  1917. 

t  See  especially  Rayleigh,  Phil.  Mag.  xxxiv,  p.  145,  1892,  by  whom  the  subject 
is  carried  much  further  than  where  Plateau  left  it.  See  also  {int.  al.)  G.  I.  Taylor, 
Proc.  R.S.  (A),  cxLvi,  p.  501,  1934;   S.  Tomotika,  ibid,  cl,  p.  322,  1935;   etc. 


V]  OF  VISCOUS  THREADS  399 

of  mathematics,  any  more  than  do  the  drops  and  jets  of  ordinary 
fluids  or  the  quickly  drawn  and  quickly  cooling  tubes  in  the  glass- 
worker's  hands. 

Plateau  says  that  in  most  hquids  the  influence  of  viscosity  is  such 
as  to  cause  the  cylinder  to  segment  when  its  length  is  about  four 
times,  or  even  six  times,  its  diameter,  instead  of  a  fraction  over 
three  times,  as  theory  would  demand  of  a  perfect  fluid.  If  we  take 
it  at  four  times,  the  resulting  spherules  would  have  a  diameter  of 
about  1-8  times,  and  their  distance  apart  would  be  about  2-2  times, 
the  original  diameter  of  the  cylinder;  and  the  calculation  is  not 
difficult  which  would  shew  how  these  dimensions  are  altered  in  the 
case  of  a  cylinder  formed  around  a  solid  core,  as  in  the  case  of  a 
spider's  web.  Plateau  also  observed  that  the  time  taken  in  the 
division  of  the  cyUnder  is  directly  proportional  to  its  diameter, 
while  varying  with  the  nature  of  the  hquid.  This  question,  of  the 
time  taken  in  the  division  of  a  cell  or  filament  in  relation  to  its 
dimensions,  has  not  so  far  as  I  know  been  enquired  into  by  biologists. 

From  the  simple  fact  that  the  sphere  is  of  aU  configurations  that 
whose  surface-area  for  a  given  volume  is  an  absolute  minimum,  we 
have  seen  it  to  be  the  one  figure  of  equilibrium  assumed  by  a  drop 
or  vesicle  when  no  disturbing  factor  is  at  hand;  but  such  freedom 
from  counter-influences  is  likely  to  be  rare,  and  neither  does  the  rain- 
drop nor  the  round  world  itself  retain  its  primal  sphericity.  For  one 
thing,  gravity  will  always  be  at  hand  to  drag  and  distort  our  drop 
or  bubble,  unless  its  dimensions  be  so  minute  that  gravity  becomes 
insignificant  compared  with  capillarity.  Even  the  soap-bubble  will 
be  flattened  or  elongated  by  gravity,  according  as  we  support  it 
from  below  or  from  above;  and  the  bubble  which  is  thinned  out 
almost  to  blackness  will,  from  its  small  mass,  be  the  one  which 
remains  most  nearly  spherical*. 

Innumerable  new  conditions  will  be  introduced,  in  the  shape  of 
comphcated  tensions  and  pressures,,  when  one  drop  or  bubble 
becomes  associated  with  another,  and  when  a  system  of  inter- 
mediate films  or  partition-walls  is  developed  between  them.  This 
subject  we  shall  discuss  later,  in  connection  with  cell-aggregates  or 
tissues,  and  we  shall  find  that  further  theoretical  considerations  are 

*  Cf.  Dewar,  On  soap-bubbles  of  long  duration,  Proc.  Roy.  Inst.  Jan.  19,  1929. 


400  THE  FORMS  OF  CELLS  [ch. 

needed  as  a  preliminary  to  any  such  enquiry.  Meanwhile  let  us 
consider  a  few  cases  of  the  forms  of  cells,  either  sohtary,  or  in  such 
simple  aggregates  that  their  individual  form  is  httle  disturbed  thereby. 
Let  us  clearly  understand  that  the  cases  we  are  about  to  consider 
are  those  where  the  perfect  sjnnmetry  of  the  sphere  is  replaced  by 
another  symmetry,  less  complete,  such  as  that  of  an  ellipsoidal  or 
cylindrical  cell.  The  cases  of  asymmetrical  deformation  or  dis- 
placement, such  as  are  illustrated  in  the  production  of  a  bud  or 
the  development  of  a  lateral  branch,  are  much  simpler;  for  here 
we  need  only  assume  a  slight  and  locahsed  variation  of  surface- 
tension,  such  as  may  be  brought  about  in  various  ways  through 
the  heterogeneous  chemistry  of  the  cell.  But  such  diffused  and 
graded  asymmetry  as  brings  about  for  instance  the  ellipsoidal  shape 
of  a  yeast-cell  is  another  matter. 

If  the  sphere  be  the  one  surface  of  complete  symmetry  and 
therefore  of  independent  equilibrium,  it  follows  that  in  every  cell 
which  is  otherwise  conformed  there  must  be  some  definite  cause  of 
its  departure  from  sphericity;  and  if  this  cause  be  the  obvious  one 
of  resistance  offered  by  a  solidified  envelope,  such  as  an  egg-shell 
or  firm  cell-wall,  we  must  still  seek  for  the  deforming  force  which 
was  in  action  to  bring  about  the  given  shape  prior  to  the  assumption 
of  rigidity.  Such  a  cause  may  be  either  external  to,  or  may  lie 
within,  the  cell  itself.  On  the  one  hand  it  may  be  due  to  external 
pressure  or  some  form  of  mechanical  restraint,  as  when  we  submit  our 
bubble  to  the  partial  restraint  of  discs  or  rings  or  more  compHcated 
cages  of  wire ;  on  the  other  hand  it  maybe  due  to  intrinsic  causes,  which 
must  come  under  the  head  either  of  differences  of  internal  pressure, 
or  of  lack  of  homogeneity  or  isotropy  in  the  surface  or  its  envelope*. 

*  A  case  which  we  have  not  specially  considered,  but  which  may  be  found  to 
deserve  consideration  in  biology,  is  that  of  a  cell  or  drop  suspended  in  a  liquid  of 
varying  density,  for  instance  in  the  upper  layers  of  a  fluid  (e.g.  sea-water)  at  whose 
surface  condensation  is  going  on,  so  as  to  produce  a  steady  density-gradient.  In 
this  case  the  normally  spherical  drop  will  be  flattened  into  an  oval  form,  with  its 
maximum  surface-curvature  lying  at  the  level  where  the  densities  of  the  drop 
and  the  surrounding  liquid  are  just  equal.  The  sectional  outline  of  the  drop  has 
been  shewn  to  be  not  a  true  oval  or  ellipse,  but  a  somewhat  complicated  quartic 
curve.  (Rice,  Phil.  Mag.  Jan.  1915.)  A  more  general  case,  which  also  may  well 
deserve  consideration  by  the  biologist,  is  that  of  a  charged  bubble  in  (for  instance) 
a  uniform  field  of  force :  which  will  expand  or  elongate  in  the  direction  of  the  lines 
of  force,  and  become  a  spheroidal  surface  in  continuous  transformation  with  the 
original  sphere. 


V]  OF  ASYMMETRY  AND  ANISOTROPY  401 

Our  formula  of  equilibrium,  or  equation  to  an  elastic  surface,  is 
P  =  jDg  +  (TjR  +  T' jR),  where  P  is  the  internal  pressure,  jo^  any 
extraneous  pressure  normal  to  the  surface,  R,  R'  the  radii  of 
curvature  at  a  point,  and  T,  T'  the  corresponding  tensions,  normal 
to  one  another,  of  the  envelope. 

Now  in  any  given  form  which  we  seek  to  account  for,  R,  R'  are 
known  quantities;  but  all  the  other  factors  of  the  equation  are 
subject  to  enquiry.  And  somehow  or  other,  by  this  formula,  we 
must  account  for  the  form  of  any  solitary  cell  whatsoever  (provided 
always  that  it  be  not  formed  by  successive  stages  of  sohdification), 
the  cyhndrical  cell  of  Spirogyra^  the  elhpsoidal  yeast-cell,  or  (as 
we  shall  see  in  another  chapter)  even  the  egg  of  any  bird.  In 
using  this  formula  hitherto  we  have  taken  it  in  a  simphfied  form, 
that  is  to  say  we  have  made  several  limiting  assumptions.  We  have 
assumed  that  P  was  the  uniform  hydrostatic  pressure,  equal  in  all 
directions,  of  a  body  of  hquid;  we  have  assumed  likewise  that  the 
tension  T  was  due  to  surface-tension  in  a  homogeneous  hquid  film, 
and  was  therefore  equal  in  all  directions,  so  that  T  =  T' ;  and  we 
have  only  dealt  with  surfaces,  or  parts  of  a  surface,  where  extraneous 
pressure,  jo„,  was  non-existent.  Now  in  the  case  of  a  bird's  egg 
the  external  pressure  p^,  that  is  to  say  the  pressure  exercised  by 
the  walls  of  the  oviduct,  will  be  found  to  be  a  very  important 
factor;  but  in  the  case  of  the  yeast-cell  or  the  Spirogym,  wholly 
immersed  in  water,  no  such  external  pressure  comes  into  play. 
We  are  accordingly  left  in  such  cases  as  these  last  with  'two 
hypotheses,  namely  that  the  departure  from  a  spherical  form  is  due 
to  inequahties  in  the  internal  pressure  P,  or  else  to  inequahties  in 
the  tension  T,  that  is  to  say  to  a  difference  between  T  and  T'. 
In  other  words,  it  is  theoretically  possible  that  the  oval  form  of  a 
yeast-cell  is  due  to  a  greater  internal  pressure,  a  greater  '"tendency 
to  grow"  in  the  direction  of  the  longer  axis  of  the  ellipse,  or 
alternatively,  that  with  equal  and  symmetrical  tendencies  to  growth 
there  is  associated  a  difference  of  external  resistance  in  respect  of 
the  tension,  and  implicitly  the  molecular  structure,  of  the  cell- wall. 
Now'  the  former  hypothesis  is  not  impossible.  Protoplasm  is  far 
from  being  a  perfect  fluid;  it  is  the  seat  of  various  internal  forces, 
sometimes  manifestly  polar,  and  it  is  quite  possible  that  the  forces, 
osmotic  and  other,  which  lead  to  an  increase  of  the  content  of  the 


402  THE  FORMS  OF  CELLS  [ch. 

cell  and  are  manifested  in  pressure  outwardly  directed  upon  its  wall 
niay  be  unsymmetrical,  and  such  as  to  deform  what  would  otherwise 
be  a  simple  sphere.  But  while  this  hypothesis  is  not  impossible, 
it  is  not  very  easy  of  acceptance.  The  protoplasm,  though  not  a 
perfect  fluid,  has  yet  on  the  whole  the  properties  of  a  fluid ;  within 
the  small  compass  of  the  cell  there  is  httle  room  for  the  development 
of  unsymmetrical  pressures;  and  in  such  a  case  as  Spirogyra,  where 
most  part  of  the  cavity  is  filled  by  watery  sap,  the  conditions  are 
still  more  obviously,  or  more  nearly,  those  under  which  a  uniform 
hydrostatic  pressure  should  be  displayed.  But  in  variations  of  T, 
that  is  to  say  of  the  specific  surface-tension  per  unit  area,  we  have 
an  ample  field  for  all  the  various  deformations  with  which  we  shall 
have  to  deal.  Our  condition  now  is,  that  (TIR  +  T' jR)  =  a  con- 
stant; but  it  no  longer  follows,  though  it  may  still  often  be  the 
case,  that  this  will  represent  a  surface  of  absolute  minimal  area. 
As  soon  as  T  and  T'  become  unequal,  we  are  no  longer  dealing 
with  a  perfectly  hquid  surface  film;  but  its  departure  from  perfect 
fluidity  may  be  of  all  degrees,  from  that  of  a  slight  non-isotropic 
viscosity  to  the  state  of  a  firm  elastic  membrane  * ;  and  it  matters 
little  whether  this  viscosity  or  semi-rigidity  be  manifested  in  the 
self-same  layer  which  is  still  a  part  of  the  protoplasm  of  the  cell, 
or  in  a  layer  which  is  completely  differentiated  into  a  distinct  and 
separate  membrane.  As  soon  as,  by  secretion  or  adsorption,  the 
molecular  constitution  of  the  surface-layer  is  altered,  it  is  clearly 
conceivable  that  the  alteration,  or  the  secondary  chemical  changes 
which  follow  it,  may  be  such  as  to  produce  an  anisotropy,  and  to 
render  the  molecular  forces  less  capable  in  one  direction  than  another 
of  exerting  that  contractile  force  by  which  they  are  striving  to  reduce 
to  a  minimum  the  surface  area  of  the  cell.  A  slight  inequality  in 
two  opposite  directions  will  produce  the  eUipsoid  cell,  and  a  great 
inequahty  will  give  rise  to  the  cylindrical  cell. 

I  take  it  therefore,  that  the  cylindrical  cell  of  Spirogyra,  or  any 
other  cyhndrical  cell  which  grows  in  freedom  from  any  manifest 
external  restraint,  has  assumed  that  particular  form  simply  by 
reason   of  the   molecular   constitution   of  its   developing   wall  or 

*  Indeed  any  non-isotropic  stiffness,  even  though  T  remained  uniform,  would 
simulate,  and  be  indistinguishable  from,  a  condition  of  non-stiffness  and  non- 
isotropic  T. 


v]  OF  ASYMMETRY  AND  ANISOTROPY  403 

membrane;  and  that  this  molecular  constitution  was  anisotropous, 
in  such  a  way  as  to  render  extension  easier  in  one  direction  than 
another.  Such  a  lack  of  homogeneity  or  of  isotropy  in  the 
cell-wall  is  often  rendered  visible,  especially  in  plant-cells,  in  the 
form  of  concentric  lamellae,  annular  and  spiral  striations,  and  the 
like.  But  there  exists  yet  another  heterogeneity,  to  help  us  account 
for  the  long  threads,  hairs,  fibres,  cylinders,  which  are  so  often 
formed.  Carl  NageH  said  many  years  ago  that  organised  bodies, 
starch-grains,  cellulose  and  protoplasm  itself,  consisted  of  invisible 
particles,  each  an  aggregate  of  many  molecules — he  called  them 
micellae;  and  these  were  isolated,  or  "dispersed"  as  we  should  say, 
in  a  watery  medium.  This  theory  was,  to  begin  with,  an  attempt  to 
account  for  the  colloid  state;  but  at  the  same  time,  the  particles 
were  supposed  to  be  so  ordered  and  arranged  as  to  render  the 
substance  anisotropic,  to  confer  on  it  vectorial  properties  as  we  say 
nowadays,  and  so  to  account  for  the  polarisation  of  light  by  a  starch- 
grain  or  a  hair.  It  was  so  criticised  by  Biitschli  and  von  Ebner  that 
it  fell  into  disrepute,  if  not  oblivion;  but  a  great  part  of  it  was  true. 
And  the  micellar  structure  of  wool,  cotton,  silk  and  similar  substances 
is  now  rendered  clearly  visible  by  the  same  X-ray  methods  as 
revealed  the  molecular  orientation,  or  lattice-structure,  of  a  crystal 
to  von  Laue. 

It  is  now  well  known  that  the  cell-wall  has  in  many  cases  a  definite  structure 
which  depends  on  molecular  assemblages  in  the  material  of  which  it  is  com- 
posed, and  is  made  visible  by  X-rays  in  the  form  of  "diffraction  patterns". 
The  green  alga  Valonia  has  very  large  bubbly  cells,  2-3  centimetres  long,  with 
cell-walls  formed,  as  usual,  of  cellulose;  this  substance  is  a  polysaccharide, 
with  long-chain  molecules  some  500  Angstrom-units,  or  say  0-05 /x  long, 
bound  together  sideways  to  form  a  multiple  sheet  or  three-dimensional  lattice. 
In  the  cell-wall  of  Valonia  one  set  of  chains  runs  round  in  a  left-handed 
spiral,  another  forms  meridians  from  pole  to  pole,  and  these  two  layers 
are  superposed  alternately  to  build  the  wall.  Hemp  has  two  layers,  both 
running  in  right-handed  spirals;  flax  two  layers,  crossing  and  recrossing  in 
spirals  of  opposite  sign.  Even  the  cytoplasm  and  its  contents  seem  to  be 
influenced  by  molecular  ''lignes  directrices,''''  corresponding  to  the  striae  of 
the  cell-wall.  Analogous  but  still  more  complicated  results  of  molecular 
structure  are  to  be  found  in  wool,  cotton  and  other  fibres*. 

*  Cf.  R.  D.  Preston,  Phil.  Trans.  (B),  ccxxiv,  p.  131,  1934:  Preston  and  Astbury, 
Proc.  R.S.  (B),  cxxii,  pp.  76-97,  1937;  and  many  other  important  papers  by 
Astbury,  van  Iterson,  Heyn,  and  others.     We  are  brought  by  them  to  a  borderland 


404  THE  FORMS  OF  CELLS  [ch. 

But  this  phenomenon,  while  it  brings  about  a  certain  departure 
from  complete  symmetry,  is  still  compatible  with,  and  coexistent 
with,  many  of  the  phenomena  which  we  have  seen  to  be  associated 
with  surface-tension.  The  symmetry  of  tensions  still  leaves  the 
cell  a  solid  of  revolution,  and  its  surface  is  still  a  surface  of  equi- 
librium. The  fluid  pressure  within  the  cy Under  still  causes  the 
film  or  membrane  which  caps  its  ends  to  be  of  a  spherical  form. 
And  in  the  young  cell,  where  the  surface  pellicle  is  absent  or  but 
little  differentiated,  as  for  instance  in  the  oogonium  of  Aclilya  or 
in  the  young  zygospore  of  Spirogyra,  we  see  the  tendency  of  the 
entire  structure  towards  a  spherical  form  reasserting  itself:  unless, 
as  in  the  latter  case,  it  be  overcome  by  direct  compression  within 
the  cyhndrical  mother-cell.  Moreover,  in  those  cases  where  the 
adult  filament  consists  of  cylindrical  cells  we  see  that  the  young 
germinating  spore,  at  first  spherical,  very  soon  assumes  with  growth 
an  elliptical  or  ovoid  form — the  direct  result  of  an  incipient  aniso- 
tropy  of  its  envelope,  which  when  more  developed  will  convert  the 
ovoid  into  a  cyhnder.  We  may  also  notice  that  a  truly  cyhndrical 
cell  is  comparatively  rare,  for  in  many  cases  what  we  call  a 
cylindrical  cell  shews  a  distinct  bulging  of  its  sides;  it  is  not  truly 
a  cyhnder,  but  a  portion  of  a  spheroid  or  ellipsoid. 

Unicellular  organisms  in  general — protozoa,  unicellular  crypto- 
gams, various  bacteria  and  the  free  isolated  cells,  spores,  ova,  etc. 
of  higher  organisms — are  referable  for  the  most  part  to  a  small 
number  of  typical  forms ;  but  there  are  many  others  in  which  either 
no  symmetry  is  to  be  recognised,  or  in  which  the  form  is  clearly 
not  one  of  equihbrium.  Among  these  latter  we  have  Amoeba  itself 
and  all  manner  of  amoeboid  organisms,  and  also  many  curiously 
shaped  cells  such  as  the  Trypanosomes  and  various  aberrant 
Infusoria.  We  shall  return  to  the  consideration  of  these;  but  in 
the  meanwhile  it  will  suffice  to  say  (and  to  repeat)  that,  inasmuch 
as  their  surfaces  are  not  equihbrium-surfaces,  so  neither  are  the 
Uving  cells  themselves  in  any  stable  equihbrium.  On  the  contrary, 
they  are  in  continual  flux  and  movement,  each  portion  of  the 

between  chemical  and  histological  structure,  where  micellae  and  long- chain  molecules 
enlarge  and  alter  our  conceptions  not  only  of  cellulose  and  keratin,  but  of  pseudopodia 
and  cilia,  of  bone  and  muscle,  and  of  the  naked  surface  of  the  cell.  See  L.  E.  R. 
Picken,  The  fine  structure  of  biological  systems,  Biol.  Reviews,  xv,  pp.  133-67,  1940. 


V]    OF  STABLE  AND  UNSTABLE  EQUILIBRIUM    405 

surface  constantly  changing  its  form,  passing  from  one  phase  to 
another  of  an  equihbrium  which  is  never  stable  for  more  than  a 
moment,  and  which  death  restores  to  the  stable  equilibrium  of  a 
sphere.  The  former  class,  which  rest  in  stable  equihbrium,  must 
fall  (as  we  have  seen)  into  two  classes — those  whose  equilibrium 
arises  from  liquid  surface-tension  alone,  and  those  in  whose  con- 
formation some  other  pressure  or  restraint  has  been  superimposed 
upon  ordinary  surface-tension. 

To  the  fact  that  all  these  organisms  belong  to  an  order  of 
magnitude  in  which  form  is  mainly,  if  not  wholly,  conditioned  and 
controlled  by  molecular  forces  is  due  the  hmited  range  of  forms 
which  they  actually  exhibit.  They  vary  according  to  varying 
physical  conditions.  Sometimes  they  do  so  in  so  regular  and 
orderly  a  way  that  w^e  intuitively  explain  them  as  "phases  of  a 
Hfe-history,"  and  leave  physical  properties  and  physical  causation 
alone:  but  many  of  their  variations  of  form  we  treat  as  exceptional, 
abnormal,  decadent  or  morbid,  and  are  apt  to  pass  these  over  in 
neglect,  while  we  give  our  attention  to  what  we  call  a  typical  or 
"characteristic"  form  or  attitude.  In  the  case  of  the  smallest 
organisms,  bacteria,  micrococci,  and  so  forth,  the  range  of  form  is 
especially  limited,  owing  to  their  minuteness,  the  powerful  pressure 
which  their  highly  curved  surfaces  exert,  and  the  comparatively 
homogeneous  nature  of  their  substance.  But  within  their  narrow 
range  of  possible  diversity  these  minute  organisms  are  protean  in 
their  changes  of  form.  A  certain  species  will  not  only  change  its 
shape  from  stage  to  stage  of  its  little  "cycle"  of  hfe;  but  it  will 
be  remarkably  different  in  outward  form  according  to  the  circum- 
stances under  which  we  find  it,  or  the  histological  treatment  to 
which  we  subject  it.  Hence  the  pathological  student,  commencing 
the  study  of  bacteriology,  is  early  warned  to  pay  little  heed  to 
differences  of  form,  for  purposes  of  recognition  or  specific  identi- 
fication. Whatever  grounds  we  may  have  for  attributing  to 
these  organisms  a  permanent  or  stable  specific  identity  (after 
the  fashion  of  the  higher  plants  and  animals),  we  can  seldom 
safely  do  so  on  the  ground  of  definite  and  always  recognisable 
form:  we  may  often  be  inclined,  in  short,  to  ascribe  to  them  a 
physiological  (sometimes  a  "pathogenic")  rather  than  a  morpho- 
logical specificity. 


406 


THE  FORMS  OF  CELLS 


[CH. 


Many  unicellular  forms,  and  a  few  other  simple  organisms,  are 
spherical,  and  serve  to  illustrate  in  the  simplest  way  the  point  at 
issue.  Unicellular  algae,  such  as  Protococcus  or  Halisphaera,  the 
innumerable  floating  eggs  of  fishes,  the  floating  unilocular  foraminifer 
Orbulina,  the  lavely  green  multicellular  Volvox  of  our  ponds,  all 
these  in  their  several  grades  of  simplicity  or  complication  are  so 
many  round  drops,  spherical  because  no  alien  forces  have  deformed 
or  mis-shapen  them.  But  observe  that,  with  the  exception  of 
Volvox,  whose  spherical  body  is  covered  wholly  and  uniformly  with 
minute  ciha,  all  the  above  are  passive  or  inactive  forms;  and  in  a 
"resting"  or  encysted  phase  the  spherical  form  is  common  and 
general  in  a  great  range  of  unicellular  organisms. 

Conversely,   we  see  that  those  unicellular  forms  which  depart 
jUBrkedly  from  sphericity — excluding  for  the  moment  the  amoeboid 

forms  and  those  provided  with  skeletons 
— are  all  cihate  or  flagellate.  Ciha  and 
flagella  are  sui  generis ;  we  know  nothing 
of  them  from  the  physical  side,  we  cannot 
reproduce  or  imitate  them  in  any  non- 
hving  drop  or  fluid  surface.  But  we  can 
easily  see  that  they  have  an  influence  on 
form,  besides  serving  for  locomotion. 
When  our  httle  Monad  or  Euglena 
develops  a  flagellum,  that  is  in  itself  an 
indication  of  asymmetry  or  "polarity," 
of  non-homogeneity  of  the  little  cell ;  and 
in  the  various  flagellate  types  the  flagellum  or  its  analogues  always 
stand  on  prominent  points,  or  ends,  or  edges  of  the  cell — on  parts, 
that  is  to  say,  where  curvature  is  high  and  surface-tension  may  be 
expected  to  be  low — for  the  product  of  surface-tension  by  mean 
curvature  tends  to  be  constant. 


Fig.  123.  A  flagellate  "monad," 
Distigma  proteus  Ehr, 
After  Saville  Kent. 


The  minute  dimensions  of  a  cilium  or  a  flagellum  are  such  that  the  molecular 
forces  leading  to  surface-tension  must  here  be  under  peculiar  conditions  and 
restraints;  we  cannot  hope  to  understand  them  by  comparison  with  a  whip- 
lash, or  through  any  other  analogy  drawn  from  a  different  order  of  magnitude. 
I  suspect  that  a  ciliary  surface  is  always  electrically  charged,  and  that 
a  point- charge  is  formed  or  induced  in  each  cilium  or  flagellum.  Just  as  we 
learn  the  properties  of  a  drop  or  a  jet  as  phenomena  proper  to  their  scale  of 
magnitude,   so  some   day   we  shall  learn  the   very  different  physical,   but 


v]  OF  CILIA  AND  FLAGELLA  407 

microcosmic,  properties  of  these  minute,  mobile,  pointed,  fluid  or  semi-fluid 
threads.* 

Cilia,  like  flagella,  tend  to  occupy  positions,  or  cover  surfaces, 
which  would  otherwise  be  unstable;  and  often  indeed  (as  in  a 
trochosphere  larva  or  even  in  a  Rotifer)  a  ring  of  cilia  seems  to 
play  the  very  part  of  one  of  Plateau's  wire  rings,  supporting  and 
steadying  the  semi-fluid  mass  in  its  otherwise  unstable  configura- 
tion. Let  us  note  here  (in  passing)  what  seems  to  be  an  analogous 
phenomenon.  Chitinous  hairs,  spines  or  bristles  are  common  and 
characteristic  structures  among  the  smaller  Crustacea,  and  more  or 
less  generally  among  the  Arthropods.  We  find  them  at  every 
exposed  point  or  corner;  they  fringe  the  sharp  edge  or  border  of 
a  limb;  as  we  draw  the  creature,  we  seem  to  know  where  to  put 
them  in !  In  short,  they  tend  to  occur,  as  the  flagella  do,  just  where 
the  surface-tension  would  be  lowest,  if  or  when  the  surface  was  in 
a  fluid  condition. 

Of  the  other  surfaces  of  Plateau,  we  find  cylinders  enough  and 
to  spare  in  Spirogyra  and  a  host  of  other  filamentous  algae  and 
fungi.  But  it  is  to  the  vegetable  kingdom  that  we  go  to  find  them, 
where  a  cellulose  envelope  enables  the  cyUnder  to  develop  beyond 
its  ordinary  limitations. 

The  unduloid  makes  its  appearance  whenever  sphere  or  cyhnder 
begin  to  give  way.  We  see  the  transitory  figure  of  an  unduloid  in 
the  normal  fission  of  a  simple  cell,  or  of  the  nucleus  itself;  and  we 
have  already  seen  it  to  perfection  in  the  incipient  headings  of  a 
spider's  web,  or  of  a  pseudopodial  thread  of  protoplasm.  A  large 
number  of  infusoria  have  unduloid  contours,  in  part  at  least;  and 
this  figure  appears  and  reappears  in  a  great  variety  of  forms.  The 
cups  of  various  Vorticellae  (Fig.  124),  below  the  ciliated  ring,  look 
like  a  beautiful  series  of  unduloids,  in  every  gradation  of  form,  from 
what  is  all  but  cylindrical  to  all  but  a  perfect  sphere;  moreover 
successive  phases  in  their  hfe-history  appear  as  mere  graded  changes 

*  It  is  highly  characteristic  of  a  cilium  or  a  flagellum  that  neither  is  ever  seen 
motionless,  unless  the  cell  to  which  it  belongs  is  moribund.  "I  believe  the  motion 
to  be  ceaseless,  unconscious  and  uncontrolled,  a  direct  function  of  the  chemical 
and  physical  environment ";  tieorge  Bidder,  in  Presidential  Address  to  Section  D, 
British  Associqtion,  1927.  Cf.  also  James  Gray,  Proc.  R.S.  (B),  xcix,  p.  398, 
1926. 


408 


THE  FORMS  OF  CELLS 


[CH. 


of  unduloid  form.  It  has  been  shewn  lately,  in  one  or  two 
instances  at  least,  that  species  of  Vorticella  may  "metamorphose" 
into  one  another :  in  other  words,  that  contours  supposed  to  charac- 
terise species  are  not  "specific"      These  VorticelUd  unduloids  are 


>^- 


Fig.  124.     Various  species  of  Vorticella. 


M 


Fig.  1 25.   Various  species  of  Salpingoeca. 


Fig.  126.     Various  species  of  Tintinnus,  Dinohryon  and  Codonella. 
After  Saville  Kent  and  others. 


not  fully  symmetrical ;  rather  are  they  such  unduloids  as  develop 
when  we  suspend  an  oil-globule  between  two  unequal  rings,  or  blow 
a  bubble  between  two  unequal  pipes.  For  our  Vorticellid  bell  hangs 
by  two  terminal  supports,  the  narrow  stallj  to  which  it  is  attached 
below,  and  the  thickened  ring  from  which  spring  its  circumoral 
cilia;    and  it  is  most  interesting  to  see  how,  when  the  bell  leaves 


OF  VARIOUS  UNDULOIDS 


409 


its  stalk  (as  sometimes  happens)  and  swims  away,  a  new  ring  of 
cilia  comes  into  being,  to  encircle  and  support  its  narrow  end. 

Similar  unduloids  may  be  traced  in  even  greater  variety  among 
other  famihes  or  genera  of  the  Infusoria.  Sometimes,  as  in  Vorticella 
itself,  the  unduloid  is  seen  in  the  contour  of  the  soft  semifluid 
body  of  the  hving  animal.  At  other  times,  as  in  Salpingoeca, 
Tintinnus,  and  many  other  genera,  we  have  a  membranous  cup 
containing  the  animal,  but  originally  secreted  by,  and  moulded 
upon,  its  semifluid  hving  surface.  Here  we  have  an  excellent 
illustration  of  the  contrast  between  the  different  ways  in  which 
such  a  structure  may  be  regarded  and  interpreted.  The  teleological 
explanation  is  that  it  is  developed  for  the  sake  of  protection, 


127.  Vaginicola. 


Fig.  128.     FolUculina. 


as  a  domicile  and  shelter  for  the  httle  organism  within.  The 
mechanical  explanation  of  the  physicist  (seeking  after  the  "efficient," 
not  the  "final"  cause)  is  that  it  owes  its  presence,  and  its  actual 
conformation,  to  certain  chemico-physical  conditions:  that  it  was 
inevitable,  under  the  given  conditions,  that  certain  constituent 
substances  present  in  the  protoplasm  should  be  drawn  by  molecular 
forces  to  its  surface  layer;  that  under  this  adsorptive  process,  the 
conditions  continuing  favourable,  the  particles  accumulated  and 
concentrated  till  they  formed  (with  the  help  of  the  surrounding 
medium)  a  pellicle  or  membrane,  thicker  or  thinner  as  the  case 
might  be;  that  this  surface  pellicPe  or  membrane  was  inevitably 
bound,  by  molecular  forces,  to  contract  into  a  surface  of  the 
least  possible  area  which  the  circumstances  permitted;  that  in 
the  present  case  the  symmetry  and  "freedom"  of  the  system 
permitted,  and  ipso  facto  caused,  this  surface  to  be  a  surface  of 
revolution;    and  that  of  the  few  surfaces  of  revolution  which,  as 


410 


THE  FORMS  OF  CELLS 


[CH. 


being  also  surfaces  minimae  areae,  were  available,  the  unduloid  was 
manifestly  the  one  permitted,  and  ipso  facto  caused,  by  the  dimen- 
sions of  the  organism  and  other  circumstances  of  the  case.  And 
just  as  the  thickness  or  thinness  of  the  pellicle 
was  obviously  a  subordinate  matter,  a  mere 
matter  of  degree,  so  we  see  that  the  actual 
outhne  of  this  or  that  particular  unduloid  is 
also  a  very  subordinate  matter,  such  as  physico- 
chemical  variants  of  a  minor  order  would  suffice 
to -bring  about;  for  between  the  various  undu- 
loids  which  the  various  species  of  Vorticella 
represent,  there  is  no  more  real  difference  than 
that  difference  of  ratio  or  degree  which  exists 
between  two  circles  of  different  diameter,  or 
two  hnes  of  unequal  length. 

In  many  cases  (of  which'  Fig.  129  is  an 
example)  we  have  a  more  or  less  unduloid  form 
exhibited  not  by  a  surrounding  pelHcle  or  shell, 
but  by  the  soft  protoplasmic  body  of  a  ciliated 
organism;  in  such  cases  the  form  is  mobile, 
and  changes  continually  from  one  to  another 
unduloid  contour  according  to  the  movements 
of  the  animal.*  We  are  deahng  here  with  no 
stable  equihbrium,  but  possibly  with  a  subtle 
problem  of  ''stream-lines,"  as  in  the  difficult 
but  beautiful  problems  suggested  by  the  form 
of  a  fish.  But  this  whole  class  of  cases,  and 
of  problems,  we  merely  take  note  of  here;  we  shall  speak  of  them 
again,  but  their  treatment  is  hard. 

In  considering  such  series  of  forms  as  these  various  unduloids  we 
are  brought  sharply  up  (as  in  the  case  of  our  bacteria  or  micrococci) 
against  the  biological  concept  of  organic  species.  In  the  intense 
classificatory  activity  of  the  last  hundred  years  it  has  come  about 
that  every  form  which  is  apparently  characteristic,  that  is  to  Say 
which  is  capable  of  being  described  or  portrayed,  and  of  being 

*  Doflein  lays  stress,  in  like  manner,  on  the  fact  that  Spirochade,  unlike 
Spirillum,  "ist  nicht  von  einer  starren  Membran  umhiillt,"  and  that  waves  of 
contraction  may  be  seen  passing  down  its  body. 


Fig.  129.  Trachelo- 
phyllum.  After 
Wreszniowski. 


V]  OF  SNOW-CRYSTALS  411 

recognised  when  met  with  again,  has  been  recorded  as  a  species — 
for  we  need  not  concern  ourselves  with  the  occasional  discussions, 
or  individual  opinions,  as  to  whether  such  and  such  a  form  deserves 
"specific  rank,"  or  be  "only  a  variety."  And  this  secular  labour 
is  pursued  in  direct  obedience  to  the  precept  of  the  Systema  Naturae 
— ''ut  sic  in  summa  confusione  rerum  apparenti,  summus  conspiciatur 
Naturae  ordo.''  In  like  manner  the  physicist  records,  and  is  entitled 
to  record,  his  many  hundred  "species"  of  snow-crystals*,  or  of 
crystals  of  calcium  carbonate.  Indeed  the  snow-crystal  illustrates  to 
perfection  how  Nature  rings  the  changes  on  every  possible  variation 
and  permutation  and  combination  of  form:  subject  only  to  the 
condition  (in  this  instance)  that  a  snow-crystal  shall  be  a  plane, 
symmetrical,  rectilinear  figure,  with  all  its  external  angles  those  of 
a  regular  hexagon.  We  may  draw  what  we  please  on  a  sheet  of 
"hexagonal  paper,"  keeping  to  its  lines;  and  when  we  repeat  our 
drawing,  kaleidoscope-fashion,  about  a  centre,  the  stellate  figure  so 
obtained  is  sure  to  resemble  one  or  another  of  the  many  recorded 
species  of  snow-crystals.  And  this  endless  beauty  of  crystalline 
form  is  further  enhanced  when  the  flakes  begin  to  thaw,  and  all 
their  feathery  outlines  soften.  But  regarding  these  "species"  of  his, 
the  physicist  makes  no  assumptions:  he  records  them  simpliciter; 
he  notes,  as  best  he  can,  the  circumstances  (such  as  temperature  or 
humidity)  under  which  each  occurs,  in  the  hope  of  elucidating  the. 
conditions  which  determine  their  formation  f;    but  above  all,  he 

*  The  case  of  the  snow-crystals  is  a  particularly  interesting  one;    for  their 
"distribution"  is  analogous  to  what  we  find,  for  instance,  among  our  microscopic 
skeletons  of  Radiolarians.     That  is  to  say,  we  may  one  day  meet  with  myriads 
of  some  one  particular  form  or  species,  and  another  day  with  myriads  of  another 
while  at  another  time  and  place  we  may  find  species  intermingled  in  all  but 
inexhaustible  variety.    Cf.  e.g.  J.  Glaisher,  Illustrated  London  News,  Feb.  17,  1855 
Q.J. M.S.  m,  pp.  179-185,  1855;    Sir  Edward  Belcher,  Last  of  the  Arctic  Voyages, 
II,  pp.  288-306  (4  plates),  1855;  William  Scoresby,  An  Account  of  the  Arctic  Regions, 
Edinburgh,  1820;    G.  Hellmann,  Schneekrystalle,  Berlin,  1893;    Bentley  and  Hum 
phreys.  Snow  Crystals,  New  York,  1931;    and  the  especially  beautiful  figures  of 
Nakaya  and  Hasikura  in  Journ.  Fac.  Sci.  Hokkaido,  Dec.  1934. 

t  Every  snow-crystal  tells,  more  or  less  plainly,  the  story  of  its  own  development. 
The  cold  upper  air  is  saturated  with  water- vapour,  but  this  is  scanty  and  rarefied 
compared  with  the  space  in  which  snow- crystallisation  is  going  on.  Hence 
crystallisation  tends  to  proceed  only  along  the  main  axes,  or  cardinal  framework, 
of  the  crystalline  structure  of  ice ;  in  so  doing  it  gives  a  visible  picture  or  actual 
embodiment  of  the  trigonal-hexagonal  space-lattice,  in  the  endless  permutations 
and  combinations  of  its  constituent  elements. 


412  THE  FORMS  OF  CELLS  [ch. 

does  not  introduce  the  element  of  time,  and  of  succession,  or  discuss 
their  origin  and  affihation  as  an  historical  sequence  of  events.  But 
in  biology,  the  term  species  carries  with  it  many  large  though  often 
vague  assumptions;  though  the  doctrine  or  concept  of  the  "per- 
manence of  species"  is  dead  and  gone,  yet  a  certain  quasi-permanency 
is  still  connoted  by  the  term.  If  a  tiny  foraminiferal  shell,  a  Lagena 
for  instance,  be  found  hving  to-day,  and  a  shell  indistinguishable 
from  it  to  the  eye  be  found  fossil  in  the  Chalk  or  some  still  more 
remote  geological  formation,  the  assumption  is  deemed  legitimate 
that  that  species  has  "survived,"  and  has  handed  down  its  minute 
specific  character  or  characters  from  generation  to  generation, 
unchanged  for  untold  milHons  of  years*.  If  the  ancient  forms  be 
Hke  to  rather  than  identical  with  the  recent,  we  still  assume  an 
unbroken  descent,  accompanied  by  the  hereditary  transmission  of 
common  characters  and  progressive  variations.  And  if  two  identical 
forms  be  discovered  at  the  ends  of  the  earth,  still  (with  occasional 
slight  reservations  on  the  score  of  possible  "homoplasy")  we  build 
hypotheses  on  this  fact  of  identity,  taking  it  for  granted  that  the 
two  appertain  to  a  common  stock,  whose  dispersal  in  space  must 
somehow  be  accounted  for,  its  route  traced,  its  epoch  determined, 
and  its  causes  discussed  or  discovered.  In  short,  the  naturaUst 
admits  no  exception  to  the  rule  that  a  natural  classification  can  only 
be  a  genealogical  one,  nor  ever  doubts  that  '"The  fact  that  we  are  able 
to  classify  organisms  at  all  in  accordance  with  the  structural  charac- 
teristics which  they  present  is  due  to  the  fact  of  their  being  related  by 
descenf\''  But  this  great  and  valuable  and  even  fundamental 
generahsation  sometimes  carries  us  too  far.  It  may  be  safe  and 
sure  and  helpful  and  illuminating  when  we  apply  it  to  such  complex 
entities — such  thousand-fold  resultants  of  the  combination  and 
permutation  of  many  variable  characters — as  a  horse,  a  lion  or  an 
eagle;  but  (to  my  mind)  it  has, a  very  different  look,  and  a  far  less 
firm  foundation,  when  we  attempt  to  extend  it  to  minute  organisms 
whose   specific   characters   are  few  and   simple,   whose  simplicity 


*  Cf.  Bergson,  Creative  Evolution,  p.  107:  "Certain  Foraminifera  have  not 
varied  since  the  Silurian  epoch.  Unmoved  witnesses  of  the  innumerable  revolu- 
tions that  have  upheaved  our  planet,  the  Lingulae  are  today  what  they  were  at 
the  remotest  times  of  the  palaeozoic  era." 

t  Ray  Lankester,  A.M.N.H.  (4),  xi,  p.  321,  1873. 


V]  OF  FORM  AND  SPECIES  413 

becomes  more  manifest  from  the  point  of  view  of  physical  and 
mathematical  analysis,  and  whose  form  is  referable,  or  largely 
referable,  to  the  direct  action  of  a  physical  force.  When  we  come 
to  the  minute  skeletons  of  the  Radiolaria  we  shall  again  find  our- 
selves dealing  with  endless  modifications  of  form,  in  which  it  becomes 
more  and  more  difficult  to  discern,  and  at  last  vain  and  hopeless  to 
apply,  the  guiding  principle  of  affihation  or  '^phylogeny.'' 

Among  the  Foraminifera  we  have  an  immense  variety  of  forms, 
which,  in  the  hght  of  surface-tension  and  of  the  principle  of  minimal 
area,  are  capable  of  explanation  and  of  reduction  to  a  small  number 
of  characteristic  types.  Many  of  them  are  composite  structures, 
formed  by  the  successive  imposition  of  cell  upon  cell,  and  these  we  shall 
deal  with  later  on ;  let  us  glance  here  at  the  simpler  conformations 
exhibited  by  the  single  chambered  or  "  monothalamic  "  genera,  and 
perhaps  one  or  two  of  the  simplest  composites. 

We  begin  with  forms  like  Astrorhiza  (Fig.  320,  p.  703),  which  are 
large,  coarse  and  highly  irregular,  and  end  with  others  which  are 
minute  and  delicate,  and  which  manifest  a  perfect  and  mathe- 
matical regularity.  The  broad  difference  between  these  two  types 
is  that  the  former  are  characterised,  like  Amoeba,  by  a  variable 
surface-tension,  and  consequently  by  unstable  equihbrium;  but  the 
strong  contrast  between  these  and  the  regular  forms  is  bridged  over 
by  various  transition-stages,  or  differences  of  degree.  Indeed,  as 
in  all  other  Rhizopods,  the  very  fact  of  the  emission  of  pseudopodia, 
which  are  especially  characteristic  of  this  group  of  animals,  is 
a  sign  of  unstable  surface-equihbrium ;  and  we  must  therefore 
consider,  or  may  at  least  suspect,  that  those  forms  whose  shells 
indicate  the  most  perfect  symmetry  and  equilibrium  have  secreted 
these  during  periods  when  rest  and  uniformity  of  surface-conditions 
contrasted  with  the  phases  of  pseudopodial  activity.  The  irregular 
forms  are  in  almost  all  cases  arenaceous,  that  is  to  say  they  have 
no  soHd  shells  formed  by  steady  adsorptive  secretion,  but  only  a 
looser  covering  of  sand  grains  with  which  the  protoplasmic  body 
has  come  in  contact  and  cohered.  Sometimes,  as  in  Ramulina,  we 
have  a  calcareous  shell  combined  with  irregularity  of  form;  but 
here  we  can  easily  see  a  partial  and  as  it  were  a  broken  regularity, 
the  regular  forms  of  sphere  and  cylinder  being  repeated  in  various 


414 


THE  FORMS  OF  CELLS 


[CH. 


parts  of  the  ramified  mass.  When  we  look  more  closely  at  the 
arenaceous  forms,  we  find  the  same  thing  true  of  them;  they 
represent,  in  whole  or  part,  approximations  to  the  surfaces  of 
equilibrium,  spheres,  cylinders  and  so  forth.  In  Aschemonella  we 
have  a  precise  rephca  of  the  calcareous  Ramuliyia;  and  in  Astrorhiza 
itself,  in  the  forms  distinguished  by  jiaturalists  as  A.  crassatina, 
what  is  described  as  the  " subsegmented  interior*"  seems  to  shew 


Fig.  130.     Various  species  of  Zagrew«.     After  Brady. 


the  natural,  physical  tendency  of  the  long  semifluid  cylinder  of 
protoplasm  to  contract  at  its  limit  of  stability  into  unduloid 
constrictions,  as  a  step  towards  the  breaking  up  into  separate 
spheres :  the  completion  of  which  process  is  restrained  or  prevented 
by  contact  with  the  unyielding  arenaceous  covering. 

Passing  to  the  typical  calcareous  Foraminifera,  we  have  the  most 
symmetrical  of  all  possible  types  in  the  perfect  sphere  of  Orbulina ; 
this  is  a  pelagic  organism,  whose  floating  habitat  gives  it  a  field  of 

♦  Brady,  Challenger  Monograph,  pi.  xx,  p.  233. 


V]  OF  HANGING  DROPS  415 

force  of  perfect  symmetry.  Save  for  one  or  two  other  forms  which 
are  also  spherical,  or  approximately  so,  like  Thurammina,  the  rest 
of  the  monothalamic  calcareous 
Foraminifera  are  all  comprised  by 
naturalists  within  the  genus 
Lagena.  This  large  and  varied 
genus  consists  of  "flask-shaped" 
shells,  whose  surface  is  that  of  an 
unduloid,  or,  hke  that  of  a  flask 
itself,  an  unduloid  combined  with 
a  portion  of  a  sphere.  We  do 
not  know  the  circumstances  under 
which  the  shell  oi  Lagena  is  formed, 
nor  the  nature  of  the  force  by 
which,  during  its.  formation,  the 
surface  is  stretched  out  into  the 
unduloid  form;  but  we  may  be 
pretty  sure  that  it  is  suspended 
vertically  in  the  sea,  that  is  to 
say  in  a  position  of  symmetry  as 
regards  its  vertical  axis,  about 
which  the  unduloid  surface  of  re- 
volution is  symmetrically  formed. 


4  5  6 

Fig.  131.   Roman  pottery,  for  comparison 

with  species  oi  Lagena.    E.g.,  1,  2,  with 

L.   sulcata;    3,    L.  orbignyana;    4,  L. 

striata;   5,  L.  crenata;  6,  L.  stelligera. 


At  the  same  time  we  have  other 
types  of  the  same  shell  in  which  the  form  is  more  or  less  flattened ; 
and  these  are  doubtless  the  cases  in  which  such  symmetry  of  position 
was  not  present,  or  was  replaced  by  a  broader,  lateral  contact  with 
the  surface  pellicle*. 

While  Orhulina  is  a  simple  spherical  drop,  Lagena  suggests  to  our 
minds  a  hanging  drop,  drawn  out  to  a  longer  or  shorter  neck  by 

*  That  the  Foraminifera  not  only  can  but  do  hang  from  the  surface  of  the 
water  is  confirmed  by  the  following  apt  quotation  which  I  owe  to  Mr  E.  Heron - 
Allen:  "Quand  on  place,  comme  il  a  ete  dit,  le  depot  provenant  du  lavage  des 
fucus  dans  un  fiacon  que  Ton  remplit  de  nouvelle  eau,  on  voit  au  bout  d'une  heure 
environ  les  animaux  [Gromia  dujardinii]  se  mettre  en  mouvement  et  commencer 
a  grimper.  Six  heures  apres  ils  tapissent  Texterieur  du  flacon,  de  sorte  que  les  plus 
eleves  sont  a  trente-six  ou  quarante-deux  millimetres  du  fond;  le  lendemain 
beaucoup  d'entre  eux,  apres  avoir  atteint  le  niveau  du  liquide,  ont  continue  a  rarnper 
a  sa  surface,  en  se  laissant  pendre  au-dessous  comme  certains  mollusques  gastero- 
podes."  (F.  Dujardin,  Observations  nouvelles  sur  les  pretendus  cephalopodes 
microscopiques,  Ann.  des  Sci.  Nat.  (2),  iii,  p.  312,  1835.) 


416  THE  FORMS  OF  CELLS  [ch. 

its  own  weight,  aided  by  the  viscosity  of  the  material.  Indeed  the 
various  hanging  drops,  such  as  Mr  C.  R.  DarUng  shews  us,  are  the 
most  beautiful  and  perfect  unduloids,  with  spherical  ends,  that  it 
is  possible  to  conceive.  A  suitable  hquid,  a  httle  denser  than  water 
and  incapable  of  mixing  with  it  (such  as  ethyl  benzoati),  is  poured 
on  a  surface  of  water.  It  spreads  over  the  surface  and  gradually 
forms  a  hanging  drop,  approximately  hemispherical;  but  as  more 
liquid  is  added  the  drop  sinks  or  rather  stretches  downwards,  still 
adhering  to  the  surface  fihn;  and  the  balance  of  forces  between 
gravity  and  surface-tension  results  in  the  unduloid  contour,  as  the 
increasing  weight  of  the  drop  tends  to  stretch  it  out  and  finally 
Sreak  it  in  two.     At  the  moment  of  rupture,  by  the  way,  a  tiny 


Fig.  132.     Large  "hanging  drops"  ot  oil.     After  Darling. 

droplet  js  formed  in  the  attenuated  neck,  such  as  we  described  in 
the  normal  division  of  a  cyhndrical  thread. 

The  thin,  fusiform,  pointed,  non-globular  Lagenas  are  less  easily 
explained.  Surface-tension,  which  tends  to  keep  the  drop  spherical, 
is  overmastered  here,  and  the  elongate  shape  suggests  the  viscous 
drag  of  a  shearing  fluid*. 

To  pass  to  a  more  highly  organised  class  of  animals,  v^e  find  the  unduloid 
beautifully  exemplified  in  the  little  flask-shaped  shells  of  certain  Pteropod 
mollusca,  e.g.  Cuvierina-\.  Here  again  the  symmetry  of  the  figure  would 
at  once  lead  us  to  suspect  that  the  creature  lived  in  a  position  of  symmetry 
to  the  surrounding  forces,  as  for  instance  if  it  floated  in  the  ocean  in  an 
erect  position,  that  is  to  say  with  its  long  axis  coincident  with  the  direction 
of  gravity;  and  this  we  know  to  be  actually  the  mode  of  life  of  the  little 
Pteropod. 

*  Cf.  G.  I.  Taylor,  The  formation  of  emulsions  in  definable  fields  of  flow,  Proc.  R.S. 
(A),  No.  858,  p.  501,  1934. 
t  Cf.  Boas,  Spolia  Atlantica,  1886,  pi.  6. 


V]         OF  RETICULATE  OR  WRINKLED  CELLS       417 

Many  species  of  Lagena  are  complicated  and  beautified  by  a 
pattern,  and  some  by  the  superaddition  to  the  shell  of  plane 
extensions  or  "wings."  These  latter  give  a  secondary,  bilateral 
symmetry  to  the  little  shell,  and  are  strongly  suggestive  of  a  phase 
or  period  of  growth  in  which  it  lay  horizontally  on  the  surface, 
instead  of  hanging  vertically  from  the  surface-film :  in  which,  that 
is  to  say,  it  was  a  floating  and  not  a  hanging  drop.  The  pattern 
is  of  two  kinds.  Sometimes  it  consists  of  a  sort  of  fine  reticulation, 
with  rounded  or  more  or  less  hexagonal  interspaces:  in  other  cases 
it  is  produced  by  a  symmetrical  series  of  ridges  or  folds,  usually 
longitudinal,  on  the  body  of  the  flask-shaped  cell,  but  occasionally 
transversely  arranged  upon  the  narrow  neck.  The  reticulated  and 
folded  patterns  we  may  consider  separately.  The  netted  pattern 
is  very  similar  to  the  wrinkled  surface  of  a  dried  pea,  or  to  the  more 
regular  wrinkled  patterns  on  poppy  and  other  seeds  and  even  pollen- 
grains.  If  a  spherical  body  after  developing  a  "skin"  begin  to 
shrink  a  little,  and  if  the  skin  have  so  far  lost  its  elasticity  as  to 
be  unable  to  keep  pace  with  the  shrinkage  of  the  inner  mass,  it  will 
tend  to  fold  or  wrinkle;  and  if  the  shrinkage  be  uniform,  and  the 
elasticity  and  flexibihty  of  the  skin  be  also  uniform,  then  the  amount 
of  foldings  will  be  uniformly  distributed  over  the  surface.  Little 
elevations  and  depressions  will  appear,  regularly  interspaced,  and 
separated  by  concave  or  convex  folds.  These  being  of  equal  size 
(unless  the  system  be  otherwise  perturbed),  each  one  wiU  tend  to 
be  surrounded  by  six  others;  and  when  the  process  has  reached  its 
limit,  the  intermediate  boundary-walls,  or  folds,  will  be  found 
converted  into  a  more  or  less  regular  pattern  of  hexagons.  To  these 
symmetrical  wrinkles  or  shrinkage-patterns  we  shall  return  again. 

But  the  analogy  of  the  mechanical  wrinkHng  of  the  coat  of  a 
seed  is  but  a  rough  and  distant  one;  for  we  are  deahng  with 
molecular  rather  than  with  mechanical  forces.  In  one  of  Darling's 
experiments,  a  little  heavy  tar-oil  is  dropped  on  to  a  saucer  of 
water,  over  which  it  spreads  in  a  thin  film  shewing  beautiful 
interference  colours  after  the  fashion  of  those  of  a  soap-bubble. 
Presently  tiny  holes  appear  in  the  film,  which  gradually  increase 
in  size  till  they  form  a  cellular  pattern  or  honeycomb,  the  oil 
gathering  together  in  the  meshes  or  walls  of  the  cellular  net.  Some 
action  of  this  sort  is  in  aU  probability  at  work  in  a  surface-film 


418  THE  FORMS  OF  CELLS  [ch. 

of  protoplasm  covering  the  shell.  As  a  physical  phenomenon  the 
actions  involved  are  by  no  means  fully  understood,  but  surface- 
tension,  diffusion  and  cohesion  play  their  respective  parts  therein*. 
The  very  perfect  cellular  patterns  obtained  by  Leduc  (to  which  we 
shall  have  occasion  to  refer  in  a  subsequent  chapter)  are  diffusion 
patterns  on  a  larger  scale,  but  not  essentially  different. 

The  folded  or  pleated  pattern  is  doubtless  to  be  explained,  in 

a  general  way,  by  the  shrinkage  of  a  surface-film  under  certain 

conditions  of  viscous  or  frictional  restraint. 

A  case  which  (as  it  seems  to  me)  is  closely 

alhed  to  that  of  our  foraminiferal  shells  is 

described  by  Quincke  t,   who  let  a   film   of 

chromatised  gelatin  or  of  resin  set  and  harden 

upon  a  surface  of  quicksilver,  and  found  that 

the  little  solid  pelHcle  had  been  thrown  into 

a  pattern  of  symmetrical  folds,  as  fine  as  a 

Fig.^  133.  diffraction  grating.     If  the  surface  thus  folded 

or  wrinkled  be  a  cyhnder,  or  any  other  figure  with  one  principal  axis 

*  This  cellular  pattern  would  seem  to  be  related  to  the  "cohesion  figures" 
described  by  Tomlinson  in  various  surface-films  {Phil.  Mag.  1861-70);  to  the 
"tesselated  structure"  on  liquid  surfaces  described  by  James  Thomson  in  1882 
{Collected  Papers,  p.  136);  and  (more  remotely)  to  the  tourhillons  cellulaires  of 
Benard,  Ann.  de  Chimie  (7),  xxiii,  pp.  62-144,  1901;  (8),  xxiv,  pp.  563-566,  1911, 
Bev.  gener.  des  Sci.  xi,  p.  1268,  1900;  cf.  also  E.  H.  Weber,  Mikroskopische  Beo- 
bachtungen  sehr  gesetzmassiger  Bewegungen  welche  die  Bildung  von  Niederschlagen 
harziger  Korper  aus  Weingeist  begleiten,  Poggend.  Ann.  xciv,  pp.  447-459,  1855; 
etc.  Some  at  least  of  TomHnson's  cohesion-figures  arise,  according  to  van  Mens- 
brugghe,  from  the  disengagement  of  minute  bubbles  of  gas,  when  a  fluid  holding 
gases  in  solution  comes  in  contact  with  a  fluid  of  lower  surface-tension.  The  whole 
phenomenon  is  of  great  interest  and  various  appearances  have  been  referred  to 
it,  in  biology,  geology,  metallurgy  and  even  astronomy:  for  the  flocculent  clouds 
in  the  solar  photosphere  shew  an  analogous  configuration.  (See  letters  by  Kerr 
Grant,  Larmor,  Wager  and  others,  in  Nature,  April  16  to  June  11,  1914;  also 
Rayleigh,  Phil.  Mag.  xxxii,  p.  529,  1916;  G.  T.  Walker,  Clouds,  natural  and 
artificial,' i?oz/a/  Inst.  8  Feb.  1935;  etc.)  In  many  instances,  marked  by  strict 
symmetry  or  regularity,  it  is  very  possible  that  the  interference  of  waves  or  ripples 
may  play  its  part  in  the  phenomenon.  But  in  the  majority  of  cases,  it  is  fairly 
certain  that  localised  centres  of  action,  or  of  diminished  tension,  are  present,  such 
as  might  be  provided  by  dust-particles  in  the  case  of  Benard's  experiment  (cf. 
infra,  p.  503). 

t  Quincke,  Ueber  physikalische  Eigenschaften  diinner  fester  Lamellen,  Sitzungsb, 
Berlin.  Akad.  1888,  p.  789;  Ueber  ansichtbare  Eigenschaften,  etc.,  Ann.  d.  Physik, 
1920,  p.  653.  Quincke  found  that  "sehr  kleine  Menge  fremder  Substanz  haben 
eine  grosse  Einfluss  auf  die  Bildung  der  Schaumwande." 


V]  OF  FLUTED  OR  PLEATED  CELLS  419 

of  symmetry,  such  as  an  ellipsoid  or  unduloid,  the  folds  will  tend 

to  be  related  to  the  ax's  of  symmetry,  and  we  may  expect  accordingly 

to  find  regular  longitudinal,  or  regular  transverse  wrinkling.     Now 

as  a  matter  of  fact  we  almost  invariably  find  in  Lagena  the  former 

condition:    that  is  to  say,  in  our  ellipsoid  or  unduloid  shell,  the 

puckering  takes  the  form  of  the  vertical  fluting  on  a  column,  rather 

than  that  of  the  transverse  pleating  of  an  accordion;    and  further, 

there  is  often  a  tendency  for  such  longitudinal  flutings  to  be  more 

or  less  locaHsed  at  the  end  of  the  eUipsoid,  or  in  the  region  where 

the  unduloid  merges  into  its  spherical  base*.     In  the  latter  region 

we  often  meet  with  a  regular  series  of  short  longitudinal  folds,  as 

in  the  forms  denominated  L.  semistriata.     All  these  various  forms 

of  surface  can  be  imitated,  or  precisely  reproduced,  by  the  art  of 

the  glass-blower;   and  they  can  be  seen  in  a  contracting  bubble  of 

saponin,  though  not  in  the  more  fluid  soap-bubble.     They  remind 

one  of  the  ribs  or  flutings  in  the  film  or  sheath  which  splashes  up 

to  envelop  a  smooth  pebble  dropped  into  a  hquid,  as  Mr  Worthington 

has  so  beautifully  shewn. 

In  Mr  Worthington's  experiment  there  appears  to  be  something 

of  the  nature  of  a  viscous  drag  in  the  surface-pellicle ;   but  whatever 

be  the  actual  cause  of  variation  of  tension,  it  is  not  difiicult  to 

see  that  there  must  be  in  general  a  tendency  towards  longitudinal 

puckering  or  "fluting"  in  the  case  of  a  thin- walled  cyhndrical  or. 

other  elongated  body,  rather  than  a  tendency  towards  transverse 

puckering,  or  "pleating."     For  let  us  suppose  that  some  change 

takes  place  involving  an  increase  of  surface-tension  in  some  small 

area  of  the  curved  wall,  and  leading  therefore  to  an  increase  of 

pressure :   that  is  to  say  let  T  become  T  +  t,  and  P  become  P  +  p. 

Our  new  equation  of  equilibrium,  then,  in  place  of  P  =  T/r  -f-  T/r', 

becomes 

T+t     T+t 
P  +  P  =  ——  +  —J-' 


*  Certain  palaeontologists  (e.g.  Haeusler  and  Spandel)  have  asserted  that  in 
each  family  or  genus  the  plain  smooth-shelled  forms  are  primitive  and  ancient, 
and  that  the  ribbed  and  otherwise  ornamented  shells  make  their  appearance  at 
later  dates  in  the  course  of  advancing  evolution  (cf.  Rhumbler,  Foraminiferen 
der  Plankton-Expedition,  1911,  p.  21).  If  this  were  true  it  would  be  of  fundamental 
importance:   but  this  book  of  mine  would  not  deserve  to  be  written. 


420  THE  FORMS  OF  CELLS  [ch. 

and  by  subtraction, 

p  =  tlr  +  tjr'. 

Now  if  r<  r\        t/r  >  tjr'. 

Therefore,  in  ordei  to  produce  the  small  increment  of  pressure  p, 
it  is  easier  to  do  so  by  increasing  tjr  than  tjr' ;  that  is  to  say,  the 
easier  way  is  to  alter  or  diminish  r.  And  the  same  will  hold  good 
if  the  tension  and  pressure  be  diminished  instead  of  increased. 

This  is  as  much  as  to  say  that,  when  corrugation  or  "ripphng" 
of  the  walls  takes  place  owing  to  small  changes  of  surface-tension, 
and  consequently  of  pressure,  such  corrugation  is  more  Hkely  to 
take  place  in  the  plane  of  r — that  is  to  say,  in  the  'plane  of  greatest 
curvature.  And  it  follows  that  in  such  a  figure  as  an  ellipsoid, 
wrinkling  will  be  most  likely  to  take  place  not  only  in  a  longitudinal 
direction  but  near  the  extremities  of  the  figure,  that  is  to  say  again 
in  the  region  of  greatest  curvature. 

The  longitudinal  wrinkUng  of  the  flask-shaped  bodies  of  our 
Lagenae,  and  of  the  more  or  less  cyhndrical  cells  of  many  other 
Foraminifera  (Fig.  134),  is  in  complete  accord  with  the  above  con- 
siderations ;  but  nevertheless,  we  soon  find  that  .our  result  is  not 
a  general  one  but  is  defined  by  certain  hmiting  conditions,  and  is 
accordingly  subject  to  what  are,  at  first  sight;  important  exceptions 
For  instance,  when  we  turn  to  the  narrow  neck  of  the  Lagena  we 
see  at  once  that  our  theory  no  longer  holds ;  for  the  wrinkling  which 
was  invariably  longitudinal  in  the  body  of  the  cell  is  as  invariably 
transverse  in  the  narrow  neck.  The  reason  for  the  difference  is  not 
far  to  seek.  The  conditions  in  the  neck  are  very  different  from 
those  in  the  expanded  portion  of  the  cell :  the  main  difference  being 
that  the  thickness  of  the  wall  is  no  longer  insignificant,  but  is  of 
considerable  magnitude  as  compared  with  the  diameter,  or  circum- 
ference, of  the  neck.  We  must  accordingly  take  it  into  account  in 
considering  the  bending  moments  at  any  point  in  this  region  of  the 
shell-wall.  And  it  is  at  once  obvious  that,  in  any  portion  of  the 
narrow  neck,  flexure  of  a  wall  in  a  transverse  direction  will  be  very 
difficult,  while  flexure  in  a  longitudinal  direction  will  be  compara- 
tively easy;  just  as,  in  the  case  of  a  long  narrow. strip  of  iron,  we 
may  easily  bend  it  into  folds  running  transversely  to  its  long  axis, 
but  not  the  other  way.     The  manner  in  which  our  little  Lagena-sheW 


V] 


OF  FLUTED  OR  PLEATED  CELLS 


421 


tends  to  fold  or  wrinkle,  longitudinally  in  its  wider  part  and  trans- 
versely or  annularly  in  its  narrow  neck,  is  thus  completely  explained. 
An  identical  phenomenon  is  apt  to  occur  in  the  little  flask-shaped 
gonangia,  or  reproductive  capsules,  of  some  of  the  hydroid  zoophytes. 
In  the  annexed  drawings  of  these  gonangia  in  two  species  of  Cam- 
panularia,  we  see  that  in  one  case  the  httle  vesicle  has  the  flask- 
shaped  or  unduloid  configuration  of  a  Lagena;  and  here  the  walls 
of  the  flask  are  longitudinally  fluted,  just  after  the  manner  we  have 
witnessed  in  the  latter  genus.  In  the  other  Campanularian  the 
vesicles  are  long,  narrow  and  tubular,  and  here  a  transverse  folding 


Fig.  134.     Nodosaria  scalaria 
Batsch. 


135.  Gonangia  of  Canjpanularians. 
(a)  C.  gracilis;  [b)  C.  grandis. 
After  AUman. 


or  pleating  takes  the  place  of  the  longitudinally  fluted  pattern; 
and  the  very  form  of  the  folds  or  pleats  is  enough  to  suggest  that 
we  are  not  dealing  here  with  a  simple  phenomenon  of  surface-tension, 
but  with  a  condition  in  which  surface-tension  and  stiffness  are  both 
present,  and  play  their  parts  in  the  resultant  form. 

An  everted  rim,  or  short  neck,  may  arise  in  various  ways  apart 
from  the  phenomenon  of  the  hanging  drop.  To  make  a  "thistle- 
head"  the  glassblower  blows  a  bubble,  and  from  that  another  one; 
after  blowing  the  latter  up  large  and  thin  he  crushes  it  to  pieces, 
and  melting  down  what  is  left  of  it  he  forms  the  rim.  I  take  it  that 
the  neck  or  rim  of  the  shell  in  Diffiugia  is  formed  in  an  analogous 
way,  in  connection  with  the  growth  of  a  new  individual  at  the  mouth 


422 


THE  FORMS  OF  CELLS 


[CH. 


of  the  first.  There  is  a  very  neat  expanded  orifice  in  the  cyst  of 
Chromulina  (Fig.  136);  it  is  doubtless  fashioned  in  just  as  simple 
a  way,  but  how  I  know  not. 

Passing  from  the  sohtary  flask-shaped  cell  of  Lagena,  but  without 
leaving  the  Foraminifera,  we  find  in  Nodosaria,  Rheophax  or  Sagrina 
constricted  cyUnders,  or  successive  unduloids,  such  as  are  repre- 
sented in  Fig.  137.     In  some  of  these,  as  in  the  arenaceous  genus 


Fig.  136.     Flask-shaped  shells  or  cysts,   a,  b,  Chromulina  and 
Deropyxis  (Flagellata);    c,  Difflugia. 


tig.  llil .     Various  species  of  Nodosaria,  Rheophax.  Sagrina.    After  Brady. 

Rheophax,  we  have  to  do  with  the  ordinary  phenomenon  of  a 
partially  segmenting  cylinder.  But  in  others,  the  structure  is  not 
developed  out  of  a  continuous  protoplasmic  cyhnder,  but,  as  we 
can  see  by  examining  the  interior  of  the  shell,  it  has  been  formed 
in  successive  stages,  beginning  with  a  simple  unduloid  ''Lagena,'' 
about  which,  after  it  solidified,  another  drop  of  protoplasm  accu- 
mulated, and  in  turn  assumed  the  unduloid  or  lagenoid  form.  The 
chains  of  interconnected  bubbles  which  Morey  and  Draper  made  many 
years  ago  of  melted  resin  are  a  similar  if  not  identical  phenomenon*. 


*  See  Sillijnan's  Journal,  ii,  p.  179,  1820;  and  cf.  Plateau,  op.  cit.  ii,  pp.  134,  461. 


V] 


OF  THE  QUIET  OF  THE  DEEP  SEA 


423 


Fishes  shew  a  vast  though  hmited  variety  of  form;  and  some  of 
the  strangest  shapes  are  found  in  the  great  depths  of  the  ocean. 
Here,  in  unchanging  temperature,  in  darkness  save  for  a  few 
phosphorescent  rays,  above  all  in  unruffled  stillness  and  eternal 


Fig.  138.  Deep-sea  fishes  (Stomiatidae).  a,Lamprotoxusflagellibarbis;  b,  Eustomias 
dactylobus;  c,  E.  parri;  d,  E.  schmidti;  e,  E.  silvescens.  After  Tate  Regan  and 
Trewavas. 


calm*,  the  conditions  of  hfe  are  strange  indeed.  In  deep-sea  fishes 
length  and  attenuation  are  common  characters  of  the  body  and  of 
its  parts.  A  barbel  below  the  hp  may  grow  to  ten  times  the 
whole  length  of  the  fish ;  it  ends,  commonly,  in  a  httle  bulb  or  blob ; 
it  may  give  off  threadlike  branches,  and  these  last  slender  filaments 

*  In  Overbeck's  jet-experiments  {supra,  p.  394)  the  water  into  which  the  jet  is 
led  must  first  stand  for  many  hours,  till  all  internal  movements  and  temperature- 
differences  are  eliminated. 


424  THE  FORMS  OF  CELLS  [ch. 

are  sometimes  finely  beaded* ;  and  slight  differences  in  the  beading 
and  branching  are  said  to  characterise  allied  species  of  fish.  Such 
a  barbel  looks  hke  a  jet  or  branching  stream  of  one  fluid  falhng 

through  another.  It  may  indeed 
be  that  in  these  quiet  depths 
growth  easily  follows  its  fines  of 
least. resistance,  and  that  in  the 
shaping  of  these  pecufiar  out- 
growths hydrodynamical  and 
capillary  forces  are  taking  the 
upper  hand. 

We  have  found  it  easy  to  illus- 
trate the  sphere,  the  cy finder  and 
the  unduloid,  three  of  the  six 
"surfaces  of  Plateau,"  all  with 
an  endless  wealth  of  illustration 
among  the  simplest  of  organisms. 
The  plane  we  need  hardly  look 
for  among  the  finite  outlines  of  a 
fluid  body;  and  the  catenoid,  also 
*a  surface  of  zero  mean  curvature, 
can  likewise  only  be  a  rare  and 
transitory  configuration.  One  last 
surface  still  remains,  namely  the 
nodoid;  and  there  also  remains 
one  very  common  but  most  re- 
markable Protozoan  configura- 
Fig.  139.    Stentor,  a  cUiate  infusorian:     tion,  that  of  the  cifiate  Infusoria, 

from  Savile  Kent.  ii.i,  ^.i,  j.     •  ,.-     r     i. 

to  the  most  characteristic  leature 

of  which  we  have  not  so  far  found  a  physical  analogue.     Here  the 

curved  contour  seems  to  enter,  re-enter,  and  disappear  within  the 

substance  of  the  body,  so  bounding  a  deep  and  twisted  space  or 

passage,  which  merges  with  the  fluid  contents  and  vanishes  within 

the   cell,   and   is   called   by   naturafists   the   "gullet."     This   very 

pecufiar  and  compficated  structure  is  only  kept  in  equifibrium,  and 

in  existence,  by  the  constant  activity  of  cifia  over  the  general  surface 

*  See,  for  instance,  C.  Tate  Regan  and  E.  Trewavas  on  The  Stomiatidae  of  the 
Dana  Expedition,  1930;   W.  Beebe,  Deep-sea  Stomiatoids,  Copeia,  Dec.  1833. 


V]  OF  THE  CILIATE  INFUSORIA  425 

of  the  body  and  very  especially  in  the  said  gullet  or  re-entrant 
portion  of  the  surface.  Now  we  have  seen  the  nodoid  to  be  a  curved 
surface,  re-entering  on  itself  and  endless;  no  method  of  support, 
by  wire-rings  or  otherwise,  enables  us  to  construct  or  reahse  more 
than  a  small  portion  of  it.  But  the  typical  cihate,  such  as  Para- 
moecium,  looks  just  hke  what  we  might  expect  a^  nodoid  surface  to 
be,  if  we  could  only  realise  it  (or  a  single  segment  of  it)  in  a  drop  of 
fluid,  and  imagine  it  to  be  kept  in  quasi-equiUbrium  by  continual 
cihary  activity.  I  suspect,  indeed,  that  here  is  nothing  more,  and* 
nothing  less,  than  a  partial  realisation  of  the  nodoid  itself;  that  the 
so-called  gullet  is  but  the  characteristic  inversion  or  "kink"  in  that 
curve;  and  that  the  ciHa,  which  normally  clothe  the  surface  and 
always  line  the  gullet,  are  needed  to  reahse  and  to  maintain  the 
unstable  equihbrium  of  the  figure.  If  this  be  so — it  is  a  suggestion 
and  no  more — we  shall  have  found  among  our  simple  organisms  the 
complete  realisation,  in  varying  abundance,  of  each  and  all  of  the 
six  surfaces  of  Plateau.  On  each  and  all  of  them  we  have  a  host 
of  beautiful  "patterns"  of  various  sorts;  all  of  them  so  beautiful 
and  so  symmetrical  that  they  ought  to  be  capable  of  geometric 
representation — and  all  waiting  for  their  interpreter ! 

From  all  these  configurations,  which  the  law  of  minimal  area 
controls  and  dominates,  Aynoeba  stands  aloof  and  alone.  The  rest 
are  all  figures  of  equilibrium,  unstable  though  it  may  sometimes  be. 
But  Amoeba  is  the  characteristic  case  of  a  fluid  surface  without  an 
equihbrium;  it  is  the  very  negation  of  stabihty.  In  composition 
it  is  neither  constant  nor  homogeneous ;  its  chemistry  is  in  constant 
flux,  its  surface  energies  vary  from  here  to  there,  its  fluid  substance 
is  drawn  hither  and  thither;  within  and  without  it  is  never  still, 
be  its  motions  swift  or  be  they  slow.  The  heterogeneity  of  its 
system  points  towards  a  maximal  surface-area,  .rather  than  a 
minimal  one ;  only  here  and  there,  in  small  portions  of  its  hetero- 
geneous substance,  do  we  see  the  rounded  contours  of  a  fluid  drop, 
in  token  of  temporary  equihbrium.  Only  when  its  heterogeneous 
reactions  quieten  down  and  the  little  living  speck  enters  on  its 
"resting-stage,"  does  the  protoplasmic  body  withdraw  itself  into  a 
sphere  and  the  law  of  area  minima  come  into  its  own.  Physically 
analogous  is  the  case  of  such  comphcated  pseudopodia,  or  "  axopodia  ", 
as  we  find  among  the  Foraminifera  and  Hehozoa :   where  the  whole 


426  THE  FORMS  OF  CELLS  [ch. 

fabric  is  in  a  flux,  and  currents  flow  and  granules  are  carried 
hither  and  thither.  Here  again  there  is  no  statical  equilibrium; 
but  surface  tension  varies,  as  does  the  chemistry  of  the  protoplasm, 
from  oiie  spot  to  another. 

The  great  oceanic  group  of  the  Radiolaria,  and  the  highly  com- 
plicated skeletons  which  they  construct,  give  us  many  beautiful 
illustrations  of  physical  phenomena,  among  which  the  efl'ects  of 
surface-tension  are  as  usual  prominent.  But  we  shall  deal  later  on 
with  these  little  skeletons  under  the  head  of  spicular  concretions. 

In  a  simple  and  typical  Heliozoan,  such  as  the  sun-animalcule, 
Actinophrys  sol,  we  have  a  "drop"  of  protoplasm,  contracted  by  its 
surface  tension  into  a  spherical  form.  Within  this  heterogeneous 
protoplasm  are  more  fluid  portions,  and  a  similar  surface-tension 
causes  these  also  to  assume  the  form  of  spherical  "vacuoles,"  or  of 
little  clear  drops  within  the  big  one;  unless  indeed  they  become 
numerous  and.  closely  packed,  in  which  case  they  run  together  and 
constitute  a  "froth,"  such  as  we  shall  study  in  the  next  chapter. 
One  or  more  of  such  clear  spaces  may  be  what  is  called  a  "con- 
tractile vacuole " :  that  is  to  say,  a  droplet  whose  surface-tension 
is  in  unstable  equilibrium  and  is  apt  to  vanish  altogether,  so  that 
the  definite  outHne  of  the  vacuole  suddenly  disappears*.  Again, 
within  the  protoplasm  are  one  or  more  nuclei,  whose  own  surface- 
tension  draws  them  in  turn  into  the  shape  of  spheres.  Outwards 
through  the  protoplasm,  and  stretching  far  beyond  the  spherical 
surface  of  the  cell,  run  stiff  linear  threads  of  modified  protoplasm, 
reinforced  in  some  cases  by  delicate  siliceous  needles.  In  either 
case  we  know  little  or  nothing  about  the  forces  which  lead  to  their 
production,  and  we  do  not  hide  our  ignorance  when  we  ascribe 
their  development  to  a  "radial  polarisation"  of  the  cell.  In  the 
case  of  the  protoplasmic  filament,  we  may  (if  we  seek  for  a  hypo- 
thesis) suppose  that  it  is  somehow  comparable  to  a  viscid  stream, 
or  "liquid  vein,"  thrust  or  spirted  out  from  the  body  of  the  cell. 
But  when  it  is  once  formed,  this  long  and  comparatively  rigid 
filament  is  separated  by  a  distinct  surface  from  the  neighbouring 

*  The  presence  or  absence  of  the  contractile  vacuole  or  vacuoles  is  one  of  the 
chief  distinctions,  in  systematic  zoology,  between  the  Heliozoa  and  the  Radiolaria. 
As  we  have  seen  on  p.  295  (footnote),  it  is  probably  no  more  than  a  physical  con- 
sequence of  the  different  conditions  of  existence  in  fresh  and  in  salt  water. 


V]  OF  THE  SUN-ANIMALCULES  427 

protoplasm,  that  is  to  say,  from  the  more  fluid  surface-protoplasm 
of  the  cell;  and  the  latter  begins  to  creep  up  the  filament,  just  as 
water  would  creep  up  the  interior  of  a  glass  tube,  or  the  sides  of 
a  glass  rod  immersed  in  the  liquid.  It  is  the  simple  case  of  a  balance 
between  three  separate  tensions:  (1)  that  between  the  filament  and 
the  adjacent  protoplasm,  (2)  that  between  the  filament  and  the 
adjacent  water,  and  (3)  that  between  the  water  and  the  protoplasm. 
Catting  these  tensions  respectively  T/^,  T^^,,  and  T^^^,,  equilibrium 
will   be   attained   when   the   angle   of  contact   between   the   fluid 

T    —  T 

protoplasm  and  the  filament  is  such  that  cos  a  =  -^~r^ — ~'  -     It  is 

evident  in  this  case  that  the  angle  is  a  very  small  one.  The  precise 
form  of  the  curve  is  somewhat  different  from  that  which,  under 
ordinary  circumstances,  is  assumed  by  a  liquid  which  creeps  up  a 
solid  surface,  as  water  in  contact  with  air  creeps  up  a  surface  of 
glass;  the  -difference  being  due  to  the  fact  that  here,  owing  to  the 
density  of  the  protoplasm  being  all  but  identical  wuth  that  of  the 
surrounding  medium,  the  whole  system  is  practically  immune  from 
gravity.  Under  normal  circumstances  the  curve  is  part  of  the 
"elastic  curve"  by  which  that  surface  of  revolution  is  generated 
which  we  have  called,  after  Plateau,  the  nodoid;  but  in  the  present 
case  it  is  apparently  a  catenary.  Whatever  curve  it  be,  it  obviously 
forms  a  surface  of  revolution  around  the  filament. 

Since  this  surface-tension  is  symmetrical  around  the  filament,  the 
latter  will  be  pulled  equally  in  all  directions;  in  other  words  the 
filament  will  tend  to  be  set  normally  to  the  surface  of  the  sphere, 
that  is  to  say  radiating  directly  outwards  from  the  centre.  If  the 
distance  between  two  adjacent  filaments  be  considerable,  the  curve 
will  simply  meet  the  filament  at  the  angle  a  already  referred  to; 
but  if  they  be  sufficiently  near  together,  we  shall  have  a  continuous 
catenary  curve  forming  a  hanging  loop  between  one  filament  and 
the  other.  And  when  this  is  so,  and  the  radial  filaments  are  more 
or  less  symmetrically  interspaced,  we  may  have  a  beautiful  system 
of  honeycomb-like  depressions  over  the  surface  of  the  organism, 
each  cell  of  the  honeycomb  having  a  strictly  defined  geometric 
configuration  (cf.  p.  710). 

In  the  simpler  Radiolaria.  the  spherical  form  of  the  entire  organism 
is  equally  well  marked;    and  here,  as  also  in  the  more  complicated 


428  THE  FORMS  OF  CELLS  [ch. 

Heliozoa  (such  as  Actinosphaerium),  the  organism  is  apt  to  be 
differentiated  into  layers,  so  constituting  sphere  within  sphere, 
whose  inter-surfaces  become  the  seat  of  adsorption,  and  the  locus 
of  skeletal  secretion.  One  layer  at  least  is  close-packed  with 
vacuoles,  forming  an  "alveolar  meshwork,"  with  the  configura- 
tions of  which  we  shall  attempt  in  another  chapter  to  correlate 
certain  characteristic  types  of  skeleton.  In  Actinosphaerium  the 
radial  filaments  pass  through  the  outer  layer,  and  seem  to  rest  on 
but  do  not  penetrate  the  layer  below;  this  must  happen  if  the 
surface-energy  between  the  one  plasma-layer  and  the  other  be  less 
than  that  between  the  filament  and  the  water  around*. 

A  very  curious  conformation  is  that  of  the  vibratile  "collar," 
found  in  Codosiga  and  the  other  "Choanoflagellates,"  and  which  we 
also  meet  with  in  the  "collar-cells"  which  line  the  interior  cavities 
of  a  sponge.  Such  collar-cells  are  always  very  minute,  and  the 
collar  is  constituted  of  a  very  dehcate  film  which  shews  an  undu- 
latory  or  ripphng  motion.  It  is  a  surface  of  revolution,  and  as  it 
maintains  itself  in  equihbrium  (though  a  somewhat  unstable  and 
fluctuating  one)  it  must  be,  under  the  restraining  circumstances  of 
its  case,  a  surface  of  minimal  area.  But  it  is  not  so  easy  to  see 
what  these  special  circumstances  are,  and  it  is  obvious  that  the 
collar,  if  left  to  itself,  must  shrink  or  shrivel  towards  its  base  and 
become  confluent  with  the  general  surface  of  the  cell ;  for  it  has  no 
longitudinal  supports  and  no  strengthening  ring  at  its  periphery. 
But  in  all  these  collar-cells,  there  stands  within  the  annulus  of  the 
collar  a  large  and  powerful  cilium  ot  flagellum,  in  constant  move- 
ment; and  by  the  action  of  this  flagellum,  and  doubtless  in  part 
also  by  the  intrinsic  vibrations  of  the  collar  itself,  there  is  set  up  a 
constant  steady  current  in  the  surrounding  water,  whose  direction 
would  seem  to  be  such  that  it  passes  up  the  outside  of  the  collar, 
down  its  inner  side,  and  out  in  the  middle  in  the  direction  of  the 
flagellum;  and  there  is  a  distinct  eddy,  in  which  foreign  particles 
tend  to  be  caught,  around  the  peripheral  margin  of  the  collar  f. 

*  Cf.  N.  K.  Koltzoff,  AnaL  Anzeiger,  xli,  p.  190,  1912. 

t  The  very  minute  size  of  Codosiga,  whose  collar  and  flagellum  measure  about 
30-40 /i,,  and  of  all  such  collar-cells,  make  the  apparently  complex  current-system 
all  the  harder  to  comprehend.  Cf.  G.  Lepage,  Notes  on  C.  botrytis,  Q.J. M.S. 
LXix,  pp.  471-508,  1925. 


V]  OF  COLLARED  CELLS  429 

When  the  cell  dies,  that  is  to  say  when  motion  ceases,  the  collar 
immediately  shrivels  away  and  disappears.  It  is  notable,  by  the 
way,  that  the  edge  of  this  little  mobile  cup  is  always  smooth,  never 
notched  or  lobed  as  in  the  cases  we  have  discussed 
on  p.  390 :  this  latter  condition  being  the  outcome 
of  a  definite  instability,  marking  the  close  of  a 
period  of  equihbrium.  But  the  vibratile  collar  of 
Codosiga  is  in  "a  steady  state,"  its  equihbrium, 
such  as  it  is,  being  constantly  renewed  and  per- 
petuated, like  that  of  a  juggler^s  pole,  by  the 
motions  of  the  system.  Somehow  its  existence  is 
due  to  the  current  motions  and  to  the  traction 
exerted  upon  it  through  the  friction  of  the  stream 
which  is  constantly  passing  by.  In  short,  I  think 
that  it  is  formed  very  much  in  the  same  way  as 
the  cup-hke  ring  of  streaming  ribbons,  which  we 
see  fluttering  and  vibrating  in  the  air-current  of  „.     ,,^ 

a    ventiktmg   fan.     If  we    turn    once   more   to 
Mr   Worthington's    Study   of  Splashes,    we   may   find   a   curious 
suggestion  of  analogy  in  the  beautiful  craters  encirchng  a  central 
jet  (as  the  collar  of  Codosiga  encircles  the  flagellum),  which  we  see 
produced  in  the  later  stages  of  the  splash  of  a  pebble. 

Another  exceptional  form  of  cell,  and  beautiful  manifestation  of 
capillarity,  occurs  in  Trypanosomes,  those  tiny  parasites  of  the 
blood  which  are  associated  with  sleeping-sickness  and  certain  other 
dire  maladies  of  beast  and  man.  These  minute  organisms  consist 
of  elongated  sohtary  cells  down  one  side  of  which  runs  a  very 
delicate  frill,  or  "undulating  membrane,"  the  free  edge  of  which  is 
seen  to  be  sHghtly  thickened,  and  the  whole  of  which  undergoes 
rhythmical  and  beautiful  wavy  movements.  When  certain  Trypano- 
somes are  artificially  cultivated  (for  instance  T.  rotatorium,  from  the 
blood  of  the  frog),  phases  of  growth  are  witnessed  in  which  the 
organism  has  no  undulating  membrane,  but  possesses  a  long  cihum 
or  "flagellum,"  springing  from  near  the  front  end,  and  exceeding 
the  whole  body  in  length*.  Again,  in  T.  lewisii,  when  it  reproduces 
by  "multiple  fission,"  the  products  of  this  division  are  Hkewise 

*  Cf.  Doflein,  Lehrbuch  der  Protozoenkunde,  1911,  p.  422. 


430 


THE  FORMS  OF  CELLS 


[CH. 


devoid  of  an  undulating  membrane,  but  are  provided  with  a  long 
free  flagellum*.  It  is  a  plausible  assumption  to  suppose  that,  as 
the  flagellum  waves  about,  it  comes  to  lie  near  and  parallel  to  the 
body  of  the  cell,  and  that  the  frill  or  undulating  membrane  is  formed 
by  the  clear,  fluid  protoplasm  of  the  surface  layer  springing  up  in 


Fig.  141.     A,  Trichomonas  muris  Hartmann;  B,  Trichomastix  serpentis  DobeU; 
C,  Trichomonas  angusta  Alexeieff.     After  Kofoid. 


Fig.  142.     A  "Trypanosome. 

a  film  to  run  up  and  along  the  flagellum,  just  as  a  soap-film  would 
form  under  similar  circumstances. 

This  mode  of  formation  of  the  undulating  membrane  or  frill 
appears  to  be  confirmed  by  the  appearances  shewn  in  Fig,  14L 
Here  we  have  three  little  organisms  closely  allied  to  the  ordinary 
Trypanosomes,  of  which  one,  Trichomastix  (B),  possesses  four 
flagella,  and  the  other  two.  Trichomonas,  apparently  three  only: 

*  Cf.  Minchin,  Introduction  to  the  Study  of  the  Protozoa,  1914,  p.  293,  Fig.  127. 


V]  OF  UNDULATING  MEMBRANES  431 

the  two  latter  possess  the  frill,  which  is  lacking  in  the  first*.  But 
it  is  impossible  to  doubt  that  when  the  frill  is  present  (as  in  A  and 
C),  its  outer  edge  is  constituted  by  the  apparently  missing  flagellum 
a,  which  has  become  attached  to  the  body  of  the  creature  at  the 
point  c,  near  its  posterior  end;  and  all  along  its  course  the  super- 
ficial protoplasm  has  been  drawn  out  into  a  film,  between  the 
flagellum  a  and  the  adjacent  surface  or  edge  of  the  body  b. 

Moreover,  this  mode  of  formation  has  been  actually  witnessed 
and  described,  though  in  a  somewhat 
exceptional  case.  The  little  flagellate 
monad  Herpetomonas  is  normally  desti- 
tute of  an  undulating  membrane,  but 
possesses  a  single  long  terminal  flagellum. 
According  to  Prof.  D.  L.  Mackinnon,  the 
cytoplasm  in  a  certain  stage  of  growth 
becomes  somewhat  "sticky,"  a  phrase 
which  we  may  in  all  probabihty  interpret 
to  mean  that  its  surface-tension  is  being 
reduced.  For  this  stickiness  is  shewn  in 
two  ways.  In  the  first  place,  the  long 
body,  in  the  course  of  its  various  bending 
movements,  is   apt  to  adhere  head  to  „ 

,    .,    .        ^  ,,        .    .  ,    ,  Fig.  143.  ^er^e^omonas  assuming 

tail   (so  to   speak),   giving   a   rounded   or  the    undulatory   membrane 

sometimes  annular  form  to  the  organism,  of  a  Trypanosome.  After 
such  as  has  also  been  described  in  certain  ^-  ^'  ^^a^^^^^non. 
species  or  stages  of  Trypanosomes.  But  again,  the  long  flagellum, 
if  it  get  bent  backwards  upon  the  body,  tends  to  adhere  to  its 
surface.  "Where  the  flagellum  was  pretty  long  and  active,  its 
efforts  to  continue  movement  under  these  abnormal  conditions 
resulted  in  the  gradual  lifting  up  from  the  cytoplasm  of  the  body 
of  a  sort  of  pseudo-undnlsitmg  membrane  (Fig.  143).  The  move- 
ments of  this  structure  were  so  exactly  those  of  a  true  undu- 
lating membrane  that  it  was  difficult  to  beheve  one  was  not  deahn^ 
with  a  small,  blunt  Trypanosome"*.     This  in  short  is  a  precise 

*  Cf.  C.  A.  Kofoid  and  Olive  Swezy,  On  Trichomonad  flagellates,  etc.,  Pr. 
Amer.  Acad,  of  Arts  and  Sci.  li,  pp.  289-378,  1915.  Also  C.  H.  Martin  and  Muriel 
Robertson,  Q.J. M.S.  lvii,  pp.  53-81,  1912. 

t  D.  L.  Mackinnon,  Herpetomonads  from  the  alimentary  tract  of  certain  dungflies, 
Parasitology,  iii,  p.  268,  1910. 


432  THE  FORMS  OF  CELLS  [ch. 

description  of  the  mode  of  development  which,  from  theoretical 
considerations  alone,  we  should  conceive  to  be  the  natural  if  not 
the  only  possible  way  in  which  the  undulating  membrane  could 
come  into  existence. 

There  is  a  genus  closely  alhed  to  Trypanosoma,  viz.  Trypanoplasma, 
which  possesses  one  free  flagellum,  together  with  an  undulating 
membrane;  and  it  resembles  the  neighbouring  genus  Bodo,  save 
that  the  latter  has  two  flagella  and  no  undulating  membrane.  In 
lijce  manner,  Trypanosoma  so  closely  resembles  Herpetomonas  that, 
when  individuals  ascribed  to  the  former  genus  exhibit  a  free 
flagellum  only,  they  are  said  to  be  in  the  "Herpetomonas  stage." 
In  short,  all  through  the  order,  we  have  pairs  of  genera  which  are 
presumed  to  be  separate  and  distinct,  viz^  Trypanosoma-Herpeto- 
monas,  Trypanoplasma-Bodo,  Trichomastix-Trichomonas,  in  which 
one  differs  from  the  other  mainly  if  not  solely  in  the  fact  that  a  free 
flagellum  in  the  one  is  replaced  by  an  undulating  membrane  in  the 
other.  We  can  scarcely  doubt  that  the  two  structures  are  essen- 
tially one  and  the  same. 

The  undulating  membrane  of  a  Trypanosome,  then,  according 
to  our  interpretation  of  it,  is  a  hquid  film  and  must  obey  the  law 
of  constant  mean  curvature.  It  is  under  curious  limitations  of 
freedom:  for  by  one  border  it  is  attached  to  the  comparatively, 
motionless  body,  while  its  free  border  is  constituted  by  a  flagellum 
which  retains  its  activity  and  is  being  constantly  thrown,  like  the 
lash  of  a  whip,  into  wavy  curves.  It  follows  that  the  membrane, 
for  every  alteration  of  its  longitudinal  curvature,  must  at  the  same 
instant  become  curved  in  a  direction  perpendicular  thereto;  it 
bends,  not  as  a  tape  bends,  but  with  the  accompaniment  of  beautiful 
but  tiny  waves  of  double  curvature,  all  tending  towards  the 
estabhshment  of  an  " equipotential  surface",  which  indeed,  as  it  is 
under  no  pressure  on  either  side,  is  really  a  surface  of  no  curvature 
at  all;  and  its  characteristic  undulations  are  not  originated  by  an 
active  mobihty  of  the  melhbrane  but  are  due  to  the  molecular  tensions 
which  produce  the  very  same  result  in  a  soap-film  under  similar 
circumstances.  Some  of  the  larger  Spirochaetes  possess  a  structure  so 
like  to  the  undulating  membrane  of  the  Trypanosomes  that  it  has  led 
some  persons  to  include  these  peculiar  allies  of  the  bacteria  among  the 
flagellate  protozoa;    but  it  would  seem  (according  to  the  weight  of 


V] 


OF  UNDULATING  MEMBRANES 


433 


evidence)  that  the  Spirochaete  membrane  does  not  undulate,  and 
possesses  no  thickened  border  or  marginal  filament  (Randfade)*. 
It  forms  a  "screw-surface,"  or  helicoid,  and,  though  we  might 
think  that  nothing  could  well  be  more  curved,  yet  its  mathematical 
properties  are  such  that  it  constitutes  a  "ruled  surface"  whose 
mean  curvature  is  everywhere  nil.  Precisely  such  a  surface,  and  of 
exquisite  beauty,  may  be  produced  by  bending  a  wire  upon  itself  so 
that  part  forms  an  axial  rod  and  part  winds  spirally  round  the  axis, 
and  then  dipping  the  whole  into  a  soapy  solution. 


Fig.  144,     Dinenympha  gracilis  Leidy. 

A  pecuUar  type  is  the  flattened  spiral  of  Dinenympha '\ ,  which 
reminds  us  of  the  cylindrical  spiral  of  a  Spirillum  among  the  bacteria. 
Here  we  have  a  symmetrical  figure,  whose  two  opposite  surfaces 
each  constitute  a  surface  of  constant  mean  curvature;  it  is  evidently 


*  For  a  discussion  of  this  obscure  lamella,  and  of  the  crista  which  seems  to 
correspond  with  it  in  other  species,  see  Doflein,  Probleme  der  Protistenkunde,  ii, 
Die  Natur  der  Spirochaeten,  Jena,  1911;  see  also  ChfFord  Dobell,  Arch.f.  Protisten- 
kunde, 1912. 

t  Leidy,  Parasites  of  the  termites,  Journ.  Nat.  Sci.,  Philadelphia,  viii,  pp.  425- 
447,  1874-81;   cf.  Savile-Kent's  Infusoria,  ii,  p.  551. 


434  THE  FORMS  OF  CELLS  [ch. 

a  figure  of  equilibrium  under  certain  special  conditions  of  restraint. 
The  cylindrical  coil  of  the  Spirillum,  on  the  other  hand,  is  a  surface 
of  constant  mean  curvature,  and  therefore  of  equilibrium,  as  truly, 
and  in  the  same  sense,  as  the  cylinder  itself. 

A  very  beautiful  "  saddle-shaped "  surface,  of  constant  mean 
curvature,  is  to  be  found  in  the  little  diatom  Cmnpylodiscus,  and 
others,   a  httle  more  complicated,  in  the  allied  genus  Surirella*. 

These  undulating  and  helicoid  surfaces  are  exactly  reproduced 
among  certain  forms  of  spermatozoa.  The  tail  of  a  spermatozoon 
consists  normally  of  an  axis  surrounded  by  clearer  and  more  fluid 
protoplasm,  and  the  axis  sometimes  splits  up  into  two  or  more 
slender  filaments.  To  surface-tension  operating  between  these  and 
the  surface  of  the  fluid  protoplasm  (just  as  in  the  case  of  the  flagellum 
of  the  Trypanosome),  I  ascribe  the  formation  of  the  undulating 
membrane  which  we  find,  for  instance,  in  the  spermatozoa  of  the 
newt  or  salamander;  and  of  the  helicoid  membrane,  wrapped  in  a 
far  closer  and  more  beautiful  spiral  than  that  which  we  saw  in 
Spirochaeta,  which  is  characteristic  of  the  spermatozoa  of  many 
birds.  The  undulatory  membrane  which  certain  ciliate  infusoria 
exhibit  is,  seemingly,  a  difl'erent  thing.  It  is  not  based  on  a  single 
marginal  flagellum,  but  consists  of  a  row  of  fine  ciha  fused  together. 
The  membrane  can  be  broken  up  by  certain  reagents  into  fibrillae, 
and — what  is  more  remarkable — a  touch  of  the  micro-dissection 
needle  may  split  it  into  a  multitude  of  cilia,  all  active  but  beating 
out  of  time;  a  moment  more  and  they  unite  again,  all  but  dis- 
appearing from  view  as  they  fuse  into  the  optically  homogeneous 
membrane.  They  unite  as  quickly  and  as  intimately  as  though 
they  were  so  many  liquid  jets,  and  they  manifestly  "partake  of 
fluidity."  Neither  they,  nor  cilia  in  general,  have  received,  nor 
seem  hkely  to  receive,  a  simple  explanation  f.  Nevertheless,  we 
may  see  a  httle  hght  in  the  darkness  after  all. 

It  would  be  overbold  to  seek  for  every  form  of  living  cell  a  parallel 
configuration  due  to  simple  capillary  forces,  as  manifested  in  drop 
or  bubble  or  jet.  And  yet,  if  the  simple  cases  of  sphere  or  cyhnder 
be  the  beginning  of  the  story,  they  assuredly  are  not  the  end.     The 

*  Van  Heurck,  Synopsis  des  Diatomees  de  Belgique,  pis.  Ixxiv,  6;   Ixxvii,  4. 

t  H.  N.  Maier,  Der  feinere  Bau  der  Wimperapparate  der  Infusorien,  Arch.  f. 
Protistenk.  ii,  p.  73,  1903;  R.  Chambers  and  J.  A.  Dawson,  Structure  of  the 
undulating  membrane  in  the  ciliate  Blepharisma,  Biol.  Bull,  xlviii,  p.  240,  1925. 


V]  OF  FLAGELLATE  CELLS  435 

pointed  and  flagellate  cell  of  a  Monad,  one  of  the  least  and  com- 
monest of  micro-organisms,  is  far  removed  from  a  simple  "drop," 
and  all  its  characters,  to  the  microscopist's  eye,  are  both  generally 
and  Specifically  those  of  a  hving  thing.  But  a  drop  of  water  falhng 
through  an  electric  field,  as  in  a  thunderstorm,  is  found  to  lengthen 
out  to  three  or  four  times  as  long  as  it  is  broad;  and  then,  if  the 
strength  of  the  field  increase  a  httle,  the  prolate  drop  becomes 
unstable,  it  grows  spindle-shaped,  and  suddenly  from  one  of  its  two 
pointed  spindle-ends  (the  positive  end  especially)  a  long  and  slender 
filament  shoots  out,  to  the  accompaniment  of  an  electrical  discharge. 
We  need  not  assert  that  the  phenomena  are  identical,  nor  that  th^ 
forces  in  action  are  absolutely  the  same.  Yet  it  is  no  small  thing  to 
have  learned  that  the  pecuhar  conformation  of  the  Httle  flagellate 
Monad  has  its  analogue  in  an  electrified  drop,  and  is  not  unique  after 
all*. 

Before  we  pass  from  the  subject  of  the  conformation  of  the 
solitary  cell  we  must  take  some  account  of  certain  other  exceptional 
forms,  less  easy  of  explanation,  and  still  less  perfectly  understood. 
Such  is  the  case,  for  instance,  of  the  red  blood-corpuscles  of  man 
and  other  vertebrates;  and  among  the  sperm-cells  of  the  decapod 
Crustacea  we  find  forms  still  more  aberrant  and  not  less  perplexing. 
These  are  among  the  comparatively  few  cells  or  cell-like  structures 
whose  form  seeins  to  be  incapable  of  explanation  by  theories  of 
surface-tension. 

In  all  the  mammalia  (save  a  very  few)  the  red  blood-corpuscles 
are  flattened  circular  discs,  dimpled  in  upon  their  two  opposite  sides. 
This  configuration  closely  resembles  that  of  an  india-rubber  ball 
when  we  pinch  it  tightly  between  finger  and  thumb  •)". 

The  form  of  the  corpuscle  is  symmetrical ;  it  is  a  sohd  of  revolu- 
tion, but  its  surface  is  not  a  surface  of  constant  mean  curvature. 
From  the  surface-tension  point  of  view,  the  blood-corpuscle  is  not 
a  surface  of  equilibrium ;  in  other  words,  it  is  not  a  fluid  drop  poised 

*  Cf.  W.  A.  Macky,  On  the  deformation  of  water-drops  in  strong  electric  fields, 
Proc.  B.S.  (A),  cxxxiii,  pp;  565-587,  1931. 

t  On  this  analogy  we  might  expect  the  double  concavity  to  pass,  with  no  great 
difficulty,  into  the  single  hollow  of  a  cup  or  bell,  and  such  a  shape  the  blood- 
corpuscles  are  said  sometimes  to  assume.  Cf.  Weidenreich,  Arch.  f.  mikr.  Anat. 
Lxvii,  1902;  and  cf.  Clerk-Maxwell  on  "dimples"  in  Tr.  R.S.E.  xxvi,  p.  11,  1870. 


436  THE  FORMS  OF  CELLS  [ch. 

in  another  liquid.  Some  other  force  or  forces  must  be  at  work  to 
conform  it,  and  the  simple  effect  of  mechanical  pressure  is  excluded, 
because  the  corpuscle  exhibits  its  characteristic  shape  while  floating 
freely  in  the  blood.  It  has  been  suggested  that  the  corpuscle  is 
perhaps  comparable  to  a  sohd  of  revolution  described  about  one  of 
Cay  ley's  equipotential  curves*,  such  as  we  have  spoken  of  briefly 
on  p.  318.  Were  the  corpuscle  a  sphere,  or  a  thin  plate,  a  gas 
diffusing  inwards  would  reach  all  parts  equally  soon ;  but  the  surface 

would  be  small  in  the  one  case  and 

-^_C  ^ ^         the  volume  in  the  other.     In  so 

far  as  the  corpuscle  resembles  or 
approaches  the  equipotential  form, 
we  might  look  on  it  as  a  com- 
Fiff  14.^  ^"  ^  promise;  but  however  advan- 
tageous such  a  shape  might  be,  and 
however  interesting  physiologically,  we  should  be  as  far  as  ever 
from  understanding  how  it  was  produced.  In  all  other  vertebrates, 
from  fishes  to  birds,  sluggish  or  active,  warm-blooded  or  cold,  the 
blood-corpuscles  have  the  simpler  form  of  a  flattened  oval  disc, 
with  somewhat  sharp  edges  and  eUipsoidal  surfaces,  and  this  again 
is  manifestly  not  a  surface  of  fluid  equilibrium.  But  there  is 
nothing  to  choose  between  the  one  type  and  the  other  in  the  way 
of  physiological  efficiency,  nor  any  apparent  need  for  a  refinement 
of  adaptive  form  in  either  of  them. 

Two  facts  are  noteworthy  in  connection  with  the  form  of  the 
mammalian  blood-corpuscle.  In  the  first  place  its  form  is  only 
maintained,  that  is  to  say  it  is  only  in  equilibrium,  in  specific  relation 
to  the  medium  in  which  it  floats.  If  we  add  water  to  the  blood, 
the  corpuscle  becomes  a  spherical  drop,  a  true  surface  of  minimal 
area  and  stable  equiUbrium;  if,  conversely,  we  add  a  little  salt,  or 
a  drop  of  glycerine,  the  corpuscle  shrinks,  and  its  surface  becomes 
puckered  and  uneven.  So  far,  it  merely  obeys  the  laws  of  diffusion; 
but  the  phenomenon  is  more  complex  than  this|.  For  the  spherical 
form  is  assumed  just  as  well  in  various  isotonic  solutions,  leaving 

♦  Cf.  H.  Hartridge,  Journ.  Physiol,  lii,  p.  Ixxxi,  1919-20;  Eric  Ponder,  Journ. 
Gen.  Physiol,  ix,  pp.  197-204,  625-629,  1925-26. 

I  Cf.  A.  Gough,  On  the  assumption  of  a  spherical  form  by  human  blood- 
corpuscles,  Biochem.  Journ.  xvni,  p.  202,  1924. 


V]  OF  BLOOD  CORPUSCLES  437 

the  volume  unchanged;  a  httle  ammonium  oxalate  impedes  or 
inhibits  the  change  of  form,  a  little  serum  brings  the  spherical 
corpuscles  back  to  biconcave  discs  again.  We  are  no  longer  deaUng 
with  simple  diffusion,  but  with  phenomena  of  a  very  subtle  kind. 

Secondly,  the  form  of  the  corpuscle  can  be  imitated  artificially 
by  means  of  other  colloid  substances.  Many  years  ago  Norris  made 
the  interesting  observation  that  drops  of  glue  in  an  emulsion  assumed 
a  biconcave  form  closely  resembling  that  of  the  mammahan  cor- 
kpuscles*;  the  glue  was  impure  and  doubtless  contained  lecithin. 
Waymouth  Reid  made  similar  emulsions  of  cholesterin  oleate,  in 
which  the  same  conformation  of  the  drops  or  particles  is  beautifully 
shewn;  and  Emil  Hatschek  has  made  somewhat  similar  biconcave 
bodies  by  dropping  gelatine  containing  potassium  ferrocyanide  into 
copper  sulphate  or  a  tannin  solution.  Here  Hatschek  beheves  that 
his  biconcave  drops  are  half -formed  vortex-rings,  arrested  by  the 
formation  of  a  semi-permeable  membrane ;  but  the  explanation  does 
not  seem  to  fit  the  blood-corpuscle.  The  cholesterin  bodies  in 
Waymouth  Reid's  experiment  are  such  as  have  a  place  of  their  own 
among  Lehmann's  "fluid  crystals "f;  and  it  becomes  at  least  con- 
ceivable that  obscure  forces  akin  t6  those  of  crystalHsation  may 
be  playing  their  part  along  with  surface-energy  in  these  strange  but 
famihar  conformations.  The  case  is  a  hard  one  in  every  way. 
From  the  physiological  point  of  view  it  is  difficult  and  coinplex^ 
enough.  For  the  surface  of  the  corpuscle  is  equivalent  to  a  semi- 
permeable membrane  t,  through  which  certain  substances  pass  freely 
but  not  others — for  the  most  part  anions  and  not  cations  §;  and 
accordingly  we  have  here  in  life  a  steady  state  of  osmotic  inequih- 
brium,  of  negative  osmotic  tension  within,  and  to  this  comparatively 
simple  cause  the  imperfect  distension  of  the  corpus(5le  may  be  due. 

*  Proc.  M.S.  ;xn,  pp.  251-257,  1862-63. 

t  Cf.  {int.  al.)  Lehmann,  Ueber  scheinbar  lebende  Kristalle  und  Myelinformen, 
Arch.f.  Entw.  Mech.  xx\^,  p.  483,  1908;   Ann.  d.  Physik,  xliv,  p.  969,  iai4. 

X  That  no  "true  membrane"  exists  has  long  been  known;  cf.  [int.  al.)  Rohring, 
KoU.  Chem.  Beihefte,  viii,  pp.  337-398,  1916.  On  the  other  hand  the  surface  of 
the  corpuscle  is  defined  by  a  monolayer,  and  very  probably  by  the  still  more  stable 
condition  of  two  "interpenetrating"  monolayers,  a  proteid  and  a  lipoid.  Cf. 
Eric  Ponder,  Phys.  Rev.  xvi,  p.  19,  1936;  and  on  "interpenetration,"  Schulman 
and  Rideal  in  Proc.  M.S.  (B),  cxxii,  pp.  29-57,  1937. 

§  Cf.  Hamburger,  Z.  f.  physikal.  Chem.  lxix,  p.  663,  1909;  Pfluger's  Archiv, 
1902,  p.  442;   etc. 


438  THE  FORMS  OF  CELLS  [ch. 

Whatever  the  forces  are  which  make  and  keep  the  man's  corpuscle 
a  dimpled  disc  and  the  frog's  a  flattened  ellipsoid,  they  seem  to  be 
of  a  powerful  kind.  When  we  submit  either  to  great  hydrostatic 
pressure,  it  tends  to  become  spherical  at  last,  the  natural  result  of 
uniform  pressure  over  its  whole  surface;  but  the  pressure  necessary 
to  bring  this  result  about  is  very  great  indeed*.  Since  the  form 
of  the  blood-corpuscle  cannot,  then,  be  rated  as  a  figure  of  equili- 
brium, we  must  be  content  to  regard  it  as  a  "steady  state";  and 
this,  moreover,  is  all  we  can  say  of  its  physico-chemical  conditional 
The  red  blood-corpuscle,  especially  the  non-nucleated  one,  is  in  no 
ordinary  sense  alive'f.  It  has  no  power  of  movement,  of  reproduction 
or  of  repair;  it  is  a  mere  haemoglobin-freighted  drop  of  protein; 
its  own  metabolism,  apart  from  its  alternate  give  and  take  of 
oxygen,  is  slight  indeed  or  absent  altogether.  But  all  the  same, 
chemical  change  is  continually  going  on;  anions  (Uke  HCO3)  pass 
freely  through  its  walls,  simple  cations  (like  Na,  K)  find  it  imper- 
meable; and  so,  between  plasma  and  corpuscles  the  conditions  are 
fulfilled  for  that  steady  osmotic  state  known  as  a  "Donnan  equi- 
Kbrium."  Somehow,  but  we  know  not  how,  a  steady  state  is 
maintained  alike  in  the  corpuscle's  osmotic  equihbrium  and  in  its 
form. 

In  mammalian  blood,  the  running  together  of  the  round  biconcave 
corpuscles  into  "rouleaux"  gives  a  well-known  and  characteristic 
picture.  When  cold,  rouleaux  are  formed  slowly,  in  warmed  plasma 
they  form  quickly  and  well,  in  salt-solution  they  do  not  form  at  all. 

*  The  whole  phenomenon  would  become  simple  and  mechanical  if  we  might 
postulate  a  stiffer  peripheral  region  to  the  corpuscle,  in  the  form  (for  instance)  of  an 
elastic  ring.  Such  an  annular  stiffening,  like  the  "  collapse-rings"  which  an  engineer 
inserts  in  a  boiler  or  the  whalebone  ring  which  a  Breton  fisherman  fits  into  his  beret, 
has  been  repeatedly  asserted  to  exist;  by  Dehler,  Arch.f.  mikr.  Anat.  xlvi,  1895;  by 
Meves,  ibid.  Lxxvii,  1911;  and  especially  by  J.  Riinnstrom,  Was  bedingt  die  Form 
und  die  Formveranderungen  derSaiigetiererythrocytcn,  Arch.  f.  Entw.  Mech.  i,  pp. 
391-409^,1922.  It  has  been  denied  at  least  as  often;  but  the  remarkable  statement 
has  been  lately  made  that  in  a  corpuscle  which  has  been  swollen  up  and  then  brought 
back  to  its  biconcave  form,  the  dimples  reappear  on  the  same  sides  as  before: 
apparently  in  "strong  evidence  for  some  sort  of  fixed  cellular  structure";  see 
R.  F.  Furchgott  and  Eric  Ponder  in  Jonrn.  Exp.  Biol.  xvii.  pp.  30-44,  117-127, 
1940.  See  also,  on  the  whole  subject,  Eric  Ponder,  The  Mammalian  Red  Cell, 
Berlin,  1934. 

t  Cf.  A.  V.  Hill,  Trans.  Faraday  Soc.  xxvi,  p.  667,  1930;  Proc.  R.S.  (B),  1930; 
K.  R.  Dixon,  in  Current  Sci.  vii,  p.  169,  1938;   etc. 


V]  OF  CERTAIN  OSMOTIC  PHENOMENA  439 

The  phenomenon,  though  a  purely  physical  one,  is  none  too  clear. 
There  is  a  difference  of  electrical  potential  between  corpuscles  and 
plasma,  and  the  charged  corpuscles  tend  to  repel  one  another;  but 
they  also  tend  to  adhere  together,  all  the,  more  when  they  meet 
broadside  on,  whether  by  actual  stickiness  or  through  surface- 
energy.  The  attractive  forces  then  overcome  the  repulsive,  and  the 
rouleau  is  formed.  But  if  the  potential  be  reduced,  and  mutual 
repulsion  reduced  with  it,  then  the  corpuscles  stick  together  just  as 
they  happen  to  meet;  rouleaux  are  no  longer  formed,  and  ordinary 
"agglutination"  takes  place.  Whatever  be  the  precise  nature  of 
the  phenomenon,  the  number  of  rouleaux  and  the  mean  number  of 


Fig.  146.     Sperm-cells  of  Decapod  Crustacea  (after  Koltzoff).     a,  Inachus  scorpio; 
b,  Galathea  sqimmijera;    c,  do.  after  maceration,  to  shew  spiral  fibrillae. 

corpuscles  in  each  is  found,  after  a  given  time,  to  obey  a  certain 
law  (Smoluchowsky's  Law),  defining  the  number  of  contacts  of 
floating  bodies  under  ordinary  physical  conditions*. 

The  sperm-cells  of  the  Decapod  Crustacea  exhibit  various  singular 
shapes.  In  the  crayfish  they  are  flattened  cells  with  stiff  curved 
processes  radiating  outwards  Hke  St  Catherine's  wheel;  in  Inachus 
there  are  two  such  circles  of  stiff  processes ;  in  Galathea  we  have  a 
still  more  comphcated  form,  with  long  and  slightly  twisted  processes. 

*  Smoluohowsky,  Ztschr.  f.  physik.  Chemie,  xcix,  p.  129,  1917;  Eric  Ponder, 
On  Rouleaux-formation,  Q.  Journ.  Exp.  Physiol,  xvi,  pp.  173-194,  1926. 


440  THE  FORMS  OF  CELLS  [ch. 

In  all  these  cases,  just  as  in  the  case  of  the  blood-corpuscle,  the 
structure  alters,  and  finally  loses,  its  characteristic  form  when  the 
constitution  of  the  surrounding  medium  is  changed*. 

Here  again,  as  in  the  blood-cOrpuscle,  we  have  to  do  with  the 
important  force  of  osmosis,  manifested  under  conditions  similar 
to  those  of  Pfeffer's  classical  experiments  on  the  plant-cell  f.  The 
surface  of  the  cell  acts  as  a  semi-permeable  membrane,  permitting 
the  passage  of  certain  dissolved  substances  (or  their  ions),  and 
including  or  excluding  others:  and  thus  rendering  manifest  and 
measurable  the  existence  of  a  definite  "osmotic  pressure."  Again, 
in  the  hen's  egg  a  delicate  yolk-membrane  separates  the  yolk  from 
the  white.  The  morphologist  looks  on  it  but  as  the  cell-wall  of 
a  vast  yolk-laden  germ-cell;  the  physiologist  sees  in  it  a  semi- 
permeable membrane,  the  seat  of  many  complex  activities.  The 
end  and  upshot  of  these  last  is  that  a  steady  difference  of  osmotic 
pressure,  the  equivalent  of  some  two  atmospheres,  is  maintained 
between  yolk  and  white ;  and  yet  there  is  no  current  flowing  through. 
Somewhere  or  other  in  the  system  there  is  a  constant  metabolic 
flux,  a  continuous  liberation  of  energy,  a  continual  doing  of  work, 
all  leading  to  the  maintenance  of  a  steady  dynamical  state,  which 
is  not  " equilibrium {." 

In  the  case  of  the  sperm-cells  of  Inachus,  certain  quantitative 
experiments  have  been  performed.  The  sperm-cell  exhibits  its 
characteristic  conformation  while  lying  in  the  serous  fluid  of  the 
animal's  body,  in  ordinary  sea-water,  or  in  a  5  per  cent,  solution 
of  potassium  nitrate,  these  three  fluids  being  all  "isotonic"  with 
one  another.  As  we  alter  the  concentration  of  potassium  nitrate, 
the  cell  assumes  certain  definite  forms  corresponding  to  definite 
concentrations  of  the  salt;  and,  as  a  further  and  final  proof  that 
the  phenomenon  is  entirely  physical,  it  is  found  that  other  salts 
produce  an  identical  effect  when  their  concentration  is  proportionate 
to  their  molecular  weight,  and  whatever  identical  effect  is  produced 

*  Cf.  N.  K.  KoltzofF,  Studien  iiber  die  Gestalt  der  Zelle,  Arch.  f.  mikrosk.  Anat. 
Lxvii,  pp.  365-572,  1905;  Biol.  Centralhl.  xxm,  pp.  680-696,  1903;  xxvi,  pp.  854- 
863,  1906;  XLvm,  pp.  345-369,  1928;  Arch.  f.  Zellforschung,  u,  pp.  1-65,  1908; 
vn,  pp.  344^23,  1911;   Anat.  Anzeiger,  xli,  pp.  183-206,  1912. 

t  W  Pfeffer,  Osmotische  Untersuchungen,  Leipzig,  1877. 

X  Cf.  J.  Straub,  Der  Unterschied  in  osmotischer  Konzentration  zwischen  Eigelb 
und  Eiklar,  Rer.  Trav.  Chim.  du  Pays-Bas,  XLvni,  p.  49,  1929. 


V] 


OF  THE  SPERM-CELLS  OF  DECAPODS 


441 


by  various  salts  in  their  respective  concentrations,  a  similarly 
identical  effect  is  produced  when  these  concentrations  are  doubled 
or  otherwise  proportionately  changed. 


KNO 


Fig.  147.     Sperm-cells  oi  Inachtis,  as  they  appear  in  saline  solutions  of 
varying  density.     After  KoltzoflF. 

Thus  the  following  table  shews  the  percentage  concentrations  of 
certain  salts  necessary  to  bring  the  cell  into  the  forms  a  and  c 
of  Fig.  147;  in  each  case  the  quantities  are  proportional  to  the 
molecular  weights,  and  in  each  case  twice  the  quantity  is  necessary 
to  produce  the  effect  of  c,  compared  with  that  which  gives  rise  to 
the  all  but  spherical  form  of  a. 

%  concentration  of  salts  in  which 

the  sperm- cell  of  Inachus 

assumes  the  form  of 


r 

^ 

a 

c 

Sodium  chloride 

0-6 

1-2 

Sodium  nitrate 

0-85 

1-7 

Potassium  nitrate 

10 

20 

Acetic  acid 

2-2 

4-5 

Cane  sugar 

50 

100 

If  we  look  then  upon  the  spherical  form  of  this  cell  as  its  true 
condition  of  symmetry  and  of  equiUbrium,  we  see  that  what  we 
call  its  normal  appearance  is  just  one  of  many  intermediate  phases 
of  shrinkage,  brought  about  by  the  abstraction  of  fluid  from  its 
interior  as  the  result  of  an  osmotic  pressure  greater  outside  than 
inside  the  cell,  and  where  the  shrinkage  of  volume  is  not  kept  pace 
with  by  a  contraction  of  the  surface-area.  In  the  case  of  the  blood - 
corpuscle,  the  shrinkage  is  of  no  great  amount,  and  the  resulting 
deformation  is  symmetrical;  such  structural  inequahty  as  may  be 
necessary  to  account  for  it  need  be  but  small.  But  in  the  case  of 
the  sperm-cells,  we  must  have,  and  we  actually  do  find,  a  somewhat 


442  THE  FORMS  OF  CELLS  [ch. 

complicated  arrangement  of  more  or  less  rigid  or  elastic  structures 

in  the  wall  of  the  cell,  which,  like  the  wire  framework  in  Plateau's 

experiments,  restrain  and  modify  the  forces  acting  on  the  drop. 

In  one  form  of  Plateau's  experiments,  instead  of  supporting  his 

drop  on  rings  or  frames  of  wire,  he  laid 
upon  its  surface  one  or  more  elastic 
coils;  and  then,  on  withdrawing  oil 
from  the  centre  of  his  globule,  he  saw 
its  uniform  shrinkage  counteracted  by 
the  spiral  springs,  with  the  result  that 
the  centre  of  each  elastic  coil  seemed 
to  shoot  out  into  a  prominence.  Just 
such    spiral    coils    are    figured    (after 

"■''•  ""-A^rKXcff.' "'""'"■  Koltzoff)  in  Fig.  148*;   and  they  may 

be  regarded  as   closely  akin  to   those 

local  thickenings  or  striations,  spiral  ard  other,  which  are  common 

in  vegetable  cells. 

Physically  speaking,  the  protoplasmic  colloids  are  neither  simple 
nor  uniform.  We  begin  by  thinking  of  our  cell  as  a  drop  of  a  homo- 
geneous fluid  and  on  this  bold  simplifying  assumption  we  account 
for  its  form  to  a  first,  and  often  to  a  near,  approximation.  For  the 
cell  is  largely  composed  of  fluid  "hydrosols,"  which  are  still  fluid 
however  viscous  they  may  be,  and  still  tend  towards  rounded, 
drop-hke  configurations.  But  it  has  also  its  "hydrogels,"  which 
shew  a  certain  tenacity,  a  certain  elasticity,  a  certain  reluctance  to 
let  their  particles  move  on  one  another;  and  of  these  are  formed 
the  scarce  distinguishable  fibrillae  within  a  host  of  highly  specialised 
cells,  the  elastic  fibres  of  a  tendon,  the  incipient  cell- walls  of  a  plant, 
the  rudiments  of  many  axial  and  skeletal  structures. 

The  cases  which  we  have  just  dealt  with  lead  us  to  another 
consideration.  In  a  semi-]:^ermeable  membrane,  through  which 
water  passes  freely  in  and  out,  the  conditions  of  a  liquid  surface  are 
greatly  modified;  in  the  ideal  or  ultimate  case,  there  is  neither 
surface  nor  surface-tension  at  all.     And  this  would  lead  us  some- 

*  As  Bethe  points  out  (Zellgestalt,  Plateausche  Fliissigkeitsfigur  und  Neuro- 
fibrille,  Armt.  Anz.  xl,  p.  209,  1911),  the  spiral  fibres  of  which  Koltzoff  speaks  must 
lie  in  the  surface,  and  not  within  the  substance,  of  the  cell  whose  conformation  is 
affected  by  them. 


V]  OF  THE  PROTOPLASMIC  COLLOIDS  443 

what  to  reconsider  our  position,  and  to  enquire  whether  the  true 
surface-tension  of  a  Hquid  film  is  actually  responsible  for  all  that 
we  have  ascribed  to  it,  or  whether  certain  of  the  phenomena  which 
we  have  assigned  to  that  cause  may  not  in  part  be  due  to  the 
contractility  of  definite  and  elastic  membranes.  But  to  investigate 
this  question,  in  particular  cases,  is  rather  for  the  physiologist:  and 
the  morphologist  may  go  his  way,  paying  little  heed  to  what  is  no 
great  difficulty.  For  in  surface-tension  we  have  the  production  of 
a  film  with  the  properties  of  an  elastic  membrane,  and  with  the 
special  peculiarity  that  contraction  continues  with  the  same  energy 
however  far  the  process  may  have  already  gone ;  while  the  ordinary 
elastic  membrane  contracts  to  a  certain  extent,  and  contracts  no 
more.  But  within  wide  limits  the  essential  phenomena  are  the 
same  in  both  cases.  Our  fundamental  equations  apply  to  both 
cases  alike.  And  accordingly,  so  long  as  our  purpose  is  morpho- 
logical, so  long  as  what  we  seek  to  explain  is  regularity  and  definite- 
ness  of  form,  it  matters  little  if  we  should  happen,  here  or  there, 
to  confuse  surface-tension  with  elasticity,  the  contractile  forces 
manifested  at  a  liquid  surface  with  those  which  come  into  play  at 
the  complex  internal  surfaces  of  an  elastic  solid. 


CHAPTER  VI 

A  NOTE  ON  ADSORPTION 

An  important  corollary  to,  or  amplification  of,  the  theory  of 
surface-tension  is  to  be  found  in  the  chemico-physical  doctrine  of 
Adsorption;  which  means,  in  a  word,  the  concentration  of  a 
substance  at  a  surface,  by  reason  of  that  surface-energy  of  which 
we  have  had  so  much  to  say*.  Charcoal,  with  its  vast  internal 
surface-area  of  carbonised  cell- walls,  is  the  commonest  and  most 
famihar  of  adsorbents,  and  of  it  Du  Bois  Reymond  first  used  the 
name.  In  its  full  statement  this  subject  becomes  very  complicated, 
and  involves  physical  conceptions  and  mathematical  treatment 
which  go  far  beyond  our  range.  But  it  is  necessary  for  us  to  take 
account  of  the  phenomenon,  even  though  it  be  in  the  most  elemen- 
tary way. 

In  the  brief  account  of  the  theory  of  surface-tension  with  which 
our  last  chapter  began,  it  was  shewn  that,  in  a  drop  of  liquid,  the 
potential  energy  of  the  system  could  be  diminished,  and  work  mani- 
fested accordingly,  in  two  ways.  In  the  first  place  we  saw  that, 
at  our  liquid  surface,  surface-tension  tends  to  set  up  an  equilibrium 
of  form,  in  which  the  surface  is  reduced  or  contracted  either  to  the 
absolute  minimum  of  a  sphere,  or  at  any  rate  to  the  least  possible 
area  which  is  permitted  by  the  various  circumstances  and  conditions ; 
and  if  the  two  bodies  which  comprise  our  system,  namely  the  drop 
of  Uquid  and  its  surrounding  medium,  be  simple  substances,  and 
the  system  be  uncomphcated  by  other  distributions  of  force,  then 
the  energy  of  the  system  will  have  done  its  work  when  this 
equihbrium  of  form,  this  minimal  area  of  surface,  is  once  attained. 
This  phenomenon  of  the  production  of  a  minimal  surface-area  we 
have  now  seen  to  be  of  fundamental  importance  in  the  external 

*  Some  define  adsorption  as  surface-condensation,  without  reference  to  the 
forces  which  produce  it;  in  other  words  they  recognize  chemical,  electrical  and 
other  forces,  including  cohesion,  as  producing  analogous  or  indistinguishable 
results:  of.  A.  P.  Mathews,  in  Physiological  Reviews,  i,  pp.  553-597,  1921. 


CH.vi]  OF  SURFACE  ENERGY  445 

morphology  of  the  cell,  and  especially  (so  far  as  we  have  yet  gone) 
of  the  solitary  cell  or  unicellular  organism. 

But  we  also  saw,  according  to  Gauss's  equation,  that  the  potential 
energy  of  the  system  will  be  diminished  (and  its  diminution  will 
accordingly  be  manifested  in  work)  if  from  any  cause  the  specific 
surface-energy  be  diminished,  that  is  to  say  if  it  be  brought  more 
nearly  to  an  equality  with  the  specific  energy  of  the  molecules  in 
the  interior  of  the  liquid  mass.  This  latter  is  a  phenomenon  of 
great  moment  in  physiology,  and,  while  we  need  not  attempt  to 
deal  with  it  in  detail,  it  has  a  bearing  on  cell-form  and  cell-structure 
which  we  cannot  afford  to  overlook. 

A  diminution  of  the  surface-energy  may  be  brought  about  in 
various  ways.  For  instance,  it  is  known  that  every  isolated  drop 
of  fluid  has,  under  normal  circumstances,  a  surface-charge  of 
electricity:  in  such  a  way  that  a  positive  or  negative  charge  (as 
the  case  may  be)  is  inherent  in  the  surface  of  the  drop,  while  a 
corresponding  charge,  of  contrary  sign,  is  inherent  in  the  imme- 
diately adjacent  molecular  layer  of  the  surrounding  medium.  Now 
the  effect  of  this  distribution,  by  which  all  the  surface  molecules 
of  our  drop  are  similarly  charged,  is  that  by  virtue  of  the  charge 
they  tend  to  repel  one  another,  and  possibly  also  to  draw  other 
molecules,  of  opposite  charge,  from  the  interior  of  the  mass;  the 
result  being  in  either  case  to  antagonise  or  cancel,  more  or  less, 
that  normal  tendency  of  the  surface  molecules  to  attract  one 
another  which  is  manifested  in  surface-tension.  In  other  words, 
an  increased  electrical  charge  concentrating  at  the  surface  of  a  drop 
tends,  whether  it  be  positive  or  negative,  to  lower  the  surface-tension. 

Again,  a  rfse  of  temperature  diminishes  surface-tension,  and 
consequently  facilitates  the  formation  of  a  bubble  or  a  froth.  It 
follows  (from  the  principle  of  Le  Chatelier)  that  foam  is  warmer  than, 
the  fluid  of  which  it  is  made,  and  the  difference  is  all  the  greater  the 
lower  the  concentration  of  the  foaming  (or  capillary-active)  substance*. 

But  a  still  more  important  case  has  next  to  be  considered.  Let 
us  suppose  that  our  drop  consists  no  longer  of  a  single  chemical 
substance,  but  contains  other  substances  either  in  suspension  or 
in  solution.     Suppose  (as  a  very  simple  case)  that  it  be  a  watery 

*  Cf.  Fr.  Schiitz,  in  Nature,  April  10,  1937.  In  the  case  of  001  per  cent,  solution 
of  saponin,  the  temperature-diflFerence  is  no  less  than  3-3°  C. 


446  A  NOTE  ON  ADSORPTION  [ch. 

fluid,  exposed  to  air,  and  containing  droplets  of  oil :  we  know  that 
the  specific  surface-tension  of  oil  in  contact  with  air  is  much  less 
than  that  of  water,  and  it  follows  that,  if  the  watery  surface  of 
our  drop  be  replaced  by  an  oily  surface  the  specific  surface-energy 
of  the  system  will  be  notably  diminished.  Now  under  these  circum- 
stances it  is  found  that  (quite  apart  from  gravity,  which  might 
cause  it  to  float  to  the  surface)  the  oil  has  a  tendency  to  be  drawn 
to  the  surface;  and  again  this  phenomenon  of  molecular  attraction 
or  adsorption  represents  work  done,  equivalent  to  the  diminished 
potential  energy  of  the  system*.  In  more  general  terms,  if  a  Hquid 
be  a  chemical  mixture,  some  one  constituent  in  which,  if  it  entered 
into  or  increased  in  amount  in  the  surface  layer,  would  have  the 
effect  of  diminishing  its  surface-tension,  then  that  constituent  will 
have  a  tendency  to  accumulate  or  concentrate  at  the  surface:  the 
surface-tension  may  be  said,  as  it  were,  to  exercise  an  attraction 
on  this  constituent  substance,  drawing  it  into  the  surface-layer, 
and  this  tendency  will  proceed  until  at  a  certain  "surface-con- 
centration" equilibrium  is  reached,  its  opponent  being  that  osmotic 
force  which  tends  to  keep  the  substance  in  uniform  solution  or 
diffusion.  In  other  words,  in  any  "two-phase"  system,  a  change 
of  concentration  at  the  boundary-surface  and  a  diminution  of 
surface-tension  there  accompany  one  another  of  necessity ;  positive 
adsorption  means  negative  surface-tension,  and  vice  versa.  Further- 
more, the  lowering  of  surface-tension  (as  by  saponin)  will  permit 
(caeteris  paribus)  an  extension  of  surface,  manifesting  itself  in 
"froth."  Thus  the  production  of  a  froth  and  the  concentration 
of  appropriate  substances  therein  are  two  sides  of  one  and  the  same 
phenomenon. 

In  the  complex  mixtures  which  constitute  the  protoplasm  of  the 
living  cell,  this  phenomenon  of  adsorption  has  abundant  play:  for 
many  of  its  constituents,  such  as  fats,  soaps,  proteins,  lecithin,  etc., 
possess  the  required  property  of  diminishing  surface-tension. 

*  The  first  instance  of  what  we  now  call  an  adsorptive  phenomenon  was 
observed  in  soap-bubbles.  Leidenfrost  was  aware  that  the  outer  layer  of  the 
bubble  was  covered  by  an  "oily"  layer  {De  aquae  communis  nonnullis  qualitatihus 
tractatus,  Duisburg,  1756).  A  hundred  years  later  Dupre  shewed  that  in  a  soap- 
solution  the  soap  tends  to  concentrate  at  the  surface,  so  that  the  surface-tension 
of  a  very  weak  solution  is  very  little  different  from  that  of  a  strong  one  {Theorie 
mdcanique  de  la  chaleur,  1869,  p.  376;   cf.  Plateau,  ii,  p.  100). 


VI]  OF  SURFACE  CONCENTRATION  447 

Moreover,  the  more  a  substance  has  the  power  of  lowering  the 
surface-tension  of  the  Hquid  in  which  it  happens  to  be  dissolved, 
the  more  will  it  tend  to  displace  another  and  less  effective  substance 
from  the  surface-layer.  Thus  we  know  that  protoplasm  always 
contains  fats,  not  only  in  visible  drops,  but  also  in  the  finest  sus- 
pension or  "colloidal  solution";  and  if  under  any  impulse,  such  for 
instance  as  might  arise  from  the  Brownian  movement,  a  droplet  of 
oil  be  brought  close  to  the  surface,  it  is  at  once  drawn  into  that 
surface  and  tends  to  spread  itself  in  a  thin  layer  over  the  whole 
surface  of  the  cell.  But  a  soapy  surface  (for  instance)  in  contact 
•with  the  surrounding  water  would  have  a  surface-tension  even  less 
than  that  of  the  film  of  oil:  and  consequently,  if  soap  be  present 
in  the  water  it  will  in  turn  be  adsorbed,  and  will  tend  to  displace 
the  oil  from  the  surface  pellicle*.  And  all  this  is  as  much  as  to 
say  that  the  molecules  of  the  dissolved  or  suspended  substance  or 
substances  will  so  distribute  themselves  throughout  the  drop  as  to 
lead  towards  an  equilibrium,  for  each  small  unit  of  volume,  between 
the  superficial  and  internal  energy;  or,  in  other  words,  so  as  to 
reduce  towards  a  minimum  the  potential  energy  of  the  system. 
This  tendency  to  concentration  at  a  surface  of  any  substance  within 
the  cell  by  which  the  surface-tension  tends  to  be  diminished,  or 
vice  versa,  constitutes,  then,  the  phenomenon  of  adsorption;  and 
the  general  statement  by  which  it  is  defined  is  known  as  the  Willard- 
Gibbs,  or  Gibbs-Thomson  lawf,  and  was  arrived  at  not  by  experi- 
mental but  by  theoretical  and  hydrodynamical  methods. 

An  assemblage  of  drops  or  droplets  offers  a  great  extension  of 
surface,  but  so  also  does  an  assemblage  of  equally  minute  cells  or 


*  This  identical  phenomenon  was  the  basis  of  Quincke's  theory  of  amoeboid 
movement  (Ueber  periodische  Ausbreitung  von  Fliissigkeitsoberflachen,  etc.,  SB. 
Berlin.  Akad.  1888,  pp.  791-806;  cf.  Pfiuger's  Archiv,  1879,  p.  136).  We  must  bear 
in  mind  that  to  describe  an  amoeboid  cell  as  "naked"  does  not  imply  that  its  outer 
layer  is  identical  with  its  internal  substance. 

t  J.  Willard  Gibbs,  Equilibrium  of  heterogeneous  substances,  Tr.  Conn.  Acad. 
Ill,  pp.  380-400,  1876,  also  in  Collected  Papers,  i,  pp.  185-218,  London,  1906; 
J.  J.  Thomson,  Applications  of  Dynamics  to  Physics  and  Chemistry,  1888  (Surface 
tension  of  solutions),  p.  190.  See  also  {int.  al.)  various  papers  by  C.  M.  Lewis, 
Phil.  Mag.  (6),  xv,  p.  499,  1908;  xvii,  p.  466,  1909;  Zeitschr.  f.  physik.  Chemie, 
Lxx,  p.  129,  1910;  Milner,  Phil.  Mag.  (6),  xm,  p.  96,  1907;  A.  B.  Macallum,  The 
role  of  surface-tension  in  determining  the  distribution  of  salts  in  living  matter, 
Trans,  loth  Int.  Congress  on  Hygiene,  etc.,  Washington,  1912;   etc. 


448  A  NOTE   ON  ADSORPTION  [ch. 

pores;  both  alike  are  "two-phase"  systems,  and  in  both  ahke  the 
phenomenon  of  adsorption  has  free  play.  The  occlusion  of  gases, 
including  water-vapour,  by  charcoal  is  a  familiar  phenomenon  of 
adsorption,  and  is  due  to  the  minuteness  of  the  pores  only  in  so  far 
as  surface-area  is  increased  and  multipUed  thereby.  For  surface- 
energy  is  surface-strain  or  surface-tension  x  surface-area,  and  is 
vastly  increased  by  minute  subdivision.  And  surface-energy  is 
such  that,  whenever  a  substance  is  introduced  into  a  two-phase 
system — which  merely  means  two  things  in  touch  (or  surface- 
contact)  with  one  another — it  is  apt  to  concentrate  itself  on  the 
surface  where  the  two  phases  meet.  Absorption  implies  uniform 
distribution,  as  when  a  gas  is  absorbed  by  a  liquid;  adsorption 
imphes  a  heterogeneous  field,  and  a  concentration  localised  on  the 
surfaces  therein. 

Among  the  many  important  physical  features  or  concomitants 
of  this  phenomenon,  let  us  take  note  at  present  that  we  need  not 
conceive  of  a  strictly  superficial  distribution  of  the  adsorbed  sub- 
stance, that  is  to  say  of  its  direct  association  with  the  surface-layer 
of  molecules  such  as  we  imagined  in  the  case  of  an  electrical  charge ; 
but  rather  of  a  progressive  tendency  to  concentrate  more  and  more, 
the  nearer  the  surface  is  approached.  Indeed  we  may  conceive 
the  colloid  or  gelatinous  precipitate  in  which,  in  the  case  of  our 
protoplasmic  cell,  the  dissolved  substance  tends  often  to  be  thrown 
down,  to  constitute  one  boundary  layer  after  another,  the  general 
effect  being  intensified  and  multiplied  by  the  repetition  of  these 
new  surfaces. 

Moreover,  it  is  not  less  important  to  observe  that  the  process 
of  adsorption,  in  the  neighbourhood  of  the  surface  of  a  hetero- 
geneous liquid  mass,  is  a  process  which  takes  time;  the  tendency 
to  surface  concentration  is  a  gradual  and  progressive  one,  and  will 
fluctuate  with  every  minute  change  in  the  composition  of  our 
substance  and  with  every  change  in  the  area  of  its  surface.  In 
other  words,  it  involves  (in  every  heterogeneous  substance)  a  con- 
tinual instabihty:  and  a  constant  manifestation  of  motion,  some- 
times in  the  mere  invisible  transfer  of  molecules,  but  often  in  the 
production  of  visible  currents,  or  manifest  alterations  in  the  form 
or  outline  of  the  system. 

Cellular  activity  is  of  necessity  associated  with  cellular  structure, 


VI]  OF  THE  BOUNDARY  STATE  449 

even  in  our  simplest  interpretation  thereof,  as  a  mere  increase  of 
surface  due  to  the  existence  and  the  multiphcation  of  cells.  In  the 
chemistry  of  the  tissues  there  may  be  substances  (catalysts  and 
others)  which  exhibit  their  proper  reactions  even  though  the  cells 
containing  them  be  disintegrated  or  destroyed ;  but  other  processes, 
oxidation  itself  among  them,  are  essentially  surface-actions,  based 
on.adsorption  at  the  vast  cell-surface  of  the  tissue*.  The  breaking- 
down  of  the  cell-walls,  the  disintegration  of  cellular  structure  in  a 
tissue,  brings  about  "a  biochemical  chaos,  a  medley  of  reactions f." 
Cells  are  not  merely  there  because  the  tissue  has  grown  by  their 
multiplication;  there  are  physico-chemical  reasons,  even  of  an 
elementary  kind,  which  render  the  morphological  phenomenon  of 
the  cell  indispensable  to  physiological  action. 

The  physiologist  deals  with  the  surface-phenomena  of  the  cell  in 
ways  undreamed  of  when  I  began  to  write  this  book.  To  begin 
with,  the  concept  of  a  surface  (in  the  old  mathematical  or  quasi- 
mathematical  sense)  no  longer  suffices  to  describe  the  boundary 
conditions  of  even  a  "naked"  protoplasmic  cell.  As  Rayleigh 
foretold,  and  as  Irving  Langmuir  has  proved,  the  "boundary-state" 
consists  of  a  layer  of  complex  molecules,  each  one  a  long  array  of 
atoms,  all  set  side  by  side  in  an  orderly  and  uniform  way.  There 
is  not  merely  a  boundary-surface  between  two  phases  (as  the  older 
colloid  chemistry  supposed)  but  a  boundary-layer,  which  itself 
constitutes  a  third  phase,  or  interphase,  and  which  part  of  the 
surface-energy  has  gone  to  the  making  of. 

Surface-energy  plays  a  leading  part  in  modern  theories  of  muscular 
contraction,  and  has  indeed  done  so  ever  since  FitzGerald  and 
d'Arsonval  indicated  a  connection  between  them  some  sixty  years 
or  more  agof.  It  plays  its  part  handsomely  (we  may  be  sure)  in  the 
electric  pile  of  the  Torpedo,  where  two  miUion  tiny  discs  present  a 

*  Many  surface-active  substances  are  known  to  be  among  the  most  active 
pharmacologically;   cf.  Michaelis  and  Rona,  Physikal.  Chemie,  1930. 

t  A.  V.  Hill,  Proc.  R.S.  (B),  cm,  p.  138^;  cf.  also  M.  Penrose  and  J.  H.  Quastel, 
on  Cell  structure  and  cell  activity,  ibid,  cvii,  p.  168. 

X  Cf.  G.  F.  FitzGerald,  On  the  theory  of  muscular  contraction,  Brit.  Ass.  Rep. 
1878;  also  in  Scientific  Writings,  ed.  Larmor,  1902,  pp.  34,  75.  A.  d'Arsonval, 
Relations  entre  I'electricite  animale  et  la  tension  superficielle,  C.R.  cvi,  p.  1740, 
1888;  A.  Imbert,  Le  mecanisme  de  la  contraction  musculaire,  deduit  de  la 
consideration  des  forces  de  tension  superficielle.  Arch,  de  Phys.  (5),  ix,  pp.  289-301, 
1897;   A.  J.  Ewart,  Protoplasmic  Streami-ng  in  Plants,  Oxford,  1903,  pp.  112-119. 


450  A  NOTE   ON  ADSORPTION  [ch. 

vast  aggregate  of  interfacial  contact.  It  gives  us  a  new  conception, 
as  Wolfgang  Ostwald  was  the  first  to  shew,  of  the  relation  of  oxygen 
to  the  red  corpuscles  of  the  blood*.  But  many  more  and  still  more 
comphcated  "film-reactions"  are  started  or  intensified  by  the  oriented 
molecules  of  the  monolayer.  The  catalytic  action  of  living  ferments 
(a  subject  vast  indeed)  is  largely  .a  question  of  modified  adsorption, 
or  of  surface-action.  The  range  of  bodies  so  adsorbed  is  extremply 
hmited;  the  specific  reactions,  which  depend  on  the  bacterium 
engaged,  are  fewer  still ;  and  sometimes  a  whole  class  of  substances 
may  be  adsorbed,  and  only  one  of  them  thrown  specifically  into 
action!.  The  physiological,  and  sometimes  lethal,  actions  of 
various  substances  are  examples  of  similar  effects.  The  chemistry 
of  the  surface-layer  in  this  cell  or  that  may  be  elucidated  by  its 
reactions  to  various  "penetrants,"  and  depends  somehow  on  the 
molecular  orientation  of  the  surfaces,  and  on  the  potentials  associated 
with  the  characteristic  electric  fields  which  we  may  suppose  to 
correspond  to  the  particular  molecular  arrangements  J. 

It  is  the  dynamic  aspect  of  the  case,  the  ingresses,  egresses  and 
metabolic  changes  associated  with  the  boundary-layer,  which  interest 
the  physiologist.  He  finds  the  monolayer  acting  in  ways  not  known 
in  a  homogeneous  hquid — and  adsorption  is  one  of  these  ways.  We 
keep  as  much  as  may  be  to  the  morphological  side  of  the  case  rather 
than  to  the  physiological,  to  the  static  side  rather  than  to  the 
dynamic,  to  the  equilibrium  attained  rather  than  to  the  energies  to 
which  it  is  due.     We  continue  to  speak  of  surface,  and  of  surface- 

*  Ueber  die  Natur  der  Bindung  der  Gase  im  Blut  und  in  seinen  Bestandteilen, 
Kolloid  Ztschr.  ii,  pp.  264-272,  1908;  cf.  Loewy.  Dissociationsspannung  des 
Oxyhaemoglobin  im  Blut,  Arch.  f.  Anat.  u.  Physiol.  1904,  p.  231.  Arrhenius 
remarked  long  ago  that  the  forces  which  produce  adsorption  are  of  the  same  order, 
and  of  the  same  nature,  as  those  which  cause  the  mutual  attractions  of  the  molecules 
of  a  gas.  Hence  the  order  is  constant  in  which  various  gases  are  adsorbed  by 
different  adsorbents.  The  question  of  the  inner  mechanism  of  the  forces  which 
result  in  surface-tension,  adsorption  and  allied  phenomena,  and  their  relation 
to  electric  charge  on  particles  or  ions,  belongs  to  the  highest  parts  of  physical 
chemistry.  Besides  countless  recent  papers,  M.  v.  Smoluchowski's  Versuch  einer 
mathematischen  Theorie  der  Koagulationskritik,  Z.f.  physik.  Chemie,  xcii,  pp.  129- 
168,  1918,  is  still  interesting. 

t  Cf.  N.  K.  Adams,  Physics  and  Chemistry  of  Surfaces,  1930.  Also  {int.  al.) 
J.  H.  Quastel,  Mechanism  of  bacterial  action.  Trans.  Faraday  Soc.  xxvi,  pp.  831- 
861,  19.30. 

+  This  is  Loeb's  so-called  "membrane-effect,"  cf.  Journ.  Biol.  Chemistry,  xxxii, 
p.  147,  1917;   and  J.  Gray,  Journ.  Physiol.  Liv,  pp.  68-78,  1920. 


VI]  OF  THE  GIBBS-THOMSON  LAW  451 

energy  and  of  adsorptive  phenomena,  in  a  somewhat  old-fashioned 
way;  but  even  with  this  simphfying  Uniitation  we  find  them  helpful, 
throwing  hght  upon  our  subject. 

In  the  first  place  our  preHminary  account,  such  as  it  is,  is  already 
tantamount  to  a  description  of  the  process  of  development  of  a 
cell-membrane,  or  cell- wall.  The  so-called  "secretion"  of  this  cell- 
wall  is  nothing  more  than  a  sort  of  exudation,  or  striving  towards 
the  surface,  of  certain  constituent  molecules  or  particles  within  the 
cell;  and  the  Gibbs-Thomson  law  formulates,  in  part  at  least,  the 
conditions  under  which  they  do  so.  The  adsorbed  material  may 
range  from  an  almost  unrecognisable  pellicle  to  the  distinctly 
differentiated  "ectosarc"  of  a  protozoon,  and  again  to  the  develop- 
ment of  a  fully-formed  cell-wall,  as  in  the  cellulose  partitions  of  a 
vegetable  tissue.  In  such  cases,  the  dissolved  and  adsorbtive 
material  has  not  only  the  property  of  lowering  the  surface-tension, 
and  hence  of  itself  accumulating  at  the 'surface,  but  has  also  the 
prope^y  of  increasing  the  viscosity  and  mechanical  rigidity  of  the 
material  in  which  it  is  dissolved  or  suspended,  and  so  of  constituting 
a  visible  and  tangible  "membrane*."  The  "zoogloea"  around  a 
group  of  bacteria  is  probably  a  phenomenon  of  the  same  order. 
In  the  superficial  deposition  of  inorganic  materials  we  see  the  same 
process  abundantly  exemphfied.  Not  only  do  we  have  the  simple 
case  of  the  building  of  a  shell  or  '"test"  upon  the  outward  surface 
of  a  Hving  cell,  as  for  instance  in  a  Foraminifer,  but  in  a  sub- 
sequent chapter,  when  we  come  to  deal  with  spicules  and  spicular 
skeletons  such  as  those  of  the  sponges  and  of  the  Radiolaria^  we 
shall  see  how  highly  characteristic   it  is  of  the  whole  process  of 


*  We  may  trace  the  first  steps  in  the  study  of  this  phenomenon  to  Melsens, 
who  found  that  thin  films  of  white  of  egg  become  firm  and  insoluble  (Sur  les 
modifications  apportees  a  I'albumine.  .  .par  faction  purement  mecanique,  C.R. 
XXXIII,  p.  247;  Ann.  de  chimie  et  de  physique  (3),  xxxiii,  p.  170,  1851);  and  Harting 
made  similar  observations  about  the  same  time.  Ramsden  investigated  the  same 
subject,  and  also  the  more  general  phenomenon  of  the  formation  of  albuminoid 
and  fatty  membranes  by  adsorption,  and  found  {i7it.  al.)  that  on  shaking  white  of  egg 
practically  all  .the  albumin  passes  gradually  into  the  froth;  cf.  his  Koagulierung 
der  Eiweisskorper  auf  mechanischer  Wege,  Arch.  f.  Anat.  u.  Phys.  {Phys.  Ahth.), 
1894,  p.  517;  Abscheidung  fester  Korper  in  Oberflachenschichten,  Z.  f.  phys.  Chern. 
XLVii,  p.  341,  1902;  Proc.  R.S.  Lxxii,  p.  156,  1904.  For  a  general  review  of  the 
whole  subject  see  H.  Zangger,  Ueber  Membranen  und  Membranfunktionen,  in 
Asher-8piro's  Ergebnisse  der  Physioloyie,  vii,  pp.  99-1(50,  1908. 


452  A  NOTE   ON  ADSORPTION  [ch. 

spicule-formation  for  the  deposits  to  be  laid  down  just  in  the 
"  interfacial "  boundaries  between  cells  or  vacuoles,  and  how  the 
form  of  the  spicular  structures  tends  in  many  cases  to  be  regulated 
and  determined  by  the  arrangement  of  these  boundaries.  The 
so-called  collenchyma,  in  which  an  excess  of  cellulose  is  laid  down 
around  the  angles  of  contact  of  adjacent  cells,  in  a  kind  of  exag- 
gerated "bourrelet,"  is  another  case  in  point*. 

No  pure  liquid  ever  forms  a  froth  or  foam.  White  of  egg  is  no 
exception  to  the  rule;  for  the  albumin  is  somehow  changed,  or 
"  denatured,"  and  becomes  a  quasi-solid  when  we  beat  it  up.  But 
in  the  frothing  liquid  there  must  always  be  some  admixture  present 
to  concentrate  on,  or  be  adsorbed  by,  the  surfaces  and  interfaces 
of  the  other ;  and  this  dispersion  must  go  on  completely  and  uni- 
formly, so  as  to  leave  the  whole  system  homogeneous.  The  resulting 
diminution  of  surface-tension  facihtates  the  subdivision  of  the 
bubbles  and  dispersion  of  the  air ;  and  the  adsorbed  surface-layer 
gives  firmness  and  stability  to  the  system.  The  sudden  increase  of 
surface  diminishes,  for  the  moment,  the  concentration,  or  "thick- 
ness" of  the  surface-layer;  the  tension  rises  accordingly,  and  the 
cycle  of  operations  begins  anewf . 

In  physical  chemistry,  a  distinction  is  usually  drawn  between  adsorption 
and  psevdo-adsorption,  the  former  being  a  reversible,  the  latter  an  irreversible 
or  permanent  phenomenon.  That  is  to  say,  adsorption,  strictly  speaking, 
implies  the  surface-concentration  of  a  dissolved  substance,  under  circumstances 
which,  if  they  be  altered  or  reversed,  will  cause  the  concentration  to  diminish 
or  disappear.  But  pseudo-adsorption  includes  cases,  doubtless  originating  in 
adsorption  proper,  where  subsequent  changes  leave  the  concentrated  substance 
incapable  of  re-entering  the  liquid  system.  It  is  obvious  that  many  (though 
not  all)  of  our  biological  illustrations,  for  instance  the  formation  of  spicules 
or  of  permanent  cell-membranes,  belong  to  the  class  of  so-called  pseudo- 
adsorption  phenomena.  But  the  apparent  contrast  between  the  two  is  in 
the  main  a  secondary  one,  and  however  important  to  the  chemist  is  of  little 
consequence  to  us. 

While  this  brief  sketch  of  the  theory  of  membrane-formation  is 
cursory  and  inadequate,  it  is  enough  to  shew  that  the  physical 
theory  of  adsorption  tends  in  part  to  overturn,  in  part  to  simphfy 

♦  Cf.  G.  Haberlandt,  Zelle  u.  Elementarorgane,  Biol.  Centralbl.  1925,  p.  263. 
t  Cf.  F.  G.  Donnan,  Some  aspects  of  the  physical  chemistry  of  interfaces,  Brit. 
Ass.  Address  (Section  B),  1923;   Nature,  Bee.  15,  22,  1923. 


VI]  OF  THE  FORMATION  OF  MEMBRANES         453 

enormously,  the  older  histological  descriptions.  We  can  no  longer 
be  content  with  such  statements  as  that  of  Strasbiirger,  that 
membrane-formation  in  general  is  associated  with  the  "activity  of 
the  kinoplasm,"  or  that  of  Harper  that  a  certain  spore-membrane 
arises  directly  from  the  astral  rays*.  In  short,  we  have  easily 
reached  the  general  conclusion  that  the  formation  of  a  cell-wall  or 
cell-membrane  is  a  chemico-physical  phenomenon,  which  the  purely 
objective  methods  of  the  biological  microscopist  do  not  suffice  to 
interpret. 

Having  reached  this  conclusion  we  may  wait  patiently,  and 
confidently,  for  more.  But  when  the  physico-chemical  nature  of 
these  phenomena  is  admitted,  and  their  dependence  on  adsorption 
recognised,  or  at  least  assumed,  we  have  still  to  remember  that  the 
chemist  himself  is  none  too  certain  of  his  ground..  He  still  finds  it 
hardj  now  and  then,  to  teU  how  far  adsorption  and  direct  chemical 
action  go  their  way  together,  what  parts  they  severally  play,  what 
shares  they  take  in  their. intimate  cooperation  f. 

If  the  process  of  adsorption,  on  which  the  formation  of  a  mem- 
brane depends,  be  itself  dependent  on  the  power  of  the  adsorbed 
substance  to  lower  the  surface-tension,  it  is  obvious  that  adsorption 
can  only  take  place  when  the  surface-tension  already  present  is 
greater  than  zero.  It  is  for  this  reason  that  films  or  threads  of 
creeping  protoplasm  shew  little  tendency,  or  none,  to  cover  them- 
selves with  an  encysting  membrane;  and  that  it  is  only  when,  in 
an  altered  phase,  the  protoplasm  has  developed  a  positive  surface- 
tension,  and  has  accordingly  gathered  itself  up  into  a  more  or  less 
spherical  body,  that  the  tendency  to  form  a  membrane  is  manifested, 
and  the  organism  develops  its  "cyst"  or  cell- wall.  The  holes  in  a 
Globigerina-shell  are  there  "to  let  the  pseudopodia  through."  They 
may  also  be  described  as  due  to  unequal  distribution  of  surface- 
energy,  such  as  to  prevent  shell-substance  from  being  adsorbed 
here  and  there,  and  at  the  same  time  inducing  a  pseudopodium  to 
emerge. 


*  Strasbiirger,  Ueber  Cytoplasmastrukturen,  etc.,  Jahrb.  f,  wiss.  Bot.  xxx, 
1897;  R.  A.  Harper,  Kerntheilung  und  freie  Zellbildung  ira  Ascus,  ibid.;  cf. 
Wilson,  The  Cell  in  Development,  etc.,  pp.  53-55. 

t  The  "adsorption  theory"  of  dyeing  is  a  case  in  point,  where  the  precise  mode, 
or  modes,  of  action  seem  still  far  from  settled. 


454  A  NOTE   ON  ADSORPTION  [ch. 

It  is  found  that  a  rise  of  temperature  greatly  reduces  the 
adsorbability  of  a  substance,  and  this  doubtless  comes,  either  in 
part  or  whole,  from  the  fact  that  a  rise  of  temperature  is  itself  a 
cause  of  the  lowering  of  surface-tension.  We  may  in  all  probability 
ascribe  to  this  fact  and  to  its  converse,  or  at  least  associate  with  it, 
such  phenomena  as  the  encystment  of  unicellular  organisms  at  the 
approach  of  winter,  or  the  frequent  formation  of  strong  shells  or 
membranous  capsules  in  "winter-eggs." 

Again,  since  a  film  or  a  froth  (which  is  a  system  of  films)  can 
only  be  maintained  by  virtue  of  a  certain  viscosity  or  rigidity  of 
the  liquid,  it  may  be  quickly  caused  to  disappear  by  the  presence 
in  its  neighbourhood  of  some  substance  capable  of  materially 
reducing  the .  surface-tension ;  for  this  substance,  being  adsorbed, 
may  displace  from  the  surface -layer  a  material  to  which  was  due 
the  rigidity  of  the  film.  In  this  way  a  "bathytonic"  substance, 
such  as  ether,  causes  most  foams  to  subside,  and  the  pouring  oil  on 
troubled  waters  not  only  calms  the  waves  but  still  more  quickly 
dissipates  the  foam  of  the  breakers.  In  a  very  different  order  of 
things,  the  breaking  up  of  an  alveolar  network,  as  at  a  certain  stage 
in  the  nuclear  division  of  the  cell,  may  be  due  in  part  to  just  such 
a  cause,  as  well  as  to  the  direct  lowering  of  surface-tension  by 
electrical  agency. 

Our  last  illustration  has  led  us  back  to  the  subject  of  a  previous 
chapter,  namely  to  the  visible  configuration  of  the  interior  of  the 
cell,  in  so  far  (at  least)  as  it  represents  a  "dispersed  system,"  coarse 
enough  to  be  visible;  and  in  connection  with  this  wide  subject  there 
are  many  phenomena  on  which  light  is  apparently  thrown  by  our 
knowledge  of  adsorption,  of  which  we  took  Httle  or  no  account  in 
our  former  discussion.  One  of  these  phenomena  is  nothing  less  than 
that  visible  or  concrete  "polarity,"  which  we  have  seen  to  be  in 
some  way  associated  with  a  dynamical  polarity  of  the  cell. 

This  morphological  polarity  may  be  of  a  very  simple  kind,  as 
when  it  is  manifested,  in  an  epithelial  cell,  by  the  outward  shape  of 
the  elongated  or  columnar  cell  itself,  by  the  essential  difference 
between  its  free  surface  and  its  attached  base,  or  by  the  presence 
in  the  neighbourhood  of  the  former  of  mucus  or  other  products  of 
the  cell's  activity.  But  in,  a  great  many  cases,  this  polarised 
symmetry  is  supplemented  by  the  presence  of  various  fibrillae,  or 


VI] 


OF  MORPHOLOGICAL  POLARITY 


455 


of  linear  arrangements  of  particles,  which  in  the  elongated  or 
"monopolar"  cell  run  parallel  with  its  axis,  but  tend  to  a  radial 
arrangement  in  the  more  or  less  rounded  or  spherical  cell.  Of  late 
years  great  importance  has  been  attached  to  these  various  linear  or 
fibrillar  arrangements,  as  they  are  seen  (after  staining)  in  the  cell- 
substance  of  intestinal  epithelium,  of  spermatocytes,  of  ganglion 
cells,  and  most  abundantly  and  frequently  of  all  in  gland  cells. 
Various  functions  have  been  assigned,  and  hard  names  given  to 
them;  for  these  structures  include  your  mitochondria*  and  your 
chondriokonts  (both  of  these  being  varieties  of  chondriosomes),  your 
Altmann's  granules,   your  microsomes,   pseudo-chromosomes,   epi- 


'M^m. 


Ih 


A  B  C 

Fig.  149.     A,    B,   Chondriosomes  in  kidney- cells,   prior  to  and   during  secretory 
activity  (after  Barratt);   C,  do.  in  pancreas  of  frog  (after  Mathews). 

dermal  fibrils  and  basal  filaments,  your  archeoplasm  and  ergasto- 
plasm,  and  probably  your  idiozomes,  plasmosomes,  and  many  other 
histological  minutiae  f. 

The  position  of  these  bodies  with  regard  to  the  other  cell- 
structures  is  carefully  described.  Sometimes  they  he  in  the 
neighbourhood  of  the  nucleus  itself,  that  is  to  say  in  proximity  to 
the  fluid  boundary  surface  which  separates  the  nucleus  from  the 


*  Mitochondria  are  threads  which  move  slowly  through  the  protoplasm,  some- 
times break  in  two,  and  often  tend  to  radiate  from  the  centrosphere  or  division-centre 
of  the  cell.  The  nucleoli  are  two  or  more  Opaque  bodies  within  the  nucleus,  which 
keep  shifting  their  position ;  within  the  cytoplasm  many  small  fatty  bodies  likewise 
move  about,  and  display  the  Brownian  oscillation, 

t  Cf.  A.  Gurwitsch,  Morphologie  und  Biologie  der  Zelle,  1904,  pp.  169-185; 
Meves,  Die  Chondriosomen  als  Trager  erblicher  Anlagen,  Arch.  'f.  mikrosk.  Anat. 
1908,  p.  72;  J.  0.  W,  Barratt,  Changes  in  chondriosomes,  etc.,  Q.J. M.S.  LVin, 
pp.  553-566,  1913,  etc.;  A.  P.  Mathews,  Changes  in  structure  of  the  pancreas  cell, 
etc.,  Journ.  Mc/rph.  xv  (Suppl.),  pp.  171-222,  1899. 


456  A  NOTE   ON  ADSORPTION  [ch. 

cytoplasm ;  and  in  this  position  they  often  form  a  somewhat  cloudy 
sphere  which  constitutes  the  Nebenkern.  In  the  majority  of  cases, 
as  in  the  epithehal  cells,  they  form  filamentous  structures,  and  rows 
of  granules,  whose  main  direction  is  parallel  to  the  axis  of  the  cell ; 
and  which  may,  in  some  cases,  and  in  some  forms,  be  conspicuous 
at  the  one  end,  and  in  some  cases  at  the  other  end  of  the  cell.  But 
I  seldom  find  the  histologists  attempting  to  explain,  or  to  correlate 
with  other  phenomena,  the  tendency  of  these  bodies  to  lie  parallel 
with  the  axis,  and  perpendicular  to  the  extremities-  of  the  cell ;  it 
is  merely  noted  as  a  peculiarity,  or  a  specific  character,  of  these 
particular  structures.  Extraordinarily  complicated  and  diverse 
functions  have  been  ascribed  to  them.  Engelmann's  "Fibrillen- 
konus,"  which  was  almost  certainly  another  aspect  of  the  same 
phenomenon,  was  held  by  him  and  by  cytologists  hke  Breda  and 
Heidenhain  to  be  an  apparatus  connected  in  some  unexplained 
way  with  the  mechanism  of  ciliary  movement.  Meves  looked  upon 
the  chondriosomes  as  the  actual  carriers  or  ti*ansmitters  of  heredity. 
Altmann  invented  a  new  aphorism,  Omne  granulum  e  granulo,  as  a 
refinement  of  Virchow's  (or  Remak's)  omnis  cellula  e  ceUula*;  and 
many  other  histologists,  more  or  less  in  accord,  accepted  the  chon- 
driosomes as  important  entities,  sui  generis,  intermediate  in  grade 
between  the  cell  itself  and  its  ultimate  molecular  components.  The 
extreme  cytologists  of  the  Munich  school,  Popoif,  Goldschmidt  and 
others,  following  Richard  Hertwig,  declaring  these  structures  to  be 
identical  with  "chromidia"  (under  which  name  Hertwig  ranked  all 
extra-nuclear  chromatin),  w^ould  assign  them  complex  functions  in 
maintaining  the  balance  between  nuclear  and  cytoplasmic  material; 
and  the  "chromidial  hypothesis,"  as  every  reader  of  cytological 
literature  knows,  has  become  a  very  abstruse  and  comphcated 
thing f.  With  the  help  of  the  " binuclearity  hypothesis"  of 
Schaudinn  and  his  school,  it  has  given  us  the  chromidial  net,  the 

♦  Virchow,  Arch.  f.  pathol.  Anat.  viii,  p.  23,  1855;  but  used,  implicitly,  by 
Remak,  in  his  paper  Ueber  extracellulare  Entstehung  thierischer  Zellen  und  iiber 
die  Vermehrung  derselben  durch  Theilung,  Muller's  Archiv,  1852,  pp.  47-57.  That 
cells  come,  and  only  come,  from  pre-existing  cells  seems  to  have  been  clearly 
understood  by  John  Goodoir,  in  1846;  see  his  Anatomical  Memoirs,  ii,  pp.  90,  389. 

f  Cf.  Clifford  Dobell,  Chromidia  and  the  binuclearity  hypotheses;  a  review  and 
a  criticism,  Q.J. M.S.  liii,  pp.  279-326,  1909;  A.  Prenant,  Les  Mitochondries  et 
I'Ergastoplasme,  Journ.  de  VAnut.  et  de  la  Physiol.  Xlvi,  pp.  217-285,  1910  (both 
with  copious  bibliography). 


VI]      OF  SOME  INTRACELLULAR  PHENOMENA     457 

chromidial  apparatus,  the  tropliochromidia,  idiochromidia,  gameto- 
chromidia,  the  protogonoplasm,  and  many  other  novel  and  original 
conceptions.  There  is  apt  to  be  confusion  between  important  and 
unimportant  things ;  and  the  very  names  are  apt  to  vary  somewhat 
in  significance  from  one  writer  to  another. 

The  outstanding  fact,  as  it  seems  to  me,  is  that  physiological 
science  has  been  heavily  burdened  in  this  matter,  with  a  jargon  of 
names  and  a  thick  cloud  of  hypotheses ;  but  from  the  physical  point 
of  view  we  see  but  little  mystery  in  the  whole  phenomenon.  For, 
on  the  one  hand,  it  is  likely  enough  that  these  various  bodies, 
by  vastly  extending  the  intra-cellular  surface-area,  may  serve 
to  increase  the  physico-chemical  activities  of  the  cell;  and,  on 
the  other  hand,  we  ascribe  their  very  existence,  in  all  probability 
and  in  general  terms,  to  the  "clumping"  together  under  surface- 
tension  of  various  constituents  .of  the  heterogeneous  cell-contents, 
and  to  the  drawing  out  of  the  little  clumps  along  the  axis  of  the  cell 
towards  one  extremity  or  the  other,  in  relation  to  osmotic  currents 
as  these  are  set  up  in  turn  in  direct  relation  to  the  phenomena  of 
surface-energy  and  of  adsorption*.  And  all  this  imphes  that  the 
study  of  these  minute  structures,  even  if  it  taught  us  nothing  else, 
at  least  surely  and  certainly  reveals  the  presence  of  a  definite  field 
of  force,  and  a  dynamical  polarity  within  the  celll. 

*  Traube  in  particular  has  maintained  that  in  differences  of  surface-tension" 
we  have  the  origin  of  the  active  force  productive  of  osmotic  currents,  and  that 
herein  we  find  an  explanation,  or  an  approach  to  an  explanation,  of  many  phenomena 
which  were  formerly  deemed  peculiarly  "vital"  in  their  character.  "Die  Diiferenz 
der  Oberflachenspannungen  oder  der  Oberflachendruck  eine  Kraft  darstellt,  welche 
als  treibende  Kraft  der  Osmose,  an  die  Stelle  des  nicht  mit  dem  Oberflachendruck 
identischen  osmotischen  Druckes  zu  setzen  ist,  etc."  (Oberflachendruck  und 
seine  Bedeutung  im  Organismus,  Pfluger's  Archiv,  cv,  p.  559,  1904.)  There  is, 
moreover,  good  reason  to  believe  that  physiological  "osmosis"  is  not  a  general 
phenomenon  common  to  this  or  that  colloid  membrane  or  dialyser,  but  depends 
{int.  al.)  on  a  specific  affinity  between  the  particular  membrane  (or  the  particular 
material  it  is  moistened  with)  and  the  substance  dialysed.  This  statement,  made 
by  Kahlenberg  in  1906  {Journ.  Phys.  Cheni^.  x,  p.  141;  also  Nature,  lxxv,  p.  430, 
1907),  has  been  confirmed  (e.g.)  by  R.  Brinkmann  and  A.  von  Szent-Gyorgyi 
in  Biochem.  Ztschr'.  cxxxix,  pp.  261-273,  1923. 

t  C.  E.  Walker,  in  an  interesting  paper  on  Artefacts  as  a  guide  to  the  chemistry 
of  the  cell,  Proc.  E.S.  (B),  cm,  pp.  397-403,  1928,  tells  how  he  took  mixtures  of 
albumen,  gelatine  and  lipins,  with  droplets  of  methyl  myristate  (with  or  without 
phosphorus)  to  act  as  nuclei;  and  found  on  treating  with  osmic  acid  that  the  lipins 
had  separated  out  and  arranged  themselves  very  much  as  do  Golgi  bodies  and 
other  structural  elements  in  ordinary  histological  preparations. 


458  A  NOTE   ON  ADSORPTION  [ch. 

Our  next  and  last  illustration  of  the  effects  of  adsorption,  which 
we  owe  to  the  work  of  the  late  Professor  A.  B.  Macallum*  of 
Montreal,  is  of  great  importance ;  for  it  introduces  us  to  phenomena 
in  regard  to  which  we  seem  to  stand  on  firmer  ground  than  in  some 
of  the  foregoing  cases,  albeit  the  whole  story  has  not  been  told. 
In  our  last  chapter  we  were  restricted  mainly,  though  not  entirely, 
to  a  consideration  of  figures  of  equilibrium,  such  as  the  sphere,  the 
cyhnder  or  the  unduloid ;  and  we  began  at  once  to  find  ourselves  in 
difficulties  when  we  were  confronted  by  departures  from  symmetry, 
even  in  such  a  simple  case  as  the  ellipsoidal  yeast-cell  and  the 
production  of  its  bud.  We  found  the  cylindrical  cell  of  Spirogyra, 
with  its  plane  partitions  or  its  spherical  ends,  a  simple  matter  to 
understand;  but  when  this  uniform  cylinder  puts  out  a  lateral 
outgrowth  in  the  act  of  conjugation,  we  have  a  new  and  very 
different  system  of  forces  to  account  for  and  explain.  The  analogy 
of  the  soap-bubble,  or  of  the  simple  hquid  drop,  was  apt  to  lead  us 
to  suppose  that  surface-tension  w^as,  on  the  -whole,  uniform  over 
the  surface  of  the  cell;  and  that  its  departures  from  symmetry  of 
form  were  due  to  variations  in  external  resistance.  But  if  we 
have  been  inchned  to  make  such  an  assumption  we  must  now 
reconsider  it,  and  be  prepared  to  deal  with  important  localised 
variations  in  the  surface-tension  of  the  cell.  For,  as  a  matter  of 
fact,  the  simple  case  of  a  perfectly  symmetrical  drop,  with  uniform 
surface,  at  which  adsorption  takes  place  with  similar  uniformity, 
is  probably  rare  in  physics,  and  rarer  still  (if  it  exist  at  all)  in  the 
fluid  or  fluid-containing  system  which  we  call  in  biology  a  cell. 
We  have  more  to  do  with  cells  whose  general  heterogeneity  of 
substance  leads  to  quahtative  diSerences  of  surface,  and  hence 
to  varying  distributions  of  surface-tension.  We  must  accordingly 
investigate  the  case  of  a  cell  which  displays  some  definite  and 
regular  heterogeneity  of  its  liquid  surface,  just  as  Amoeba  displays 
a  heterogeneity  which  is  complex,  irregular  and  continually 
fluctuating  in  amount  and  distribution.  Such  heterogeneity  as  we 
are  speaking  of  must  be  essentially  chemical,  and  the  preliminary 
problem  is  to  devise  methods  of  "  microchemical "  analysis,  which 
shall  reveal  localised  accumulations  of  particular  substances  within 

*  See  his  Methoden  u.  Ergebnisse  der  Mikrochemie  in  der  biologischen  Forschung; 
Asher-Spiro's  Ergebnisse,  vn,  1908. 


VI]  OF  MACALLUM'S  EXPERIMENTS  459 

the  narrow  limits  of  a  cell  in  the  hope  that,  their  normal  effect  on 
surface-tension  being  ascertained,  we  may  then  correlate  with  their 
presence  and  distribution  the  actual  indications  of  varying  surface- 
tension  which  the  form  or  movement  of  the  cell  displays.  In 
theory  the  method  is  all  that  we  could  wish,  but  in  practice  we 
must  be  content  with  a  very  limited  apphcation  of  it;  for  the 
substances  which  have  such  action  as  we  are  looking  for,  and 
which  are  also  actual  or  possible  constituents  of  the  cell,  are  very 
numerous,  while  the  means  are  very  seldom  at  hand  to  demonstrate 
their  precise  distribution  and  locahsation.  But  in  one  or  two  cases 
we  have  such  means,  and  the  most  notable  is  in  connection  with 
the  element  potassium.  As  Macallum  has  shewn,  this  element  can 
be  revealed  in  very  minute  quantities  by  means  of  a  certain  salt, 
a  nitrite  of  cobalt  and  sodium*.  This  salt  penetrates  readily  into 
the  tissues  and  into  the  interior  of  the  cell ;  it  combines  with 
potassium  to  form  a  sparingly  soluble  nitrite  of  cobalt,  sodium  and 
potassium;  and  this,  on  subsequent  treatment  with  ammonium 
sulphide,  is  converted  into  a  characteristic  black  precipitate  of 
cobaltic  sulphide  f. 

By  this  means  Macallum  demonstrated,  years  ago,  the  unexpected 
presence  of  potassium  (i.e.  of  chlorides  or  other  potassium  salts) 
accumulated  in  particular  parts  of  various  cells,  both  sohtary  cells 
and  tissue  cells  J ;  and  he  arrived  at  the  conclusion  that  the  localised 
accumulations  in  question  were  simply  evidences  of  concentration  of 
the  dissolved  potassium  salts,  formed  and  locaUsed  in  accordance 
with  the  Gibbs-Thomson  Law.  For  potassium  (as  we  now  know) 
has  a  much  higher  ionic  velocity  than  sodium ;   and  accordingly  the 

*  On  the  distribution  of  potassium  in  animal  and  vegetable  cells,  Journ.  Physiol. 
XXXII,  p.  95,  1905.  (The  only  substance  at  all  likely  "to  be  confused  with  potassium 
in  this  reaction  is  creatine.) 

t  The  reader  will  recognise  a  fundamental  difference,  and  contrast,  between 
such  experiments  as  those  of  Macallum 's  and  the  ordinary  staining  processes  of 
the  histologist.  The  latter  are  (as  a  general  rule)  merely  empirical,  while  the  former 
endeavour  to  reveal  the  true  microchemistry  of  the  cell.  "On  pent  dire  que  la 
microchimie  n'est  encore  qu'a  la  periode  d'essai,  et  que  I'avenir  de  I'histologie 
et  specialement  de  la  cytologic  est  tout  entier  dans  la  microchimie":  A.  Prenant, 
Methodes  et  resultats  de  la  microchimie,  Journ.  de  VAnat.  et  de  la  Physiol,  xlvi, 
pp.  343-404,  1910.  There  is  an  interesting  paper  by  Brunswick,  on  the  Limitations 
of  microchemical  methods  in  biology,  in  Die  Naturwissenschaften,  Nov.  2,  1923. 

X  It  is  always  conspicuously  absent,  as  are  chlorides  and  phosphates  in  general, 
from  the  nuclear  substance. 


460 


A  NOTE   ON  ADSORPTION 


[CH. 


K-ioris  reach  and  occupy  the  adsorbing  surfaces  of  the  cell-membranes 
out  of  all  proportion  to  their  abundance  in  the  external  media*. 
And  we  may  take  it  also  that  our  potassium  salts,  hke  inorganic 
substances  in  general,  tend  to  raise  the  surface-tension,  and  will  be 
found  concentrated,  therefore,  at  a  portion  of  the  surface  where  the 
tension  is  weakf.    , 


iM2^ 


Fig.  150.  Adsorptive  concentration  of  potassium  salts  in  (1)  a  cell  of  Pleurocarpus 
about  to  conjugate;  (2)  conjugating  cells  of  Mesocarpus;  (3)  sprouting  spores 
of  Equisetum.     After  Macallum. 

In  Professor  Macallum's  figure  (Fig.  150,  1)  of  the  little  green 
alga  Pleurocarpus ,  we  see  that  one  side  of  the  cell  is  beginning  to 
bulge  out  in  a  wide  convexity.  This  bulge  is,  in  the  first  place, 
a  sign  of  weakened  surface-tension  on  one  side  of  the  cell,  which  as 
a  whole  had  hitherto  been  a  symmetrical  cylinder;  in  the  second 
place,  we  see  that  the  bulging  area  corresponds  to  the  position  of 
a  great  concentration  of  the  potassium  salt;  while  in  the  third  place, 

*  Cf.  A.  B.  Macallum,  Address  to  Section  I,  Brit.  Ass.  1910;  Oberflachen- 
spannung  und  Lebenserscheinungen,  in  Asher-Spiro's  Ergehnisse  der  Physiologie, 
XI,  pp.  598-688,  1911;  also  his  important  paper  on  Tonic  mobility  as  a  factor  in 
influencing  the  distribution  of  potassium  in  living  matter,  Proc.  R.S.  (B),  civ, 
pp.  440-458,  1929;  cf.  E.  F.  Burton,  Trans.  Faraday  Soc.  xxvi,  p.  677,  1930. 
•  t  Ii^  accordance  with  the  "principle  of  Le  Chatelier,"  which  is  in  fact  a  corollary 
to  the  Gibbs-Thomson  Law. 


VI]  OF  MACALLUM'S  EXPERIMENTS  461 

from  the  physiological  point  of  view,  we  call  the  phenomenon 
the  first  stage  in  the  process  of  conjugation.  In  the  figure  of 
Mesocarpus  (a  close  ally  of  Spirogyra),  we  see  the  same  phenomenon 
admirably  exemplified  in  a  later  stage.  From  the  adjacent  cells 
distinct  outgrowths  are  being  emitted,  where  the  surface-tension  has 
been  weakened:  just  as  the  glass-blower  warms  and  softens  a  small 
part  of  his  tube  to  blow  out  the  softened  area  into  a  bubble  or' 
diverticulum;  and  in  our  Mesocarpus  cells  (besides  a  certain 
amount  of  potassium  rendered  visible  over  the  boundary  which 
separates  the  green  protoplasm  from  the  cell-sap),  there  is  a  very 
large  accumulation  precisely  at  the  point  where  the  tension  of  the 
originally  cylindrical  cell  is  weakening  to  produce  the  bulge. 
But  in  a  still  later  stage,  when  the  boundary  between  the  two 
conjugating  cells  is  lost  and  the  cytoplasm  of  the  two  cells  becomes 
fused  together,  then  the  signs  of  potassium  concentration  quickly 
disappear,  the  salt  becoming  generally  diffused  through  the  now 
symmetrical  and  spherical  "zygospore." 

In  a  spore  of  Equisetum,  while  it  is  still  a  single  cell,  no 
localised  concentration  of  potassium  is  to  be  discerned;  but  as 
soon  as  the  spore  has  divided  by  an  internal  partition  into  two 
cells,  the  potassium  salt  is  found  to  be  concentrated  in  the  smaller 
one,  and  especially  towards  its  outer  wall  which  is  marked  by  a 
pronounced  convexity.  As  this  convexity  (which  corresponds 
to  one  pole  of  the  now  asymmetrical,  or  quasi-ellipsoidal  spore) 
grows  out  into  the  root-hair,  the  potassium  salt  accompanies  its 
growth  and  is  concentrated  under  its  wall.  The  concentration  is, 
accordingly,  a  concomitant  of  the  diminished  surface-tension  which 
is  manifested  in  the  altered  configuration  of  the  system. 

The  Acinete  protozoa  obtain  their  food  through  suctorial  tentacles 
extruded  from  the  surface  of  the  cell :  their  extrusion  being  doubtless 
due  to  a  local  diminution  of  surface-tension.  A  dense  concentration 
of  potassium  reveals  itself,  accordingly,  in  the  surface-film  of  each 
tiny  tentacle.  As  the  tentacles  are  withdrawn  their  potassium 
diffuses  into  the  cytoplasm;  when  retraction  is  complete  it  is  again 
found  in  surface-concentration,  but  the  surface-films  on  which  it 
now  concentrates  are  the  surfaces  of  the  protein-spherules  (or  "food- 
vacuoles")  within  the  body  of  the  cell. 

In  the  case  of  ciliate  or  flagellate  cells,  there  is  to  be  found  a 


462  A  NOTE   ON  ADSORPTION  [ch. 

characteristic  accumulation  of  potassium  at  and  near  the  base  of 
the  ciUa.  The  relation  of  ciliary  movement  to  surface-tension* 
lies  beyond  our  range,  but  the  fact  which  we  have  just  mentioned 
throws  light  upon  the  frequent  or  general  presence  of  a  little 
protuberance  of  the  cell-surface  just  where  a  flagellum  is  given  oif 
(cf.  p.  406),  and  of  a  little  projecting  ridge  or  fillet  at  the  base  of 
an  isolated  row  of  cilia,  such  as  we  find  in  Vorticella. 

Yet  another  of  Professor  Macallum's  demonstrations,  though  its 
interest  is  mainly  physiological,  will  help  us  somewhat  further  to 
comprehend  what  is  impHed  in  our  phenomenon.  In  a  normal  cell 
of  Spirogyra,  a  concentration  of  potassium  is  revealed  along  the 
whole  surface  of  the  spiral  coil  of  chlorophyll-bearing,  or  "chromato- 
phoral,"  protoplasm,  the  rest  of  the  cell  being  wholly  destitute  of 
that  substance :  the  inference  being  that  at  this  particular  boundary, 
between  chromatophore  and  cell-sap,  the  surface-tension  is  small 
in  comparison  with  any  other  interfacial  surface  within  the  system. 
And  again,  in  certain  minute  Chytridia-like  fungi,  parasitic  on 
Spirogyra  and  the  like,  the  potassium-reaction  helps  to  trace  the 
delicate  haustoria  of  the  parasite  in  their  course  within  the  host-cell 
— a  clear  indication  of  low  surface-tension  at  the  surface  between. 

Now  as  Macallum  points  out,  the  presence  of  potassium  is  known 
to  be  a  factor,  in  connection  with  the  chlorophyll-bearing  proto- 
plasm, in  the  synthetic  production  of  starch  from  COg  under  the 
influence  of  sunlight;  but  we  are  left  in  some  doubt  as  to  the 
consecutive  order  of  the  phenomena.  For  the  lowered  surface- 
tension,  indicated  by  the  presence  of  the  potassium,  may  be  itself 
a  cause  of  the  carbohydrate  synthesis;  while  on  the  other  hand, 
this  synthesis  may  be  attended  by  the  production  of  substances 
(e.g.  formaldehyde)  which  lower  the  surface-tension,  and  so  conduce 
to  the  concentration  of  potassium.  All  we  know  for  certain  is  that 
the  several  phenomena  are  associated  with  one  another,  as  ap- 
parently inseparable  parts  or  inevitable  concomitants  of  a  certain 
complex  aqtionf. 

*  Cf.  J.  Gray,  The  mechanism  of  ciliary  movement,  Proc.  U.S.  (B),  1922-24. 

t  The  distribution  of  potassium  within  plant-cells  is  more  complicated  than  it 
seemed  at  first  to  be;  but  it  is  still  the  general  if  not  the  invariable  rule  to  find  it 
associated  (by  adsorption)  with  one  boundary-surface  or  another.  Cf.  E.  S. 
Dowding,  Regional  and  seasonal  distribution  of  potassium  in  plant  tissues,  Ann. 
Bot.  xxxix,  pp.  459-470, 1925.    The  whole  ciuestion,  first  adumbrated  by  Macallum, 


VI]  OF  MACALLUM'S  EXPERIMENTS  463 

And  now  to  return,  for  a  moment,  to  the  question  of  cell-form. 
When  we  assert  that  the  form  of  a  cell  (in  the  absence  of  mechanical 
pressure)  is  essentiall>  ..^pendent  on  surface-tension,  and  even  when 
we  make  the  preliminary  assumption  that  protoplasm  is  essentially 
a  fluid,  we  are  resting  our  belief  on  a  general  consensus  of  evidence, 
rather  than  on  comphance  with  any  one  crucial  definition.  The 
simple  fact  is  that  the  agreement  of  cell-forms  with  the  forms  which 
physical  experiment  and  mathematical  theory  assign  to  liquid 
surfaces  under  the  influence  of  surface-tension  is  so  frequently  and 
often  so  typically  manifested  that  we  are  led,  or  driven,  to  accept 
the  surface-tension  hypothesis  as  generally  apphcable  and  as  equi- 
valent to  a  universal  law.  The  occasional  difficulties  or  apparent 
exceptions  are  such  as  to  call  for  further  enquiry,  but  fall  short  of 
throwing  doubt  on  the  hypothesis.  Macallum's  researches  introduce 
a  new  element  of  certainty,  a  "nail  in  a  sure  place,"  when  they 
demonstrate  that  in  certain  movements  or  changes  of  form  which 
we  should  naturally  attribute  to  weakened  surface-tension,  a 
chemical  concentration  which  would  naturally  accompany  such 
weakening  actually  takes  place.  They  further  teach  us  that  in  the 
cell  a  chemical  heterogeneity  may  exist  of  a  very  marked  kind, 
certain  substances  being  accumulated  here  and  absent  there,  within 
the  narrow  bounds  of  the  system. 

"  Such  localised  accumulations  can  as  yet  only  be  demonstrated  in 
the  case  of  a  very  few  substances,  and  of  a  single  one  in  particular ; 
and  these  few  are  substances  whose  presence  does  not  produce,  but 
whose  concentration  tends  to  follow^  a  weakening  of  surface-tension. 
The  physical  cause  of  the  localised  inequalities  of  surface-tension 
remains  unknown.  We  may  assume,  if  we  please,  that  they  are 
due  to  the  prior  accumulation,  or  local  production,  of  bodies  which 
have  this  direct  effect;  though  we  are  by  no  means  limited  to  this 
hypothesis.  But  in  spite  of  some  remaining  difficulties  and  un- 
certainties, we  have  arrived  at  the  conclusion,  as  regards  unicellular 
organisms,  that  not  only  their  general  configuration  but  also  their 

is  part  of  the  general  subject  of  ionic  regulation,  which  has  since  become  a  matter 
of  great  physiological  importance;  cf.  [int.  al.)  D.  A.  Webb,  Ionic  regulation  in 
Carcinus  mnenns,  Pror.  R,S.  (B),  cxxix,  pp.  107-130,  1940,  and  many  works 
quoted  therein.  It  is  curious  and  interesting  that  Macallum's  first  work  on  unequal 
ionic  distribution  in  the  tissues  and  Donnan's  fundamental  conception  of  the  Donnan 
equilibrium  {Journ.  Chem,.  Sdv.  xcix,  p.  1554,  1911)  came  just  at  the  same  time. 


464  A  NOTE   ON  ADSORPTION  [ch.  vi 

departures  from  symmetry  may  be  correlated  with  the  molecular 
forces  manifested  in  their  fluid  or  semi-fluid  surfaces. 

Looking  at  the  physiological  side,  rather  than  at  the  morphological 
which  is  more  properly  our  own,  we  see  how  very  important  a 
cellular  system  is  bound  to  be,  even  in  respect  of  its  surface-area 
alone.  The  order  of  magnitude  of  the  cells  which  constitute  our 
tissues  is  such  as  to  give  a  relation  of  surface  to  volume  far  beyond 
anything  in  all  the  structures  or  mechanisms  devised  and  fabricated 
by  man.  At  this  extensive  surface,  capillary  energy,  a  form  of 
energy  scarcely  utilised  by  man,  plays  a  large  predominant  part  in 
the  energetics  of  the  organism.  Even  the  warm-blooded  animal  is 
not  in  reality  a  heat-engine;  working  as  it  does  at  almost  constant 
temperatures  its  output  of  energy  is  bound,  by  the  principle  of 
Carnot,  to  be  small.  Nor  is  it  an  electrostatic  machine,  nor  yet  an 
electrodynamic  one.  It  is  a  mechanism  in  which  chemical  energy 
turns  into  surface-energy,  and,  working  hand  in  hand,  the  two  are 
transformed  into  mechanical  energy,  by  steps  which  are  for  the 
most  part  unknown*. 

We  are  led  on  by  these  considerations  to  reflect  on  the  molecular, 
rather  than  the  histological,  structure  of  the  cell.  We  have  already 
spoken  in  passing  of  "  monomolecular  layers,"  such  as  Henri  Devaux 
imagined  some  thirty  years  ago,  and  afterwards  obtained  f,  and 
such  as  Irving  Langmuir  has  lately  made  his  own.  The  free  surface 
of  every  liquid  (provided  the  form  and  symmetry  of  its  molecules 
permit)  presents  a  single  layer  of  oriented  molecules.  Such  a  surface 
h  no  mere  limit  or  simple  boundary;  it  becomes  a  region  of  great 
importance  and  peculiar  activity  in  certain  cases,  when,  for  instance, 
protein  molecules  of  vast  complexity  are  concerned.  It  is  then 
a  morphological  field  with  a  molecular  structure  of  its  own,  and  a 
dynamical  field  with  energetics  of  its  own.  It  becomes  a  frontier 
where  this  alien  molecule  may  be  excluded  and  that  other  be  passed 
through:  where  some  must  submit  to  mere  adsorption,  and  others 
sufl'er  chemical  change.  In  a  word,  we  begin  to  look  on  a  surface-layer 
or  membrane,  visible  or  invisible,  as  a  vastly  important  thing,  a  place 
of  delicate  operations,  and  a  field  of  peculiar  and  potent  activity. 

*  Lippmann  imagined  a  moteur  electrocapillaire,  unique  in  the  history  of 
mechanical  invention.     Cf.  Berthelot,  Rev.  Sci.  Dec.  7,  1913. 

t  CL{int.  al.)  H.  Devaux,  La  structure  moleculaire  de  la  cellule  vegetale,  Bull. 
Soc.  Bot.  de  France,  lxxv,  p.  88,  1928. 


CHAPTER  VII 

THE  FORMS  OF  TISSUES  OR  CELL-AGGREGATES 

We  pass  from  the  solitary  cell  to  cells  in  contact  with  one  another 
— to  what  we  may  call  in  the  first  instance  "cell-aggregates," 
through  which  we  shall  be  led  ultimately  to  the  study  of  complex 
tissues.  In  this  part  of  our  subject,  as  in  the  preceding  chapters, 
we  shall  have  to  consider  the  effect  of  various  forces;  but,  as  in 
the  case  of  the  soUtary  cell,  we  shall  probably  find,  and  we  may  at 
least  begin  by  assuming,  that  the  agency  of  surface-tension  is 
especially  manifest  and  important.  The  effect  of  this  surface-tension 
will  manifest  itself  in  surfaces  mininiae  areae:  where,  as  Plateau 
was  always  careful  to  point  out,  we  must  understand  by  this 
expression  not  an  absolute  but  a  relative  minimum,  an  area,  that 
is  to  say,  which  approximates  to  an  absolute  minimum  as  nearly  as 
the  circumstances  and  material  exigencies  of  the  case  permit. 

There  are  certain  fundamental  principles,  or  fundamental  equa- 
tions, besides  those  we  have  already  considered,  which  we  shall  need 
in  our  enquiry;  for  instance,  the  case  which  we  briefly  touched 
on  (on  p.  426)  of  the  angle  of  contact  between  the  protoplasm 
and  the  axial  filament  in  a  Heliozoan,  we  shall  now  find  to  be  but 
a  particular  case  of  a  general  and  elementary  theorem. 

Let  us  re-state  as  follows,  in  terms  of  Energy,  the  general 
principle  which  underlies  the  theory  of  surface-tension  or  capillarity*. 

When  a  fluid  is  in  contact  with  another  fluid,  or  with  a  solid  or 
with  a  gas,  a  portion  of  the  total  energy  of  the  system  (that,  namely, 
which  we  call  surface  energy)  is  proportional  to  the  area  of  the 
surface  of  contact;  it  is  also  proportional  to  a  coefficient  which  is 
specific  for  each  particular  pair  of  substances  and  is  constant  for 
these,  save  only  in  so  far  as  it  may  be  modified  by  changes  of 
temperature  or  of  electrical  charge.  Equihbrium,  which  is  the 
condition  of  minimum  potential  energy  in  the  system,  will  accordingly 

*  See  Clerk  Maxwell's  famous  article  on  "Capillarity"  in  the  ninth  edition  of 
the  Encyclopedia  Britannica,  revised  by  Lord  Rayleigh  in  the  tenth  edition. 


466  THE  FORMS  OF  TISSUES  [ch. 

be  obtained,  caeteris  paribus,  by  the  utmost  possible  reduction  of 
the  surfaces  in  contact. 

When  we  have  three  bodies  in  contact  with  one  another  the  same 
is  true,  but  the  case  becomes  a  Httle  more  complex.  Suppose  a 
drop  of  some  fluid,  A,  to  float  on  another  fluid,  J5,  while  both  are 
exposed  to  air,  C  Here  are  three  surfaces  of  contact,  that  of  the 
drop  with  the  fluid  on  which  it  floats,  and  those  of  air  with  the  one 
and  other  of  these  two;  and  the  whole  surface-energy,  E,  of  the 
system   consists  of.  three'  parts  resident  in  these   three   surfaces, 


Fig.  151. 

or  of  three  specific  energies,  -S^^,  -E'^ic  ^bc-  "^^^  condition  of 
equihbrium,  or  minimal  potential  energy,  will  be  reached  by  con- 
tracting those  surfaces  whose  specific  energy  happens  to  be  large 
and  extending  those  where  it  is  small — contraction  leading  to  the 
production  of  a  "drop,'f  and  extension  to  a  spreading  "film." 
Floating  on  water,  turpentine  gathers  into  a  drop,  olive-oil  spreads 
out  in  a  film;  and  these,  according  to  the  several  specific  energies, 
are  the  ways  by  which  the  total  energy  of  the  system  is  diminished 
and  equihbrium  attained. 

A  drop  will  continue  to  exist  provided  its  own  two  surface-energies 
exceed,  per  unit  area,  the  specific  energy  of  the  water-air  surface 
around:   that  is  to  say,  provided  (Fig.  151) 

^AB  +  ^AC  >  ^BC 

But  if  the  one  fluid  happen  to  be  oil  and  the  other  water,  then  the 
combined  energy  per  unit-area  of  the  oil-water  and  the  oil-air 
surfaces  together  is  less  than  that  of  the  water-air  surface : 

E     >  E    4-  E 

Hence  the  oil-air  and  oil-water  surfaces  increase,  the  air-water 
surface  contracts  and  disappears,  the  oil  spreads  over  the  water, 
and  the  "drop"  gives  place  to  a  "film,"  In  both  cases  the  total 
surface-area  is  a  minimum  under  the  circumstances  of  the  case,  and 
always  provided  that  no  external  force,  such  as  gravity,  complicates 
the  situation. 


VII]  OF  FLOATING  DROPS  467 

The  surface-energy  of  which  we  are  speaking  here  is  manifested 
in  that  contractile  force,  or  tension,  of  which  we  have  had  so  much 
to  say*.  In  any  part  of  the  free  water-surface,  for  instance,  one 
surface-particle  attracts  another  surface-particle,  and  the  multi- 
tudinous attractions  result  in  equilibrium.  But  a  water-particle  in 
the  immediate  neighbourhood  of  the  drop  may  be  pulled  outwards, 


Pig.  152. 

so  to  speak,  by  another  water-particle,  but  find  none  on  the  other 
side  to  furnish  the  counter-pull;  the  pull  required  for  equiUbrium 
must  therefore  be  provided  by  tensions  existing  in  the  other  two 
surfaces  of  contact.  In  short,  if  we  imagine  a  single  particle  placed 
at  the  very  point  of  contact,  it  will  be  drawn  upon  by  three  different 
forces,  whose  directions  lie  in  the  three  surface-planes  and  whose 

*  It  can  easily  be  proved  (by  equating  the  increase  of  energy  stored  in  an 
increased  surface  with  the  work  done  in  increasing  that  surface),  that  the  tension 
measured  per  unit  breadth,  T^^,  is  equal  to  the  energy  per  unit  area,  EfO,-  Surface- 
tensions  are  very  diverse  in  magnitude, -but  all  are  positive;  Clerk  Maxwell 
conceived  the  existence  of  negative  surface-tensions,  but  could  not  point  to  any 
certain  instance.  When  blood-serum  meets  a  solution  of  common  salt,  the  two 
fluids  hasten  to  mix,  long  streamers  of  the  one  running  into  the  other;  this 
remarkable  phenomenon,  first  observed  by»Almroth  Wright  {Proc.  R.S.  (B), 
xcii,  1921)  and  called  by  him  "  pseudopodial  intertraction,"  was  described  by 
Schoneboom  (ibid.  (A),  ci,  1922)  as  a  case  of  negative  surface-tension.  But  it 
is  a  diffusion-phenomenon  rather  than  a  capillary  one. 


468  THE  FORMS  OF  TISSUES  [ch. 

magnitudes  are  proportional  to  the  specific  tensions  characteristic 
of  the  three  "  interfacial "  surfaces.  Now  for  three  forces  acting  at 
a  point  to  be  in  equihbrium  they  must  be  capable  of  representation, 
in  magnitude  and  direction,  by  the  three  sides  of  a  triangle  taken 
in  order,  in  accordance  with  the  theorem  of  the  Triangle  of  Forces. 
So,  if  we  know  the  form  of  our  drop  as  it  floats  on  the  surface 
(Fig.  152),  then  by  drawing  tangents  P,  R,  from  0  (the  point  of 
mutual  contact),  we  determine  the  three  angles  of  our  triangle,  and 
know  therefore  the  relative  magnitudes  of  the  three  surface-tensions 
proportional  to  its  sides.  Conversely,  if  we  know  the  three  tensions 
acting  in  the  directions  P,  R,  S  (viz.  Tab,  T^c,  T^^)  we  know  the  three 
sides  of  the  triangle,  and  know  from  its  three  angles  the  form  of 
the  section  of  the  drop.  All  points  round  the  edge  of  the  drop  being 
under  similar  conditions,  the  drop  must  be  circular  and  its  figure 
that  of  a  sohd  of  revolution*. 

The  principle  of  the  triangle  of  forces  is  expanded,  as  follows, 
in  an  old  seventeenth-century  theorem,  called  Lamy's  Theorem : 

//  three  forces  acting  at  a  point  be  in  equilibrium,  each  force  is 
proportional  to  the  sine  of  the  angle  contained  between  the  directions  of 
the  other  two.     That  is  to  say  (in  Fig.  152) 

P :  R  :  S  =  sin  (f) :  sin  p  :  sin  s,     . 

P         R         S 

or  ^-7  =  -^ — ==-■ — • 

sm  (f)      sm  p      sm  s" 

And  from  this,  in  turn,  we  derive  the  equivalent  formulae  by  which 
each  force  is  expressed  in  terms  of  the  other  two  and  of  the  angle 
between  them:  viz. 

P^  =  R^  +  S^+  2RS  cos  </.,  etc. 

From  this  and  the  foregoing,  we  learn  the  following  important 
and  useful  deductions: 

(1)   The  three  forces  can  only  be  in  equihbrium  when  each  is  less 

*  Bubbles  have  many  beautiful  properties  besides  the  more  obvious  ones.  For 
instance,  a  floating  bubble  is  always  part  of  a  sphere,  but  never  more  than  a 
hemisphere;  in  fact  it  is  always  rather  less,  and  a  very  small  bubble  is  considerably 
less,  than  a  hemisphere.  Again,  as  we  blow  up  a  bubble,  its  thickness  varies 
inversely  as  the  square  of  its  diameter;  the  bubble  becomes  a  hundred  and  fifty 
times  thinner  as  it  grows  from  an  inch  in  diameter  to  a  foot.  In  an  actual  calculation 
we  must  always  take  account  of  the  tensions  on  both  surfaces  of  each  film  or 
membrane. 


VII]  OF  FLOATING  DROPS  469 

than  the  sum  of  the  other  two ;  otherwise  the  triangle  is  impossible. 
In  the  case  of  a  drop  of  olive-oil  on  a  clean  water-surface,  the  relative 
magnitudes  of  the  three  tensions  (at  15°  C.)  are  nearly  as  follows: 

Water-air  surface  59 

Oil-air  „  25 

Oil- water       ,,  16 

No  triangle  having  sides  of  these  relative  magnitudes  is  possible, 
and  no  such  drop  can  remain  in  existence*. 

(2)  The  three  surfaces  may  be  all  ahke :  as  when  two  soap-bubbles 
are  joined  together  on  either  side  of  a  partition-film.  The  three 
tensions  then  are  all  co-equal,  and  the  three  angles  are  co-equal; 
that  is  to  say,  when  three  similar  liquid  surfaces,  or  films,  meet 
together,  they  always  do  so  at  identical  angles  of  120°.  Whether 
our  two  conjoined  soap-bubbles  be  equal  or  unequal,  this  is  still 
the  invariable  rule;  because  the  specific  tension  of  a  particular 
surface  is  independent  of  form  or  magnitude. 

(3)  If  all  three  surfaces  be  different,  as  when  a  fluid  drop  Ues 
between  water  and  air,  the  three  surface- tensions  will  (in  all  likeli- 
hood) be  different,  and  the  two  surfaces  of  the  drop  will  differ  in 
their  amount  of  curvature. 


Fig,  153. 

(4)  If  two  only  of  the  surfaces  be  alike,  then  two  of  the  angles 
will  be  ahke  and  the  other  will  be  unlike;  and  this  last  will  be  the 
difference  between  360°  and  the  sum  of  the  other  two.  A  particular 
case  is  when  a  film  is  stretched  between  sohd  and  parallel  walls, 
like  a  soap-film  within  a  cylindrical  tube.  Here,  so  long  as  no 
external  pressure  is  applied  to  either  side,  so  long  as  both  ends  of 
the  tube  are  open  or  closed,  the  angles  on  either  side  of  the  film 
will  be  equal,  that  is  to  say  the  film  will  set  itself  at  right  angles  to 
the  sides.    Many  years  ago  Sachs  laid  it  down  as  a  principle,  which 

*  Nevertheless,  if  the  water-surface  be  contaminated  by  ever  so  thin  a  film  of 
oil,  the  oil-drop  may  be  made  to  float  upon  it.  See  Rayleigh  on  Foam,  Collected 
Works,  m,  p.  351. 


470  THE  FORMS  OF  TISSUES  [ch. 

has  become  celebrated  in  botany  under  the  name  of  Sachs's  Rule, 
that  one  cell- wall  always  tends  to  set  itself  at  right  angles  to  another 
cell- wall.  But  this  rule  only  applies  to  the  case  we  have  just 
illustrated;  and  such  validity  as  it  possesses  is  due  to  the  fact  that 
among  plant-tissues  it  commonly  happens  that  one  cell-wall  has 
become  soUd  and  rigid  before  another  partition- wall  impinges  upon  it. 
(5)  Another  important  principle .  arises,  not  out  of  our  equations 
but  out  of  the  general  considerations  which  led  to  them.  We  saw  in 
the  soap-bubble  that  at  and  near  the  point  of  contact  between  our 
several  surfaces,  there  is  a  continued  balance  of  forces,  carried  (so 
to  speak)  across  the  interval;  in  other  words,  there  is  physical 
continuity  between  one  surface  and  another  and  it  follows  that  the 
surfaces  merge  one  into  another  by  a  continuous  curve.     Whatever 


A^ 


Fig.  154.     Plateau's  bourrelet, 

in  an  algal  filament.   After  a  b 

Berthold.  Fig.  155. 

be  the  form  of  our  surfaces  and  whatever  the  angle  between  them, 
a  small  intervening  curved  surface  is  always  there  to  bridge  over 
the  Hne  of  contact;  and  this  Uttle  fillet,  or  "bourrelet,"  as  Plateau 
called  it,  is  big  enough  to  be  a  common  and  conspicuous  feature  in 
the  microscopy  of  tissues  (Fig.  154).  A  similar  "bourrelet"  is 
clearly  seen  at  the  boundary  between  a  floating  bubble  and  the  liquid 
on  which  it  floats:  in  which  case  it  constitutes  a  "masse  annulaire," 
whose  mathematical  properties  and  relation  to  the  form  of  the 
nearly  hemispherical  bubble  have  been  investigated  by  van  der 
Mensbrugghe*.  The  superficial  vacuoles  in  Actinophrys  or  Actino- 
sphaerium  present  an  identical  phenomenon. 

(6)  It  is  a  curious  efl'ect,  or  consequence,  of  the  bourrelet  that 
a  "horizontal"  soap-film  is  never  either  horizontal  or  plane.  For 
the  bourrelet  at  its  edge  is  deformed  by  gravity,  and  the  film  is 
correspondingly  inclined  upwards  where  it  meets  it  (Fig.  155  6). 

*  Cf.  Plateau,  op.  cit.  p.  366. 


VII 


OF  PLATEAU'S  BOURRELET 


471 


(7)  The  bourrelet,  a  fluid  mass  connected  with  a  fluid  film,  is  no 
mere  passive  phenomenon  but  has  its  active  influence  or  dynamical 
effect.  This  was  pointed  out  by  Willard  Gibbs*,  and  Plateau's 
bourrelet  is  more  often  called,  nowadays,  "Gibbs's  Ring."  The  ring 
is  continuous  in  phase  with  the  interior  of  the  film,  and  fluid  is 
sucked  into  it  from  the  latter,  which  thins  rapidly;  and  this, 
becoming  a  more  potent  factor  of  unrest  than  gravity  itself,  leads 
presently  to  the  rupture  of  the  film.  Plateau's  explanation  of  his 
bourrelet  as  a  ''surface  of  continuity"  is  thus  but  a  part,  and  a 
small  part  of  the  story. 

(8)  In  the  succulent,  or  parenchymatous,  tissue  of  a  vegetable, 
the  cells  have  their  internal  corners  rounded  off  (Fig.  156)  in  a  way 
which  might  suggest  the  bourrelet,  but  comes  pf  another  cause. 


Fig.  156.    Parenchyma  of  maize ;   shewing  intercellular  spaces. 

Where  the  angles  are  rounded  off  the  cell-walls  tend  to  split  apart 
from  one  another,  and  each  cell  seems  tending  to  withdraw,  as  far 
as  it  can,  into  a  sphere;  and  this  happens,  not  when  the  tissue  is 
young  and  the  cell-walls  tender  and  quasi-fluid,  but  later  on,  when 
cellulose  is  forming  freely  at  the  surface  of  the  cell.  The  cell- walls 
no  longer  meet  as  fluid  films,  but  are  stiffening  into  pelHcles;  the 
cells,  which  began  as  an  association  of  bubbles,  are  now  so  many 
balls,  in  solid  contact  or  partial  detachment;  and  flexibility  and 
elasticity  have  taken  the  place  of  the  capillary  forces  of  an  earlier 
and  more  liquid  phase  f. 


*  Collected  Works,  i,  p.  309. 
t  J.  H.  Priestley,  Cell-growth , 
pp.  54-81,  1929. 


,  in  the  flowering  plant,  New  Phytologist,  xxviii, 


472  THE  FORMS  OF  TISSUES  [ch. 

(9)  Statically  though  not  dynamically,  that  is  to  say  as  a  line  or 
surface  of  continuity  in  Plateau's  sense,  our  bourrelet  is  analogous 
to  the  accumulation  of  sand  seen  where  two  nodal  lines  cross  in  a 
Chladni  figure:  "Vers  les  endroits  oii  des  lignes  nodales  se  coupent, 
elles  s'elargissent  toujours,  de  sorte  que  la  forme  des  parties  vibrantes 
pres  de  ces  endroits  n'est  pas  angulaire  mais  plus  ou  moins  arrondie, 
sou  vent  en  forme  d'hyperbole*."  And  in  somewhat  remoter  analogy, 
we  may  look  on  the  three  corpora  Arantii  as  so  many  bourrelets, 
helping  to  fill  the  angles  where  three  semilunar  valves  meet  at  the 
base  of  the  great  arteries. 

We  may  now  illustrate  some  of  the  foregoing  principles,  con- 
stantly bearing  in  mind  the  principles  set  forth  in  our  chapter  on 
the  Forms  of  Cells,  and  especially  those  relating  to  the\pressure 
exercised  by  a  curved  film. 


Fig.  157. 

Let  us  look  for  a  moment  at  the  case  presented  by  the  partition- 
wall  in  a  double  soap-bubble.  As  we  have  just  seen,  the  three  films 
in  contact  (viz.  the  outer  walls  of  the  two  bubbles  and  the  partition- 
wall  between)  being  all  composed  of  the  same  substance  and  being 
all  ahke  in  contact  with  air,  the  three  tensions  must  be  equal,  and 
the  three  films  must,  in  all  cases,  meet  at  co-equal  angles  of  120°.  But 
unless  the  two  bubbles  be  of  precisely  equal  size,  and  therefore  of 
equal  curvature,  the  tangents  to  the  spheres  will  not  meet  the  plane 
of  their  circle  of  contact  at  equal  angles,  and  the  partition-wall  will 
of  necessity  be  a  curved,  and  indeed  a  spherical,  surface;  it  is  only 
plane  when  it  divides  two  equal  and  symmetrical  cells.  It  is 
obvious,  from  the  symmetry  of  the  figure,  that  the  centres  of  the 
two  bubbles  and  of  the  partition  between  are  all  on  one  and  the 
same  straight  line. 

The  two  bubbles  exert  a  pressure  inwards  which  is  inversely 

♦  E.  F.  F.  Chladni,  Traite  d'acoustique,  1809,  p.  127. 


VII 


OF  CELL-PARTITIONS 


473 


proportional  to  their  radii:  that  is  to  say,  p:  p' ::  Ijr :  l/r';  and  the 
partition-wall  must,  for  equilibrium,  exert  a  pressure  (P)  which  is 
equal  to  the  difference  between  these  two  pressures,  that  is  to  say, 
P  =  l/R  =  Ijr'  —  Ijr  =  (r  —  r')lrr'.  It  follows  that  the  curvature  of 
the  partition  must  be  just  such  as  is  capable  of  exerting  this  pressure, 
that  is  to  say,  R  =  rr'j(r  —  /).  The  partition,  then,  is  a  portion  of 
a  spherical  surface,  whose  radius  is  equal  to  the  product,  divided 
by  the  difference,  of  the  radii  of  the  two  bubbles;  if  the  two  bubbles 
be  equal,  the  radius  of  curvature  of  the  partition  is  infinitely  great, 
that  is  to  say  the  partition  is  (as  we  have  already  seen)  a  plane 
surface. 


Fig.  158. 


In  the  typical  case  of  an  evenly  divided  cell,  such  as  a  double 
and  co-equal  soap-bubble  (Fig.  158),  where  partition- wall  and  outer 
walls  are  identical  with  one  another  and  the  same  air  is  in  contact 
with  them  all,  we  can  easily  determine  the  form  of  the  system. 
For,  at  any  point  of  the  boundary  of  the  partition,  P,  the  tensions 
being  equal,  the  angles  QPP',  RPP',  QPR  are  all  equal,  and  each 
is,  therefore,  an  angle  of  120°.  But  PQ,  PR  being  tangents,  the 
centres  of  the  two  spheres  (or  circular  arcs  in  the  figure)  lie  on  lines 
perpendicular  to  them;  therefore  the  radii  CP,  C'P  meet  at  an 
angle  of  60°,  and  CPC  is  an  equilateral  triangle.  That  is  to  say, 
the  centre  of  each  circle  Hes  on  the  circumference  of  the  other;   the 


474  THE  FORMS  OF  TISSUES  [ch. 

partition  lies  midway  between  the  two  centres;    and  the  diameter 

OP  a/3 

of  the  partition-wall,  PP\  is  -^^  =  sin  60°  =  — -  =  0-866  times  the 

diameter  of  each  of  the  two  cells.  This  gives  us,  then,  the  form 
of  a  combination  of  two  co-equal  spherical  cells  under  uniform 
conditions. 

By  integrating  between  the  known  values  of  the  meridian  section  and  the 
plane  partition,  we  should  find  each  half  of  the  double  cell  (or  soap-bubble) 
to  be  equal  to  27/32  of  a  complete  sphere.  Therefore  the  radius  of  curvature 
of  each  half  of  the  divided  bubble  is  greater  than  that  of  a  sphere  of  equal 
volume  in  the  ratio  of: 

\^32  :  \^27  =  2 .  ^4  :  3  =  1058  :  1  =  1  :  0-945. 

And  the  radius  of  the  original  sphere,  before  division,  is  to  the  radius  of 
each  half,  or  each  product  of  cell-division,  as 

^54:^^32  =  3.^^2:2.^^4  =  1191:  1  =  1:0-84. 

In  the  case  of  three  co-equal  and  united  bubbles  (to  which  case  we  shall 
presently  return),  each  is  approximately  five-sevenths  of  a  whole  sphere: 
and  their  radii,  therefore,  are  to  the  radius  of  the  whole  sphere  as 

^7  :  \^5  =  1  :  0-893  =  1  :  (0-945)2. 

When  two  co-equal  bubbles  coalesce,  the  internal  pressure,  du5  to  the 
tension  of  the  wall  and  varying  inversely  as  its  radius  of  curvature,  is 
diminished  in  the  ratio  of  1  :  0-945,  or  say  5i  per  cent.  And  we  begin  to  see, 
in  the  case  of  three  bubbles,  that  the  process  proceeds  in  a  geometrical 
progression,  each  new  coalescence  increasing  the  radius  of  curvature  and 
diminishing  the  internal  pressure,  by  a  constant  fraction  of  the  whole.  This 
and  other  simple  corollaries  may  perchance,  some  day,  be  found  useful  to  the 
biologist. 

In  the  case  of  unequal  bubbles,  the  curvature  of  their  partition- 
wall  is  easily  determined,  and  is  shewn  in  Fig.  159.  The  three 
films  meeting  in  P  being  (as  before)  identical  films,  the  three 
tangents,  PQ,  PR,  PS,  meet  at  co-equal  angles  of  120°,  and  PS 
produced  bisects  the  angle  QPR.  PQ,  PR  are  tangents  perpen- 
dicular to  the  radii  CP,  C'P\  and  C"P,  the  radius  of  the  spherical 
partition  PP\  is  found  by  drawing  a  perpendicular  to  PS  in  P. 
The  centre  C"  is,  by  the  symmetry  of  the  figure,  in  a  straight  line 
with  C,  C. 

Whether  the  partition  be  or  be  not  a  plane  surface,  it  is  obvious 
that  its  line  of  junction  with  the  rest  of  the  system  lies  in  a  plane, 


VIl] 


OF  SACHS'S  RULE 


475 


and  is  at  right  angles  to  the  axis  of  symmetry.  The  actual  curvature 
of  the  partition- wall  is  easily  seen  in  optical  section ;  but  in  surface 
view  the  line  of  junction  is  projected  as  a  plane  (Fig.  160),  perpen- 
dicular to  the  axis,  and  this  appearance  has  helped  to  lend  support 
and  authority  to  "Sachs's  Rule." 


Fig.  159.  Fig.  160. 

As  soon  as  the  tensions  of  the  cell- walls  become  unequal,  whether 
from  changes  in  their  own  substance  or  in  the  substances  with 
which  they  are  in  contact,  then  the  form  alters.  If  the  tension 
along  the  partition  P  diminishes,  the  partition  itself  enlarges  and 
the  angle  QPR  increases:  until,  when  the  tension  p  is  very  small 
compared  with  q  or  r,  the  whole  figure  becomes  a  sphere,  and  the 
partition-wall,  dividing  it  into  two  hemispheres,  stands  at  right 
angles  to  the  outer  wall.  This  is  the  case  w^hen  the  outer  wall  of 
the  cell  is  practically  sohd.     On  the  other  hand,  if  p  begins  to 


Fig.  161. 
increase  relatively  to  q  and  r,  then  the  partition-wall  contracts, 
and  the  two  adjacent  cells  become  larger  and  larger  segments  of 
a  sphere,   until  at  length  the  system  becomes  divided  into  tw^o 
separate  cells. 

To  put  the  matter  still  more  simply,  let  the  annexed  diagrams 
(Fig.  161)  represent  a  system  of  three  films,  one  being  a  partition- 


476 


THE  FORMS  OF  TISSUES 


[CH. 


wall  running  between  the  other  two;  and  where  the  partition  t 
meets  the  outer  wall  TT\  let  the  several  tensions,  or  the  tractions 
exerted  on  a  point  at  their  meeting-place,  be  proportional  to  T,  T 
and  t.     Let  a,  ^,  y  be,  as  in  the  figure,  the  opposite  angles.     Then: 

(1)  If  T  be  equal  to  T' ,  and  t  be  relatively  insignificant,  the 
angles  a,  ^  will  be  of  90°. 

(2)  If  r  =  T\  but  be  a  little  greater  than  t,  then  t  will  exert 
an  appreciable  traction,  and  a,  ^  will  be  more  than  90°,  say  for 
instance,  100°. 

(3)  lfT=T'  =  t,  then  a,  ^,  y  will  all  equal  120°. 


Fig.  I(i2.     Part  of  a  dragonfly's  wing. 


The  outer  walls  of  the  two  cells  on  either  side  of  the  partition 
will  be  straight,  as  well  as  continuous,  in  the  first  case,  and  more 
or  less  curved  in  the  other  two.  We  have  a  vivid  illustration  (if  a 
somewhat  crude  one)  of  the  first  case  in  a  section  of  honey :  where 
the  waxen  walls,  which  meet  one  another  at  120°,  meet  the  wooden 
sides  of  the  box  at  90°. 

The  wing  of  a  dragon-fly  shews  a  seemingly  complicated  system 
of  veins  which  the  foregoing  considerations  help  much  to  simplify. 
The  wing  is  traversed  by  a  few  strong  "veins,"  or  ribs,  more  or 
less  parallel  to  one  another,   between  which  finer  veins  make  a 


VII]  OF  CELL-PARTITIONS  477 

mesh  work  of  "cells,"  these  lesser  veins  being  all  much  of  a  muchness, 
and  exerting  tensions  insignificant  compared  with  those  of  the 
greater  veins.  Where  (a)  two  ribs  run  so  near  together  that  only 
one  row  of  cells  lies  betw^een,  these  cells  are  quadrangular  in  form, 
their  thin  partitions  meeting  the  ribs  at  right  angles  on  either  side. 
Where  (6)  two  rows  of  cells  are  intercalated  between  a  pair  of  ribs, 
one  row  fits  into  the  other  by  angles  of  120°,  the  result  of  co-equal 
tensions;  but  both  meet  the  ribs  at  right  angles,  as  in  the  former 
case.  W^here  (c)  the  cell-rows  are  numerous,  all  their  angles  in 
common  tend  to  be  co-equal  angles  of  120°,  and  the  cells  resolve, 
consequently^  into  a  hexagonal  meshw^ork. 

Many  spherical  cells,  such  as  Protococcus,  divide  into  two  equal 
halves,  separated  by  a  plane  partition.  Among  other  lower  Algae 
akin  to  Protococcus,  such  as  the  Nostocs 

and  Oscillatoriae,  in  which  the  cells  are    a  OCOOOTyCOD 
embedded  in  a  gelatinous  matrix,  we 
find  a  series  of  forms  such  as  are  re- 
presented in  Fig.  163,  which  various  OOCjOOOO 

conditions  depend,  according  to  what         r^rv" — ^z- v^  I 

we  have   already   learned,   upon   the         ^^--^  AJ 

relative  magnitudes  of  the  tensions  at 

the  surface  of  the  cells  and  the  boundary    q  fTTTHnnnrTT'T^ 

between  them.     In  some  cases  (Fig. 

163,    B)   the    cells   remain   spherical, 

because  they  are  merely  embedded  in    D      1    1    I    1    I    I  I  I  I  I  I  I 

the  matrix,   with   no  other   physical  Fig.  163.    Filaments,  or  chains  of 

continuity  between  them;     even  two         cells,  in  various  lower  Algae. 

soap-bubbles  do  not  tend   to   unite,         \f:l  J"T-'    ^fln'^nT'"' 

^  '(C)  Rivulana;  (D)  Oscillatoria. 

unless  their  surfaces  be  moist  or  we 

put  a  drop  of  soap-solution  between  them.  In  certain  other  cases, 
the  system  consists  of  a  relatively  thick- walled  tube,  subdivided  by 
more  dehcate  partitions,  which  latter  then  tend  (as  in  D)  to  become 
plane  septa,  set  at  right  angles  to  the  walls.  Or  again,  side- walls 
and  septa  may  be  all  alike,  or  nearly  so ;  and  then  the  configuration 
(as  in  C,  on  Fig.  163)  is  that  of  a  linear  cluster  of  soap-bubbles*. 
In  the  spores  of  liverworts,  such  as  Pellia,  the  first  partition 

*  Cf.  Dewar,  Studies  on  liquid  films,  Proc.  Roy.  Inst.  1918,  p.  359. 


478 


THE  FORMS  OF  TISSUES 


[CH. 


(the  equatorial  partition  in  Fig.  165  a)  divides  the  spore  into  two 
equal  halves,  and  is  therefore  a  plane  surface  normal  to  the  surface 
of  the  cell.  But  the  next  partitions  arise  near  to  either  end  of  the 
original  spherical  or  elliptical  cell,  and  each  of  these  latter  will 
likewise  tend  to  set  itself  normally  to  the  cell-wall — at  least  the 


Fig.  164. 

angles  on  either  side  of  the  partition  will  tend  to  be  identical,  and 
their  magnitude  will  depend  on  the  relative  tensions  of  the  cell- wall 
and  the  partition.  The  angles  will  be  right  angles  if  the  cell-wall  is 
solid  or  nearly  so  when  the  partition  is  formed;  but  they  will  be 
somewhat  greater,  if  (in  all  probability)  rigidity  of  the  cell-wall 
has  not  been  quite  attained.     In  either  case  the  partition  itself  will 


Fig.  165.    Early  development  of  a  liverwort  (Pellia).    After  Wildeman. 

be  part  of  a  spherical  surface,  whose  curvature  will  now  correspond 
to  the  difference  of  pressures  in  the  two  chambers  (or  cells)  which 
it  serves  to  separate. 

We  have  innumerable  cases,  near  the  tip  of  a  growing  filament 
for  instance,  where  in  like  manner  the  partition-wall  which  cuts  off 
the  terminal,  more  or  less  conical,  cell  constitutes  a  spherical  lens- 


VII]  OF  CELL-PARTITIONS  479 

shaped  surface,  set  normally  to  the  adjacent  walls;  and  the  centre 
of  curvature  is  the  meeting-point  of  two  tangents  to  the  cone.  We 
find  such  a  lenticular  partition  at  the  tips  of  the  branches  of  many 
Florideae;  in  Dictyota  dichotoma,  as  figured  by  Reinke,  we  have 
a  succession  of  them.  And  by  the  way,  where,  in  such  cases  as 
these,  the  tissues  happen  to  be  very  transparent,  we  often  have  a 
puzzUng  confusion  of  lines  (Fig.  166);  one  being  the  optical  section 


© 


® 


© 

Fig.  166.     CeWs  oi  Dictyota.  Fig.  167.     Terminal  and  other  cells 

After  Reinke.  of  Chara. 

of  the  curved  partition- wall,  the  other  being  the  straight  linear 
projection  of  its  outer  edge  to  which  w^  have  already  referred.  In 
the  conical  terminal  cell  of  Chara,  we  have  the  same  lens-shaped 
curve;  but  a  little  lower  down,  w^here  the  sides  of  the  shoot  are 
approximately  parallel,  we  have  flat  transverse  partitions,  and  the 
form  of  the  cells  is,  more  or  less,  w^hat  we  have  been  led  to  expect 
in  the  simple  case  of  successive  transverse  partitions  (Fig.  167). 

In  the  young  antheridia  of  Chara  (Fig.  168),  and  in  the  geo- 
metrically similar  case  of  the  sporangium  (or  conidiophore)  of 
Mucor,  we  easily  recognise  the  hemispherical  form 
of  the  septum  which  shuts  off  the  large  spherical 
cell  from  the  cylindrical  filament.  Here,  in  the  first 
phase  of  development,  we  should  have  to  take 
into  consideration  the  different  pressures  exerted 
by  the  single  curvature  of  the  cyUnder  and  the 
double  curvature  of  its  spherical  cap  (p.  371) ;  a  id 
we  should  find  that  the  partition  would  have  a 
somewhat  low  curvature,  with  a  radius  less  than 
the  diameter  of  the  cylinder,  which  it  would  have  Fig.  168.  Young 
exactly  equalled  but  for  the  additional  pressure  anthendium  of 
inwards  which  it  receives  from  the  curvature  of 
the   large   surrounding   sphere.     But   as   the   latter     continues   to 


480 


THE  FORMS  OF  TISSUES 


[CH. 


grow  its  curvature  decreases,  and  so  likewise  does  the  inward 
pressure  of  its  surface ;  and  accordingly  the  little  convex  partition 
bulges  out  more  and  more. 

In  the  ordinary  meristematic  tissue  of  a  plant,  the  new  partition- 
wall  within  a  dividing  cell  will  generally  meet  the  old  walls  at  right 
angles  to  begin  with,  because  its  tension  is  usually  small  compared 
to  what  theirs  has  become.  But  as  the  system  grows  and  the  old 
wall  strengthens,  the  tensions  of  all  three  walls  become  approxi- 
mately the  same;  and  they  tend  towards  a  new  position  of  equi- 
librium, in  which  (as  seen  in  optical  section)  they  meet  as  before, 
at  co-equal  angles  of  120°*. 


A  B 

Fig.  169.     Cambium  cells  after  division,  altering  from  A  to  B. 

The  biological  facts  which  the  foregoing  considerations  go  far  to 
explain  and  account  for  have  been  the  subject  of  much  argument 
and  discussion  on  the  part  of  the  botanists.  Let  me  recapitulate, 
in  a  very  few  words,  the  history  of  this  long  discussion. 

Some  seventy  years  ago,  Hofmeister  laid  it  down  as  a  general, 
but  purely  empirical,  law  that  "The  partition- wall  stands  always 
perpendicular  to  what  was  previously  the  principal  direction  of 
growth  in  the  cell" — or,  in  most  cases,  perpendicular  to  the  long 
axis  of  the  cellf.  This  contains  an  important  truth;  for  it  is  as 
much  as  to  say  that  the  cell  tends  to  be  divided  by  the  smallest 

*  J.  H.  Priestley,  Studies. .  .of  cambium  activity,.  New  Phytologist,  xxix,  p.  101. 
1930.  Cf.  also  J.  J.  Beijer,  Vermehrung  der  radialen  Reihen  in  Cambium,  Rec. 
de  trav.  hot.  Need,  xxiv,  pp.  631-786,  1927. 

Hofmeister,  Pringsheini's  Jahrh.  iii,  p.  272,  1863;  Hdb.  d.  physiol.  Bot.  i, 
p.  129,  1867;  etc.  Hofmeister  adds  the  somewhat  curious  qualification:  "Wohl- 
bemerkt,  nicht  senkrecht  zum  grossten  Durchmesser  der  Zelle,  der  mit  der  Richtung 
des  starksten  Wachstums  nicht  zusammenfallen  braucht,  und  in  sehr  viel  Fallen 
in  der  That  auch  nicht  mit  ihr  zusammenfallt." 


VII]  OF  CELL-PARTITIONS  481 

partition  capable  of  doing  so.  Ten  years  later,  Sachs  formulated 
his  rule  of  "rectangular  section,"  declaring  that  in  all  tissues, 
however  complex,  the  cell-walls  cut  one  another  (at  the  time  of 
their  formation)  at  right  angles*.  Years  before,  Schwendenlfer  had 
found  in  the  final  results  of  cell-division  a  universal  system  of 
"orthogonal  trajectoriesf;  and  this  idea  Sachs  further  developed, 
introducing  complicated  systems  of  confocal  elhpses  and  hyperbolae, 
and  distinguishing  between  pericHnal  walls  whose  curves  approxi- 
mate to  the  peripheral  contours,  radial  partitions  which  cut  these 
at  an  angle  of  90°,  and  finally  anticlines,  which  stand  at  right  angles 
to  the  other  two. 

Reinke  (in  1880)  was  the  first  to  throw  doubt  upon  this  explana- 
tion. He  pointed  out  cases  where  the  angle  was  not  a  right  angle, 
but  very  definitely  an  acute  one;  and  he  saw  in  the  commoner 
rectangular  symmetry  merely  what  he  called  a  necessary,  but 
secondary,  result  of  growth  {. 

Within  the  next  few  years  a  number  of  botanical  writers  were 
content  to  point  out  further  exceptions  to  Sachs's  rule§,  and  in 
some  cases  to  show  that  the  curvatures  of  the  partition-walls, 
especially  such  cases  of  lenticular  curvature  as  we  have  described, 
were  by  no  means  accounted  for  by  either  Hofmeister  or  Sachs; 
while  within  the  same  period,  Sachs  himself,  and  also  Rauber, 
attempted  to  extend  the  main  generalisation  to  animal  tissues^. 
The  simple  fact  is  that  Sachs's  rule  is  limited  to  those  many 
caaes    where    one   cell-wall   grows    stiff    or    solid    before    another 

*  Sachs,  Ueber  die  Anordnung  d.  Zellen  in  jiingsten  Pflanzentheilen,  Verh.  pkys.- 
med.  Gesellsch.  Wurzhurg,  xi,  pp.  219-242,  1877;  Ueber  Zellenanordnung  u. 
Waehstum,  ibid,  xii,  1878;  cf.  Arb.  bot.  Inst.  Wurzburg,  ii,  1882;  Ueber  die  durch 
Wachstum  bedingte  Verschiebung  kleinster  Theilchen  in  trajeetorisehen  Curven, 
Monatsb.  k.  Akad.  Wiss.  Berlin,  1880;  Physiology  of  Plants,  chap,  xxvii,  Oxford, 
1887. 

t  Schwendener,  Bau  u.  Wachstum  des  Flechtenthallus,  Naturf.  Gesellsch.  Zurich, 
1,860,"  pp.  272-296. 

J  Reinke,  Lehrbuch  d.  Botanik,  1880,  p.  519;  Kienitz-Gerloff,  Botan.  7Ag.  1878, 
p.  58,  had  already  shewn  some  exceptions  "to  Sachs's  rules,  and  ascribed  them, 
vaguely,  to  "heredity."  It  was  a  time  when  heredity  overruled  everything,  and 
when  Sachs  himself  spoke  of  the  difficulty  of  demonstrating  the  causes  of  any 
morphological  phenomenon  in  any  other  way  than  "genetically":  Textbook.  1882, 
p.  201. 

§    E.g.,  Leitgeb,  V niersuchungen  iiber  die  Lebermoose,  ii,  p.  4,  Graz,  1881. 

•i  Rauber,  Neue  Grundlegungen  zur  Kenntniss  der  Zelle,  Morvhol.  Jahrb.  viii, 
pp.  279,  334,  1882. 


482  THE  FORMS  OF  TISSUES  [ch. 

impinges  upon  it;  and,  suuject  to  this  limitation,  the  rule  is  strictly 
true. 

While  thes^  writers  regarded  the  form  and  arrangement  of  the 
cell-wdls  as  a  biological  phenomenon,  with  little  if  any  direct 
relation  to  ordinary  physical  laws,  or  wioh  but  a  vague  reference 
to  "mechanical  conditions,"  the  physical  side  of  the  case  was 
soon  urged  by  others,  with  more  or  less  force  and  cogency.  Indeed 
the  general  resemblance  between  a  cellular  tissue  and  a  "froth" 
had  been  pointed  out  long  before.  Robert  Hooke  described  the 
cells  within  the  shaft  of  a  feather  as  forming  "a  kind  of  solid  or 
hardened  froth,  or  a  congeries  of  very  small  bubbles,"  and  Grew 
described  a  parenchyma  as  made  by  "fermentation",  "as  we  see 
Bread  in  Baking",  and  again  as  being  "much  the  same  thing,  as  to 
its  construction  which  the  froth  of  beer  or  eggs  is."  Later  on,  within 
the  days  of  the  cell-theory,  Melsens  made  an  "artificial  tissue"  by 
blowing  into  a  solution  of  white  of  egg*. 

In  1886,  Berthold  pubhshed  his  Protoplasmamechanik,  in  which  he 
definitely  adopted  the  principle  of  "minimal  areas,"  and,  following 
on  the  lines  of  Plateau,  compared  the  forms  of  many  cells  and  the 
arrangement  of  their  partitions  with  those  assumed  under  surface- 
tension  by  a  system  of  "weightless  films."  But,  as  Klebs|  pointed 
out,  in  reviewing  the  book,  Berthold  was  so  cautious  as  to  stop  short 
of  attributing  the  biological  phenomena  to  a  mechanical  cause. 
They  remained  for  him,  as  they  had  done  for  Sachs,  so  many 
"phenomena  of  growth,"  or  "properties  of  protoplasm." 

In  the  same  year,  but  while  still  unacquainted,  apparently,  with 
Berthold's  work,  Leo  Errera  pubhshed  a  short  but  very  striking 
article  J  in  which  he  definitely  ascribed  to  the  cell-wall  (as 
Hofmeister  had  already  done)  the  properties  of  a  semi-liquid  film, 
and  drew  from  this  as  a  logical  consequence  the  deduction  that  it 
must  assume  the  various  configurations  which  the  law  of  minimal 
areas  imposes  on  the  soap-bubble.  So  what  we  may  call  Errera's 
Law  is  formulated  as  follows:    A  cell- wall,  at  the  moment  of  its 

*  C.R.  xxxin,  p.  247,  1851;  Ann.  de  chimie  et  de  phys.  (3),  xxxra,  p.  170,  1851 ; 
Bull.  R.  Acad.  Belg.  xxiv,  p.  531,  1857. 

t  Georg  Klebs,  Biol.  Centralbl.  vn,  pp.  193-201,  1887. 

X  L.  Errera,  Sur  une  condition  fondaraentale  d'equilibre  des  cellules  vivantes, 
C.R.  cm,  p.  822,  1886;  Bull.  Soc.  Beige  de  Microscopie,  xm,  Oct.  1886:  Recueil 
d'ceuvres  {Physiologie  gdnirale),  1910,  pp.  201-205. 


VII]  OF  ERR  ERA'S  LAW  483 

formation,  tends  to  assume  the  form  which  would  be  assumed  under 
the  same  conditions  by  a  Uquid  fihn  destitute  of  weight*. 

Soon  afterwards  Chabryf,  discussing  the  segmentation  of  the 
Ascidian  egg,  indicated  many  ways  in  which  cells  and  cell-partitions 
repeat  the  surface-tension  phenomena  of  the  soap-bubble.  He 
came  to  the  conclusion  that  some,  at  least,  of  the  embryological 
phenomena  were  purely  physical,  and  the  same  line  of  investigation 
and  thought  was  pursued  and  developed  by  RobertJ  in  connection 
w^ith  the  embryology  of  the  Mollusca.  Driesch  also,  in  a  series  of 
papers,  continued  to  draw  attention  to  capillary  phenomena  in  the 
segmenting  cells  of  various  embryos,  and  came  to  the  conclusion, 
startling  to  the  embryologists  of  the  time,  that  the  mode  of 
segmentation  was  of  little  importance  as  regards  the  final  result  §. 

Lastly  de  Wildeman^f,  in  a  somewhat  wider  but  also  vaguer 
generahsation  than  Errera's,  declared  that  "The  form  of  the  cellular 
framework  of  plants  and  also  of  animals  depends,  in  its  essential 
features,  upon  the  forces  of  molecular  physics." 

Let  us  return  to  our  problem  of  the  arrangement  of  partition 
films.  When  we  have  three  bubbles  in  contact,  instead  of  two  as 
in  the  case  already  considered,  the  phenomenon  is  strictly  analogous 
to  the  former  case.  The  three  bubbles  are  separated  by  three 
partition  surfaces,  whose  curvature  will  depend  upon  the  relative 
size  of  the  spheres,  and  which  will  be  plane  if  the  latter  are  all  of 
equal  size;  but  whether  plane  or  curved,  the  three  partitions  will 
meet  one  another  at  angles  of  120°,  in  an  axial  fine.  Various 
pretty  geometrical  corollaries  accompany  this  arrangement.  For 
instance,  if  Fig.  170  represent  the  three  associated  bubbles  in  a 

*  There  was  no  lack  of  hearty  antagonism  to  Berthold  and  Errera's  views. 
Cf.  (e.g.)  Zimmermann,  Beitr.  z.  Morphologie  und  Physiologie  der  Pflanzenzelle, 
Tubingen,  1891;  Jost,  Vorlesungen  iiber  Pflanzenphysiologie,  1904,  p.  329,  etc.; 
Giesenhagen,  Studien  iiber  Zelltheilungen  im  Pjianzenreiohe,  1905.  Cf.  also  K. 
Habermehl,  Die  mechanische  Ursache  fiir  die  xegelmdssige  Anordnung  der  Teilungs- 
wdnde  in  Pflanzenzellen  (Inaug.  Diss;),  Kaiserslautern,  1909. 

t  L.  Chabry,  Embryologie  des  Ascidiens,  J.  Anat.  et  Physiol,  xxm:,  p.  266,  1887. 

X  H.  Robert,  Embryologie  des  Troques,  Arch,  de  Zool.  exper.  et  gen.  (3),  x,  1892. 

§  "Dass  der  Furchungsmodus  etwas  fiir  das  Zukiinftige  unwesentliches  ist," 
Z.  /.  w.  Z.  LV,  1893,  p.  37.  With  this  statement  compare,  or  contrast,  that  of 
Conklin,  quoted  on  p.  5;   cf.  also  p,  287  (footnote). 

Tl  E.  de  Wildeman,  Etudes  sur  I'attache  des  cloisons  cellulaires,  Mem.  Couronn. 
de  VAcad.  R.  de  Belgique,  mi,  84  pp.,  1893-94. 


484 


THE  FORMS  OF  TISSUES 


[CH. 


Fig.  170. 


Fig.  171. 


VII]  OF  CLUSTERED  BUBBLES  485 

plane  drawn  through  their  centres,  c,  c',  c"  (or  what  is  the  same 
thing,  if  it  represent  the  base  of  three  bubbles  resting  on  a  plane), 
then  the  lines  uc,  uc",  or  sc,  sc',  etc.,  drawn  to  the  centres  from  the 
points  of  intersection  of  the  circular  arcs,  will  always  enclose  an 
angle  of  60°.  Again  (Fig.  171),  if  we  make  the  angle  c"uf  equal 
to  60°,  and  produce  uf  to  meet  cc"  in/,  /  will  be  the  centre  of  the 
circular  arc  which  constitutes  the  partition  Ou;  and  further,  the 
three  points  /,  g,  h,  successively  determined  in  this  manner,  will  lie 
on  one  and  the  same  straight  line.  In  the  case  of  three  co-equal 
bubbles  (as  in  Fig.  170,  B),  it  is  obvious  that  the  hnes  joining  their 
centres  form  an  equilateral  triangle:  and  consequently,  that  the 
centre  of  each  circle  (or  sphere)  lies  on  the  circumference  of  the 
other  two;  it  is  also  obvious  that  uf  is  now  parallel  to  cc'\  and 
accordingly  that  the  centre  of  curvature  of  the  partition  is  now 
infinitely  distant,  or  (as  we  have  already  said)  that  the  partition 
itself  is  plane. 

The  mathematician  will  find  a  more  elegant  way  of  dealing  with  our 
spherical  bubbles  and  their  associated  interfaces  by  the  method  of  spherical 
inversion,  (i)  Take  three  planes  through  a  line,  cutting  one  another  at  60°, 
and  invert  from  any  point,  arid  you  have  the  case  of  two  spherical  bubbles 
fused,  with  their  interface  also  spherical,  (ii)  Take  the  six  planes  projecting 
the  edges  of  a  regular  tetrahedron  from  its  centre,  and  you  get  by  inversion 
the  case  of  the  three  unequal  bubbles  and  their  three  interfaces,  (iii)  Take 
these  same  planes  with  a  bubble  added  centrally  (thus  adding  a  spherical 
tetrahedron),  and  inversion  gives  the  general  case  of  four  fused  bubbles  and 
their  six  spherical  partitions. 

When  we  have  four  bubbles  meeting  in  a  plane  (Fig.  172),  they 
would  seem  capable  of  arrangement  in  two  symmetrical  ways: 
either  (a)  with  four  partition-walls  intersecting  at  right  angles, 
or  (6)  with  five  partitions  meeting,  three  and  three,  at  angles  of  120°. 
The  latter  arrangement  is  strictly  analogous  to  the  arrangement  of 
three  bubbles  in  Fig.  170.  Now,  though  both  of  these  figures  might 
seem,  from  their  apparent  symmetry,  to  be  figures  of  equilibrium, 
yet  in  point  of  fact  the  latter  turns  out  to  be  of  stable  and  the 
former  of  unstable  equilibrium.  If  we  try  to  bring  four  bubbles 
into  the  form  (a),  that  arrangement  endures  only  for  an  instant; 
the  partitions  glide  upon  one  another,  an  intermediate  wall  springs 
into  existence,  and  the  system  assumes  the  form  (6),  with  its  two 


486  THE  FORMS  OF  TISSUES  [ch. 

triple,  instead  of  one  quadruple,  conjunction.  In  like  manner,  when 
four  billiard-balls  are  packed  close  upon  a  table,  two  tend  to  come 
together  and  separate  the  other  two. 

Let  us  epitomise  the  Law  of  Minimal  Areas  and  its  chief  clauses 
or  corollaries  in  the  particular  case  of  an  assemblage  of  fluid  films, 
as  was  first  done  by  Lamarle*.  Firstly  and  in  general:  In  every 
liquid  system  of  thin  films  in  stable  equihbrium,  the  sum  of  the 
areas  of  the  films  is  a  minimum.  From,  observation  and  experience, 
rather  than  by  demonstration,  it  follows  that  (2)  the  area  of  each 
is  a  minimum  under  its  own  limiting  conditions;  and  further  that 
(3)  the  mean  curvature  of  any  film  is  constant  throughout  its  whole 
area,  null  when  the  pressures  are  equal  on  either  side  and  in  other 


Fig.  172.     A,  an  unstable  arrangement  of  four  cells  or  bubbles.     B,  the  normal  and 
stable  configuration,  showing  the  polar  furrow. 

cases  proportional  to  their  difference.  Less  obvious,  very  important, 
and  hkewise  subject  (but  none  too  easily)  to  rigorous  mathematical 
proof,  are  the  next  two  propositions,  both  of  which  had  been  laid 
down  empirically  by  Plateau:  (4)  the  films  meeting  in  any  one  edge 
are  three  in  number;  (5)  the  crests  or  edges  meetirg  in  any  one 
corner  are  four  in  number,  neither  more  nor  less.  Lastly,  and 
following  easily  from  these:  (6)  the  three  films  meeting  in  a  crest 
or  edge  do  so.  at  co-equal  angles,  and  the  same  is  true  of  the  four 
edges  meeting  in  a  corner. 

Wherever  we  have  a  true  cellular  complex,  an  arrangement  of 
cells  in  actual  physical  contact  by  means  of  their  intervening 
boundary  walls,  we  find  these  general  principles  in  force;  we  must 
only  bear  in  mind  that,  for  their  easy  and  perfect  recognition,  we 

*  Ernest  Lamarle,  Sur  la  stabilite  des  systemes  liquides  en  lames  minces,  Mem. 
de  VAcad.  R.  de  Belgique,  xxxv,  xxxvr,  1864-67. 


VII]  OF  LAMARLE'S  LAW  487 

must  De  able  to  view  the  object  in  a  plane  at  right  angles  to  the 
boundary  walls.  For  instance,  in  any  ordinary  plane  section  of  a 
vegetable  parenchyma,  we  recognise  the  appearance  of  a  "froth," 
precisely  resembling  that  which  we  can  construct  by  imprisoning 
a  mass  of  soap-bubbles  in  a  narrow  vessel  with  flat  sides  of  glass; 
in  both  cases  we  see  the  cell-walls  everywhere  meeting,  by  threes, 
at  angles  of  120°,  irrespective  of  the  size  of  the  individual  cells: 
whose  relative  size,  on  the  other  hand,  determines  the  curvature  of 
the  partition-walls.  On  the  surface  of  a  honey-comb  we  have 
precisely  the  same  conjunction,  between  cell  and  cell,  of  three 
boundary  walls,  meeting  at  120°.  In  embryology,  when  we  examine 
a  segmenting  egg,  of  four  (or  more)  segments,  we  find  in  like  manner, 
in  the  majority  of  cases  if  not  in  all,  that  the  same  principle  is  still 
exemplified.  The  four  segments  do  not  meet  in  a  common  centre, 
but  each  cell  is  in  contact  with  two  others ;  and  the  three,  and  only 
three,  common  boundary  walls  meet  at  the  normal  angle  of  120°. 
A  so-called  polar  furrow"^,  the  visible  edge  of  a  vertical  partition- 
wall,  joins  (or  separates)  the  two  triple  contacts,  precisely  as  in 
Fig.  172,  B,  and  so  gives  rise  to  a  diamond-shaped  figure,  which 
was  recognised  more  than  a  hundred  years  ago  (in  a  newt  or 
salamander)  by  Rusconi,  and  called  by  him  a  tetracitula. 

That  four  cells,  contiguous  in  a  plane,  tend  to  meet  in  a  lozenge 
with  three-way  junctions  and  a  "polar  furrow"  between  the  cells,  is 
a  geometrical  theorem  of  wide  bearing.  The  first  four  cells  in 
a  wasp's  nest  shew  it  neither  better  nor  worse  than  do  those  of 
a  segmenting  ovum,  or  the  ambulacral  plates  of  a  sea-urchin 
or  the  oosphere  of  Oedogonium  giving  birth  to  its  four  zoo- 
spores!. Going  farther  afield  for  an  illustration,  we  find  it  in  the 
molecules  of  a  viscous  liquid  under  shear:    where  a  group  of  four 

*  It  was  so  termed  by  Conklin  in  1897,  in  his  paper  on  Crepidula  (Journ.  Morph. 
XIII,  1897).  It  is  the  Querfurche  of  Rabl  (Morph.  Jahrh.  v,  1879);  the  Polarfurche 
of  ().  Hertwig  (Jen.  Zeitschr.  xiv,  1880);  the  Brechuugslinie  of  Rauber  (Neue 
Grundlage  zur  Kenntniss  der  Zelle,  Morph.  Jahrb.  viir,  1882);  and  the  cross-line  of 
T.  H.  Morgan  (1897).     It  is  carefully  discussed  by  Robert,  op.  rit.  p.  307  seq. 

t  Speaking  of  the  complicated  polygonal  patterns  in  the  test  of  the  protozoon 
genus  Peridinium,  Barrows  says:  "In  the  experience  of  the  writer  no  case  has 
been  found  in  which  four  sutures  actually  meet  at  one  point.  Cases  which  at  first 
sight  appeared  as  such,  upon  loser  analysis  in  a  favourable  position  have  been 
resolved  into  two  junction-points  of  three  sutures  each,  etc."  On  skeletal  variation 
in  the  genus  Peridinium,  Univ.  Calif.  Puhl.  1918,  p.  463. 


488  THE  FORMS  OF  TISSUES  [ch. 

molecules  is  supposed  to  slip  from  one  lozenge-configuration  to  an 
opposite  one,  passing  on  the  way  through  the  simple  cross  or  square 
— a  configuration  of  "higher  energy"  and  less  stable  equilibrium*. 

The  sohd  geometry  of  this  four-celled  figure  is  not  without 
interest.  If  the  two  polar  furrows  (the  one  above  and  the  other 
below)  run  criss-cross,  the  whole  is  a  more  or  less  flattened  and 
distorted  spherical  tetrahedron.  If  they  run  parallel,  then  it  is  a 
four-sided  lozenge  with  two  curved  quadrilateral  faces,  and  two 
bilateral  faces  each  bounded  by  two  curved  edges,  Hke  the  "liths" 


Fig.  173.  Examples  of  the  "polar  furrow".  A,  Pollen-grains  (tetrads)  of  Neottia. 
B,  Egg  of  hookworm  {Ankylostoma).  C,  First  cells  of  a  wasp's  nest  (Polistes). 
(From  Packard,  after  Saussure.)  D,  Four-celled  stage  of  Volvox:  from  Janet. 
E,  Hair  of  Salvia,  after  Hanstein. 

of  an  orange  I .  In  either  case  the  lozenge-configuration  is  under 
some  restraint  to  keep  its  four  cells  in  a  plane;  for  a  tetrahedral 
pile,  or  pyramid,  of  four  spheres  would  be  the  simplest  arrangement 
of  all. 


The  polar  furrow  and  the  partition  of  which  it  forms  an  edge  are,  like  all 
the  edges  and  partitions  in  our  associated  cells,  perfectly  definite  in  dimensions 
and  position;  and  to  draw  them  to  scale,  in  projection,  is  a  simple  matter. 
Taking  the  simplest  case,  when  the  radii  of  all  four  cells  are  equal  to  one  another, 
let  c,  c\  c"  and  c'"  be  the  centres  of  the  four  cells,  Fig.  174.     The  centres  of 

*  Cf.  J.  D.  Bernal,  Proc.  R.S.  (A),  No.  914,  p.  321,  1937. 

t  The  geometer  seldom  takes  account  of  such  two-sided  surfaces  or  facets; 
but  in  groups  of  cells  or  bubbles  they  are  of  common  occurrence,  and  in  the  theory 
of  polyhedra  they  fit  in  without  difficulty  with  the  rest  (cf.  infra,  p.  737). 


VI  [] 


OF  THE  POLAR  FURROW 


489 


any  two  are  related  precisely  as  though  two  cells  only  were  conjoined;  the 
centres  of  three  contiguous  cells  (as  c,  c'  and  c")  are  related  as  though  three 
only  were  concerned;  and  the  centres  of  two  opposite  cells  are  situated 
symmetrically  to  one  another.  This  is  as  much  as  to  say  that  if  there  be 
two  bubbles  in  contact  the  addition  of  a  third  does  not  disturb  their  symmetry; 
and  if  there  be  three  in  contact,  the  addition  of  a  fourth  leaves  the  first  three 
likewise  in  statu  quo.  Thus  the  triangle  cc'c"  is  equilateral,  as  we  already 
know.  The  partition  so  bisects  the  side  cc",  and  the  angle  cc'c";  and  the 
point  o  is  the  centre  of  gravity  of  the  triangle.     Therefore  oj)  —  \oc"  and 


Fig.  174.     The  geometric  symmetry  of  a  system  of  four  cells. 

Again,  in  the  triangle  cpc",  where  cc"  =  r,  pc"  —  ^r,  and  oo'  (the  polar 

f urrow)  = -7:^  r.    Once  again,  in  the  triangle   soc",   sc  =  r;    and   so   (one   of 
V3 

2 
the  partitions)  = -^  r  =  twice  00'.    The  length  of  the  polar  furrow,  then,  as 

V.3 
seen  in  vertical  projection  in  a  system  of  four  co-equal  cells,  is  (theoretically) 
just  one- half  that  of  the  four  intercellular  partitions,  and  very  nearly  three- 
fifths  that  of  a  c(4l-radius. 

It  is  worth  while  to  remark  that  the  universal  phenomenon  of  a 
polar  furrow  gives  an  appearance  of  bilateral  symmetry  to  every 
egg  or  embryo  in  its  four-celled  stage,  no  matter  to  what  kind  or 
class  or  organism  it  belongs. 


490  THE  FOKxAlS  OF  TISSUES  [ch. 

In  the  four-celled  stage  of  the  frog's  egg,  Rauber  (an  exception- 
ally careful  observer)  shews  us  three  alternative  modes  in  which 
the  four  cells  may  be  found  to  be  conjoined  (Fig.  175).  In  A  we 
have  the  commonest  arrangement,  which  is  that  which  we  have 
just  studied  and  found  to  be  the  simplest  theoretical  one;  that 
namely  where  a  straight  polar  furrow  intervenes,  and  where  the 
partition- walls  are  conjoined  at  its  extremities,  three  by  three. 
In  B,  we  have  again  a  polar  furrow,  which  is  now  seen  to  be  a 
portion  of  the  first  "segmentation-furrow"  by  which  the  egg  was 
originally  divided  into  two;  the  four-celled  stage  being  reached  by 
the  appearance  of  the  two  transverse  furrows.  In  this  case,  the 
polar  furrow  is  seen  to  be  sinuously  curved,  and  Rauber  tells  us  that 
its  curvature  gradually  alters;  as  a  matter  of  fact,  it,  or  rather  the 


-f- 


Fig.  175.     Various  conjunctions  of  the  first  four  cells  in  a 
frog's  egg.     After  Rauber. 

partition-wall  corresponding  to  it,  is  gradually  setting  itself  into  a 
position  of  equilibrium,  that  is  to  say  of  equiangular  contact  with 
its  neighbours,  which  position  is  already  attained  or  nearly  so  in 
A.  In  C  we  have  a  very  different  condition,  with  which  we  shall 
deal  in  a  moment. 

The  polar  furrow  may  be  longer  or  shorter,  and  it  may  be  so 
minute  as  to  be  not  easily  discernible;  but  it  is  quite  certain  that 
no  simple  and  homogeneous  system  of  fluid  films  such  as  we 'are 
deahng  with  is  in  equilibrium  without  its  presence.  In  the  accounts 
given,  however,  by  embryologists  of  the  segmentation  of  the  egg, 
while  the  polar  furrow  is  depicted  in  the  great  majority  of  cases, 
there  are  others  in  which  it  has  not  been  seen  and  some  in  which 
its  absence  is  definitely  asserted*.     The  cases  where  four  cells  lying 

*  Thus  Wilson  declared  (Journ.  Morph.  viii,  1895)  that  in  Amphioxus  the  polar 
furrow  was  occasionally  absent,  and  Driesch  took  occasion  *to  criticise  and  to 
throw  doubt  upon  the  statement  {Arch.  f.  Entw.  Mech.  i,  p.  418,  1895). 


vii]  OF  THE  POLAR  FURROW  491 

in  one  plane  meet  in  a  point,  such  as  were  frequently  figured  by  the 
older  embryologists,  are  hard  to  verify  and  sometimes  not  easy  to 
believe.  Considering  the  physical  stability  of  the  other  arrange- 
ment, the  great  preponderance  of  cases  in  which  it  is  known  to 
occur,  the  difficulty  of  recognising  the  polar  furrow  in  cases  where 
it  is  very  small  and  unless  it  be  specially  looked  for,  and  the  natural 
tendency  of  the  draughtsman  to  make  an  all  but  symmetrical 
structure  appear  wholly  so,  I  was  wont  to  attribute  to  error  or 
imperfect  observation  all  those  cases  where  the  junction-hnes  of  four 
cells  are  represented  (after  the  manner  of  Fig.  172,  A)  as  a  simple 
cross*.  As  a  matter  of  fact,  the  simple  cross  is  no  very  rare  pheno- 
menon, even  in  the  frog's  egg;  but  it  is  a  transitory  one,  and 
unstable.  Viscosity  and  friction  may  enable  it  to  endure  for  a 
while,  but  the  partitions  inevitably  shift  into  the  stable,  three-way, 
configuration.  In  such  a  case,  the  polar  furrow  manifests  itself 
slowly  and  as  it  were  laboriously;  but  in  the  more  fluid  soap-bubble 
it  does  so  in  the  twinkling  of  an  eye. 

While  a  true  four-rayed  intersection,  or  simple  cross,  is  theoretic- 
ally impossible  save  as  a  transitory  and  unstable  condition,  there 
is  another  configuration  which  may  closely  simulate  it,  and  which 
is  common  enough.  There  are  plenty  of  faithful  representations  of 
segmenting  eggs  in  which,  instead  of  the  triple  junctions  and  polar 
furrow,  the  four  cells  (and  also  their  more  numerous  successors)  are 
represented  as  rounded  off,  and  separated  from  one  another  by  an 
empty  space,  or  by  a  little  drop  of  extraneous  fluid,  evidertly  not 
directly  miscible  with  the  fluid  surface  of  the  cells.  Such  is  the 
case  in  the  obviously  accurate  figure  which  Rauber  gives  (Fig. 
175,  C)  of  his  third  mode  of  conjunction  in  the  four-celled  stage  of 
the  frog's  egg.  Here  Rauber  is  most  careful  to  point  out  that  the 
furrows  do  not  simply  "cross,"  or  meet  in  a  point,  but  are  separated 
by  a  little  space,  which  he  calls  the  Polgrilbchen,  and  asserts  to  be 
constantly  present  whensoever  the  polar  furrow,  or  Brechungslinie, 
is  not  to  be  discerned.  This  little  interposed  space  with  its  con- 
tained drop  of  fluid  materially  alters  the  case,  and  implies  a  new 

*  The  same  remark  was  made  long  atgo  by  Driesch:  "Das  so  oft  schematisch 
gezeichnete  Vierzellenstadium  mit  zwei  sieh  in  zwei  Punkten  scheidende  Medianen 
kann  man  wohl  getrost  aus  der  Rcihe  des  Existierenden  streichen"  (Entw,  mech. 
Studien.  Z.  f.  w.  Z.  liii,  p.  166,  1892).  Cf.  also  his  Math,  mechanische  Bedeutung 
morphologischer  Probleme  der  Biologic,  Jena,  59  pp.,  1891. 


492  THE  FORMS  OF  TISSUES  [ch. 

condition  of  theoretical  and  actual  equilibrium.  For  on  the  one 
hand,  we  see  that  now  the  four  intercellular  partitions  do  not  meet 
one  another  at  all',  but  really  impinge  upon  four  new  and  separate 
partitions,  which  constitute  interfacial  contacts  not  between  cell 
and  cell,  but  between  the  respective  cells  and  the  intercalated 
drop.  And  secondly,  the  angles  at  which  these  four  little  surfaces 
meet  the  four  cell -partitions  will  be  determined,  in  the  usual  way, 
by  the  balance  between  the  respective  tensions  of  these  several 
surfaces.  In  an  extreme  case  (as  in  some  pollen-grains)  it  may  be 
found  that  the  cells  under  the  observed  circumstances  are  not  truly 
in  surface  contact:  that  they  are  so  many  drops  which  touch  but 
do  not  "wet"  one  another,  and  which  are  merely  held  together 


by  the  pressure  of  the  surrounding  envelope.  But  even  supposing 
that  they  are  in  actual  fluid  contact,  the  case  from  the  point  of 
view  of  surface-tension  presents  no  difficulty.  In  the  case  of  the 
conjoined  soap-bubbles,  we  were  dealing  with  similar  contacts  and 
with  equal  surface-tensions  throughout  the  system;  but  in  the 
system  of  protoplasmic  cells  which  constitute  the  segmenting  egg 
we  must  make  allowance  for  inequality  of  tensions,  between  the 
surfaces  where  cell  meets  cell  and  where  on  the  other  hand  cell- 
surface  is  in, contact  with  the  surrounding  medium — generally  water 
or  one  of  the  fluids  of  the  body.  Remember  that  our  general  con- 
dition is  that,  in  our  entire  system,  the  sum  of  the  surface  energies 
is  a  minimum ;  and,  while  this  is  attained  by  the  sum  of  the  surfaces 
being  a  minimum  in  the  case '  where  the  energy  is  uniformly 
distributed,  it  is  not  necessarily  so  under  non-uniform  conditions. 
In  the  diagram  (Fig.  176),  if  the  energy  per  unit  area  be  greater 
along  the  contact  surface  cc',  where  cell  meets  cell,  than  along  ca 


VII]  OF  EPIDERMAL  TISSUES  493 

or  c6,  where  cell-surface  is  in  contact  with  the  surrounding  medium, 
these  latter  surfaces  will  tend  to  increase  and  the  surface  of  cell- 
contact  to  diminish.  In  short  there  will  be  the  usual  balance  of 
forces  between  the  tension  along  the  surface  <x\  and  the  two 
opposing  tensions  along  ca  and  ch.  If  the  former  be  greater  than 
either  of  the  other  two,  the  outside  angle  will  be  less  than  120°; 
and  if  the  tension  along  the  surface  cc  be  as  much  or  more  than 
the  sum  of  the  other  two,  then  the  drops  will  merely  touch  one 
another,  save  for  the. possible  effect  of  external  pressure.  This  is 
the  explanation,  in  general  terms,  of  the  peculiar  conditions  ob- 
taining in  Nostoc  and  its  allies  (p.  477),  and  it  also  leads  us  to  a 
consideration  of  the  general  properties  and  characters  of  a  super- 
ficial or  "epidermal"  layer*. 

While  the  inner  cells  of  the  honeycomb  are  symmetrically 
situated,  sharing  with  their  neighbours  in  equally  distributed 
pressures  or  tensions,  and  therefore  all  tending  closely  to  identity 
of  form,  the  case  is  obviously  different  with  the  cells  at  the  borders 
of  the  system.  So  it  is  with  our  froth  of  soap-bubbles  f.  The 
bubbles,  or  cells,  in  the  interior  of  the  mass  are  all  ahke  in  general 
character,  and  if  they  be  equal  in  size  are  alike  in  every  respect: 
as  we  see  them  in  projection  their  sides  are  uniformly  flattened, 
and  tend  to  meet  at  equal  angles  of  120°.  But  the  bubbles  which 
constitute  the  outer  layer  retain  their  spherical  surfaces  (just  as 
in  the  cells  of  a  honeycomb),  and  these  still  tend  to  meet  the 
partition-walls  connected  with  them  at  constant  angles  of  120°. 
This  outer  layer  of  bubbles,  which  forms  the  surface  of  our  froth, 
constitutes  after  a  fashion  what  we  should  call  in  botany  an 
"epidermal"  layer.     But  in  our  froth  of  soap-bubbles  we  have,  as 

*  A  surface-layer  always  tends  to  have,  ipso  facto,  a  character  of  its  own :  a  "skin  " 
has  such  and  such  characteristics  just  because  it  is  a  skin.  The  "Beilby  layer" 
on  a  metallic  surface  is,  in  its  own  special  way;-  a  consequence  of  its  own  externality. 

f  A  froth  is  a  collocation  of  bubbles  containing  air;  or  in  the  language  of  colloid 
chemistry,  an  emulsion  with  air  for  its  disperse  phase.  The  power  of  forming 
a  froth  is  not  the  same  as  that  of  forming  isolated  bubbles;  for  some  liquids,  such 
as  a  solution  of  saponin,  of  gum  arable,  of  albumin  itself,  give  a  copious  and  lasting 
froth,  but  we  find  it  hard  to  blow  even  a  single  tiny  bubble  with  any  of  them. 
Something  more  than  surface-tension  seems  necessary  for  the  production  and  main- 
tenance of  a  film :  perhaps  a  certain  amount  of  viscosity,  to  resist  the  tendency  of 
surface-tension  to  tear  the  film  asunder. 


494 


THE  FORMS  OF  TISSUES 


[CH. 


a  rule,  the  same  kind  of  contact  (that  is  to  say,  contact  with  air) 
both  within  and  without  the  bubbles;  while  in  our  hving  cell,  the 
outer  wall  of  the  epidermal  cell  is  exposed  to  air  on  the  one  side, 
but  is  in  contact  with  the  protoplasm  of  the  cell  on  the  other :  and 
this  involves  a  difference  of  tensions,  so  that  the  outer  walls  and 
their  adjacent  partitions  need  no  longer  meet  at  precisely  equal 
angles  of  120°.  Moreover  a  chemical  change,  due  perhaps  to 
oxidation  or  possibly  also  to  adsorption,  is  very  apt  to  affect  the 
external  wall  and  lead  to  the  formation  of  a  "cuticle";  and  this 
process,  as  we  have  seen,  is  tantamount  to  a  large  increase  of  tension 
In  t^at  outer  wall,  and  will  cause  the  adjacent  partitions  to  impinge 
upon  it  at  angles  more  and  more  nearly  approximating  to  90°:  the 
bubble-hke,  or  spherical,  surfaces  of  the  individual  cells  being  more 


Fig.  177.     A  froth,  with  its  outer  and  inner  cells  or  vesicles, 

and  more  flattened  in  consequence.  Lastly,  the  chemical  changes 
which  affect  the  outer  walls  of  the  superficial  cells  may  extend  in 
greater  or  less  degree  to  their  inner  walls  also :  with  the  result  that 
these  cells  will  tend  to  become  more  or  less  rectangular  throughout, 
and  will  cease  to  dovetail  into  the  interstices  of  the  next  subjacent 
layer.  These  then  are  the  general  characters  which  we  recognise 
in  an  epidermis ;  and  we  now  perceive  that  its  fundamental  character 
simply  is  that  it  lies  outside,  and  that  its  physical  characteristics 
follow,  as  a  matter  of  course,  from  the  position  which  it  occupies 
and  from  the  various  consequences  which  that  situation  entails. 

In  the  young  shoot  or  growing  point  of  a  flowering  plant  botanists 
(following  Hanstein)  find  three  cell-layers,  and  call  them  dermatogen, 
periblern  and  plerome.  The  first  is  an  epidermis,  such  as  we  have 
just  described.  Its  cells  grow  long  as  the  shoot  grows  long;  new 
partitions  cross  the  lengthening  cell  and  tend  to  lie  at  right  angles 
to  its  hardening  walls;  and  this  epidermis,  once  formed,  remains 
a  single  superficial  layer.     The  next  few  layers,  the  so-called  peri- 


VII]  OF  EPIDERMAL  TISSUES  495 

blem,  are  compressed  and  flattened  between  the  epidermis  with  its 
tense  cuticle  and  the  growing  mass  within ;  and  mider  this  restraint 
the  cell-layers  of  the  periblem  also  continue  to  divide  in  their  own 
plane  or  planes.  But  the  cells  of  the  inner  mass  or  plerome,  lying 
in  a  more  homogeneous  field,  tend  to  form  "  space-fiUing "  poly- 
hedra,  twelve-  or  perhaps  fourteen-sided  according  to  the  freedom 
which  they  enjoy.  In  a  well-known  passage  Sachs  declares  that 
the  behaviour  of  the  cells  in  the  growing  point  is  determined  not 
by  any  specific  characters  or  properties  of  their  own,  but  by  their 
position  and  the  forces  to  which  they  are  subject  in  the  system  of 
which  they  are  a  part*.  This  was  a  prescient  utterance,  and  is 
abundantly  confirmed f. 

We  have  hitherto  considered  our  cells,  or  our  bubbles,  as  lying 
in  a  plane  of  symmetry,  and  have  only  considered  their  appearance 
as  projected  on  that  plane;  but  we  must  also  begin  to  consider 
them  as  sohds,  whether  they  lie  in  a  plane  (hke  the  four  cells  in 
Fig.  172),  or  are  heaped  on  one  another,  like  a  froth  of  bubbles  or 
a  pile  of  cannon-balls.  We  have  still  much  to  do  with  the  study 
of  more  complex  partitioning  in  a  plane,  and  we  have  the  whole 
subject  to  enter  on  of  the  solid  geometry  of  bodies  in  "close 
packing,"  or  three-dimensional  juxtaposition. 

The  same  principles  which  account  for  the  development  of 
hexagonal  symmetry  hold  true,  as  a  matter  of  course,  not  only  of 
cells  (in  the  biological  sense),  but  of  any  bodies  of  uniform  size  and 
originally  circular  outline,  close-packed  in  a  plane;  and  hence  the 
hexagonal  pattern  is  of  very  common  occurrence,  under  widely 
varying  circumstances.  The  curious  reader  may  consult  Sir  Thomas 
Browne's  quaint  and  beautiful  account,  in  the  Garden  of  Cyrus,  of 
hexagonal,  and  also  of  quincuncial,  symmetry  in  plants  and  animals, 
which  "doth  neatly  declare  how  nature  Geometrizeth,  and  observeth 
order  in  all  things," 

We  come  back  to  very  elementary  geometry.  The  first  and 
simplest  of  all  figures  in  plane  geometry  (wdth  which  for  that  reason 
Euclid  begins  his  book)  is  the  equilateral  triangle;  because  three 
straight  lines  are  the  least  number  which  enclose  two-dimensional 

*  Lectures  on  the  Physiology  of  the  Plant,  Oxford,  1887,  p.  460,  etc. 
t  Cf.  J.  H.  Priestley  in  Biol.  Reviews,  m,  pp.  1-20,  1928;   U.  Tetley  in  Ann.  Bot. 
L,  pp.  522-557,  1936;   etc. 


496  THE  FORMS  OF  TISSUES  [ch. 

space,  and  three  equal  sides  make  the  simplest  of  triangles.  But 
it  by  no  means  follows  that  equilateral,  or  any  other,  triangles 
combine  to  form  the  simplest  of  polygonal  associations  or  patterns. 
On  the  other  hand,  three  straight  lines  meeting  in  a  point  are  the 
least  number  by  which  we  can  subdivide  or  partition  two-dimensional 
space ;  the  simplest  case  of  all  is  when  the  three  partitions  meet  at 
co-equal  angles,  and  a  pattern  of  hexagons,  so  produced,  is,  geo- 
metrically speaking,  the  siiT^plest  of  all  ways  in  which  a  surface  can 
be  subdivided — the  simplest  of  all  two-dimensional  '"space-fiUing" 
patterns.  So  it  comes  to  pass  that  we  meet  with  a  pattern  of 
hexagons  here  and  there  and  again  and  again,  in  all  sorts  of  plane 
symmetrical  configurations,  from  a  soapy  froth  to  the  retinal 
pigment,  from  the  cells  of  the  honeycomb  to  the  basaltic  columns 
of  Staffa  and  the  Giant's  Causeway. 

We  pass  to  solid  geometry,  and  arrive  by  similar  steps  at  an 
analogous  result.  Four  plane  sides  are  now  the  least  number  which 
enclose  space,  and  (next  to  the  sphere  itself)  the  regular  tetrahedron 
is  the  first  and  simplest  of  solids;  but  its  simplicity  is  that  of  a 
solitary  or  isolated  figure,  and  tetrahedra  do  not  combine  to  fill  space 
at  all.  But  as  the  partitioning  of  an  equilateral  triangle  was  the 
first  step  towards  the  symmetrical  partitioning  of  two-dimensional 
space,  so  we  draw  from  the  regular  tetrahedron  a  first  lesson  in  the 
partitioning  of  space  of  three  dimensions ;  and  as  three  lines  meeting 
in  a  point  w«ere  needed  to  partition  two-dimensional  space,  so  here, 
for  three-dimensional  space,  we  need  four.  The  simplest  case  is,  as 
before,  when  these  meet  at  co-equal  angles,  but  we  do  not  see  quite 
so  easily  what  those  four  co-equal  angles  are. 

For  as  the  centreof  symmetry  of  our  equilateral  triangle  was  defined 
by  three  lines  bisecting  its  three  angles  and  meeting  one  another  in 
a  point  at  co-equal  angles  of  120°,  so  in  our  regular  tetrahedron  four 
straight  fines,  running  symmetrically  inwards  from  the  four  corners, 
meet  in  a  point  at  co-equal  angles,  and  again  define  the  centre  of 
symmetry.  If  we  make  (as  Plateau  made)  a  wire  tetrahedron,  and 
dip  it  into  soap-solution,  we  find  that  a  film  has  attached  itself  to 
each  of  the  six  wires  which  constitute  the  httle  tetrahedral  cage; 
that  these  six  films  meet,  three  by  three,  in  four  edges;  and  that 
these  four  edges  meet  at-  co-equal  angles  in  a  point,  which  is  the 
centi'oid,  or  centre  of  symmetry,  or  centre  of  gravity,  of  the  system. 


VII 


OF  TETRAHEDRAL  SYMMETRY 


497 


This  is  the  centre  of  symmetry  not  only  for  our  tetrahedron,  but  for 
any  close-packed  tetrahedral  aggregate  of  co-equal  spheres ;  we  meet 
with  it  over  and  over  again,  in  a  pile  of  cannon-balls,  a  froth  of 
soap-suds,  a  parenchyma  of  cells,  or  the  interior  of  the  honeycomb. 
Moreover,  in  the  actual  demonstration  by  soap-films  of  this  tetra- 
hedral symmetry,  we  see  reahsed  all  the  main  criteria  laid  down  by 
Plateau  and  by  Lamarle  for  a  system  minimae  areae:  three  films 
and  no  more  meet  in  an  edge;  four  fluid  edges  and  no  more  meet 
in  a  point,  just  as  three  wire  edges  and  one  fluid  edge  met  in  a 
point  at  each  corner  of  the  experimental  figure.  Lastly,  the  sym- 
metry of  the  whole  configuration  is  such  that  the  three  fluid  films 


Vh 


178.    A  regular  tetrahedron,  with  its  centre  of  symmetry. 


meeting  in  an  edge,  or  the  four  fluid  edges  meeting  in  a  point,  all 
do  so  at  co-equal  angles. 

In  the  plane  configuration  we  saw  without  more  ado  that  the 
angles  of  symmetry  were  the  co-equal  angles  of  120°;  but  the  four 
co-equal  angles  between  the  four  edges  which  meet  at  the  centre  of 
our  tetrahedron  require  a  little  more  consideration.  If  in  our  figure 
of  a  regular  tetrahedron  (Fig.  178)  o  be  the  centroid,  and  we  produce 
ao  to  p,  the  centre  of  the  opposite  side^  bed,  it  may  be  shewn  that 
the  fine  ap  is  so  divided  that  ao  =  Sop  and  ao  =  bo  =  co  =  do. 
For  let  four  equal  weights  be  put  at  the  four  corners  of  the  tetra- 
hedron, a,  6,  c,  d.  The  resultant  of  the  three  at  b,  c,  d  is  equivalent 
to  3 If  at  p,  the  centre  of  symmetry  of  the  equilateral  triangle. 


498  THE  FORMS  OF  TISSUES  [ch. 

The  resultant  of  all  four  is  equal  to  the  resultant  of  W  at  a,  and 
3Pf  at  p;  it  lies,  therefore,  on  the  straight  line  ap,  and  at  the 
point  0,  such  that  ao  =  '6of.  Therefore,  in  the  triangle  pod,  as 
in  the  other  three  similar  triangles  in  the  figure,  cos  pod  =  1/3, 
and  cos  aod  =  — 1/3.  Our  tables  tell  us  that  the  angle  pod,  whose 
trigonometrical  value  is  the  very  simple  one  of  cos  pod  ^  1/3,  has; 
in  degrees  and  minutes  to  the  nearest  second,  the  seemingly  less 
simple  value  of  70°  31' 43";  and  its  supplement,  the  angle  aod, 
has  the  corresponding  value  of  109°  28'  16". 

This  latter  angle,  then,  of  109°  28'  16",  or  very  nearly  109  degrees 
and  a  half,  is  the  angle  at  which,  in  this  and  throughout  every  other 
three-dimensional  system  of  liquid  films,  the  edges  of  the  partition- 
walls  meet  one  another.  It  is  the  fundamental  angle  in  simple 
homogeneous  partitioning  of  three-dimensional  space.  It  is  an 
angle  of  statical  equilibrium,  an  angle  of  close-packing,  an  angle 
of  repose.  In  the  simplest  of  carbon-compounds,  the  molecule  of 
marsh-gas  (CH4),  we  may  be  sure  that  this  angle  governs  the 
arrangement  of  the  H-atoms;  it  determines  the  relation  of  the 
carbon-atoms  one  to  another  in  a  diamond — simplest  of  crystal- 
lattices;  it  defines  the  intersections  of  the  bubbles  in  a  froth,  and 
of  the  cells  in  the  honeycomb  of  the  bee. 

It  is  sometimes  called  the  "tetrahedral  angle";  it  might  be  better 
called  (for  a  reason  we  shall  see  presently)  "Maraldi's  angle."  The 
whole  story  is  less  a  physical  than  a  mathematical  one;  for  the 
phenomena  do  not  depend  on  surface  tension  nor  on  any  other 
physical  force,  but  on  such  relations  between  surface  and  volume 
as  are  involved  in  the  properties  of  space.  If  we  take  four  little 
elastic  balloons,  half  fill  them  with  air,  smear  them  with  glycerine 
to  lessen  friction,  place  them  in  a  bottle  and  exhaust  the  air  therein, 
they  wull  expand,  adjust  themselves  together,  and  group  themselves 
in  a  tetrahedral  configuration,  whose  partition  walls,  edges  and 
centre  of  symmetry  are  just  those  of  our  experiment  of  the  soap-films. 

This  characteristic  angle,  though  it  leads  in  ordinary  angular  measurement 
to  an  endless  decimal  of  a  second,  is  nevertheless  a  very  simple  and  perfectly 
definite  magnitude.  It  is  a  strange  property  of  Number  that  it  fails  to  express 
certain  simple  and  definite  magnitudes,  such  as  tt,  or  y/2,  or  Vs,  or  this  four- 
fold angle  made  by  four  lines  meeting  symmetrically  in  three-dimensional  space. 
It  is  not  these  magnitudes  that  are  peculiar,  it  is  Number  itself  that  is  so !  In 
all  of  these  cases  we  have  to  import  a  new  symbol;   and  in  this  case,  when  we 


viij  OF  HEXAGONAL  SYMMETRY  499 

draw  it  not  from  arithmetic  but  from  trigonometry,  and  define  our  angle  as 
cos  —  ^ ,  nothing  can  or  need  be  more  precise  or  siinpler.  We  may  put  the 
same  thing  a  little  differently,  and  say  that  Number  itself  fails  us,  now  and 
then,  to  express  what  we  want,  although  we  have  all  the  ten  digits  and  their 
apparently  endless  permutations  at  our  command.  In  such  a  deadlock,  we 
have  only  to  bring  one  new  symbol,  one  new  quantity,  into  use;  and  at  once 
a  wide  new  field  is  open  to  us. 

Out  of  these  two  angles — the  Maraldi  angle  of  109°  etc.,  and  the 
plane  angle  of  120° — we  may  construct  a  great  variety  of  figures, 


Fig.  179.     Diagram  of  hexagonal  cells.     After  Bonanni. 

plane  and  solid,  which  become  still  more  complex  and  varied  when 
we  consider  associations  of  unequal  as  well  as  of  co-equal  cells,  and 
thereby  admit  curved  as  well  as  plane  intercellular  partitions.  Let 
us  consider  some  examples  of  these,  beginning  with  such  as  w^e  need 
only  consider  in  reference  to  a  plane. 

Let  us  imagine  a  system  of  equal  cylinders,  or  equal  spheres,  in 
contact  with  one  another  in  a  plane,  and  represented  in  section  by 
the  equal  and  contiguous  circles  of  Fig.  179.  I  borrow  my  figure 
from  an  old  Italian  naturalist,  Bonanni  (a  contemporary  of  Borelli, 
of  Ray  and  Willoughby,  and  of  Martin  Lister),  who  dealt  with  this 
matter  in  a  book  chiefly  devoted  to  molluscan.  shells*. 

*  A.  P.  P.  Bonanni,  Kicreatione  delV  occhlo  e  delta  mente,  nelV  Osservatione  delle 
Chiocciole,  Roma,  1681. 


500  THE  FORMS  OF  TISSUES  [ch. 

It  is  obvious,  as  a  simple  geometrical  fact,  that  each  of  these 
co-equal  circles  is  in  contact  with  six  others  around.  Imagine  the 
whole  system  under  some  uniform  stress — of  pressure  caused  by- 
growth  or  expansion  within  the  cells,  or  due  to  some  uniformly 
applied  constricting  pressure  from  without.  In  these  cases  the  six 
points  of  contact  between  the  circles  in  the  diagram  will  be  extended 
into  lines,  representing  surfaces  of  contact  in  the  actual  spheres  or 
cylinders;  and  the  equal  circles  of  our  diagram  will  be  converted 
into  regular  and  co-equal  hexagons.  The  result  is  just  the  same 
so  far  as  form  is  concerned — so  long  as  we  are  concerned  only  with 
a  morphological  result  and  not  with  a  physiological  process — what- 
ever be  the  force  which  brings  the  bodies  together.  For  instance, 
the  cells  of  a  segmenting  egg,  lying  within  their  vitelhne  membrane 
or  within  some  common  film  or  ectoplasm,  are  pressed  together  as 
they  grow,  and  suffer  deformation  accordingly;  their  surface  tends 
towards  an  area  minima,  but  we  need  not  even  enquire,  in  the  first 
instance,  whether  it  be  surface-tension,  mechanical  pressure,  or  what 
not  other  physical  force,  which  is  the  cause  of  the  phenomenon*. 

The  production  by  mutual  interaction  of  polygons,  which  be- 
come regular  hexagons  when  conditions  are  perfectly  symmetrical, 
is  beautifully  illustrated  by  Benard's  tourbillons  cellulaires,  and  also 
in  some  of  Leduc's  diffusion  experiments.  In  these  latter,  a 
solution  of  gelatine  is  allowed  to  set  on  a  plate  of  glass,  and  little 
drops  of  weak  potassium  ferrocyanide  are  then  let  fall  at  regular 
intervals  upon  the  gelatine.  Immediately  each  little  drop  becomes 
the  centre  of  a  system  of  diffusion  currents,  and  the  several  systems 
conflict  with  and  repel  one  another;  so  that  presently  each  little 
area  becomes  the  seat  of  a  to-and-fro  current  system,  outwards  and 
back  again,  until  the  concentration  of  the  field  becomes  equalised 
and  the  currents  cease.  When  equihbrium  is  attained,  and  when 
the  gelatin-layer  is  allowed  to  dry,  we  have  an  artificial  tissue  of 

*  The  following  is  one  of  many  curious  corollaries  to  the  principle  of  close- 
packing  here  touched  upon.  A  circle  surrounded  by  six  similar  circles,  the  whole 
bounded  by  a  circle  of  three  times  the  radius  of  the  original  one,  forms  a  unit,  so 
to  speak,  next  in  order  after  the  circle  itself.  A  round  pea  or  grain  of  shot  will 
pass  through  a  hole  of  its  own  size;  but  peas  or  shot  will  not  rtin  out  of  a  vessel 
through  a  hole  less  than  three  times  their  own  diameter.  There  can  be  no  freedom 
of  motion  among  the  close -packed  grains  when  confronted  by  a  smaller  orifice. 
Cf.  K.  Takahasi,  Sci.  Papers  Inst.  Chem.,  etc.,  Tokio,  xxvi,  p.  19.  1935. 


VII 1 


OF  HEXAGONAL  SYMMETRY 


501 


Fig.   180.     An  "artificial  tissue,"  formed  by  coloured  drops  of  sodium  chloride 
solution  diffusing  in  a  less  dense  .solution  of  the  same  salt.     After  Leduc. 


Fig.  181.     An  artificial  cellular  tissue,  formed  by  the  diffusion  in  gelatine  of 
drops  of  a  solution  of  potassium  ferrocyanide.     After  Leduc. 


502  THE  FORMS  OF  TISSUES  [ch. 

hexagonal  "cells,"  which  simulate  an  organic  parenchyma  very 
closely;  and  by  varying  the  experiment  in  ways  which  Leduc 
describes,  we  may  imitate  various  forms  of  tissue,  and  produce  cells 
with  thick  walls  or  with  thin,  cells  in  close  contact  or  with  wide 
intercellular  spaces,  cells  with  plane  or  with  curved  partitions,  and 
so  forth. 

James  Thomson  (Kelvin's  elder  brother)  had  observed  nearly 
sixty  years  ago  a  curious  "tesselated  structure"  on  a  Hquid  surface, 
to  wit,  the  soapy  water  of  a  wash-tub.  The  eddies  and  streaks  of 
swirling  water  settled  down  into  a  cellular  configuration,  which 
continued  for  hours  together  to  alter  its  details;  small  areoles 
disappeared,  large  ones  grew  larger,  and  subdivided  into  small  ones 
again.  With  few  and  transitory  exceptions  three  partitions  and  no 
m^  re  met  at  every  node  of  the  mesh  work ;  and  (as  it  seems  to  me) 
the  subsequent  changes  were  all  due  to  such  shifting  of  the  lines  as 
tended  to  make  the  three  adjacent  angles  more  and  more  nearly 
co-equal  with  one  another :  the  obvious  effect  of  this  being  to  make 
the  pattern  more  and  more  regularly  hexagonal*. 

In  a  not  less  homely .  experiment,  hot  water  is  poured  into  a 
shallow  tin  and  a  layer  of  milk  run  in  below ;  on  blowing  gently  to 
cool  the  water,  holes,  more  or  less  close-packed  and  evenly  inter- 
spaced, appear  in  the  milk.  They  shew  how  cooling  has  taken  place, 
so  to  speak,  in  spots,  and  the  cooled  water  has  descended  in  isolated 
columns  f. 

Benard's  "tourbillons  cellulaires"J,  set  up  in  a  thin  hquid  layer, 

*  James  Thomson,  On  a  changing  tesselated  structure  in  certain  liquids,  Proc. 
Glasgow  Phil.  Soc.  1881-82;  Coll.  Papers,  p.  136— a  paper  with  which  M.  Benard 
was  not  acquainted,  but  see  Benard's  later  note  in  Ann.  <Ie  Chim.  Dec.  1911. 

t  See  Graham's  paper,  quoted  below. 

%  H.  Benard,  Les  tourbillons  cellulaires  dans  une  nappe  hquide.  Rev.  gener.  des 
Sciences,  xii,  pp.  1261-1271,  1309-1328,  1900;  Ann.  Chimie  et  Physique  (7),  xxm, 
pp.  62-144,  1901 ;  ibid.  1911.  Quincke  had  seen  much  the  same  long  before:  An7i.  d. 
Phys.  cxxxix,  p.  28,  1870.  The  "figures  of  de  Heen"  are  an  analogous  electrical 
phenomenon;  cf.  P.  de  Heen,  Les  tourbillons  et  les  projections  de  I'ether,  Bull. 
Acad,  de  Bruxelles  (3)  xxxvii,  p.  589,  1899;  A.  Lafay,  Ann.  de  Physique  (10),  xiii, 
pp.  349-394,  19.30.  These  various  phenomena,  all  leading  to  a  pattern  of  hexagons, 
have  often  been  studied  mathematically:  cf.  Rayleigh,  Phil.  Mag.  xxxii,  pp.  529- 
546, 1916,  Coll.  Papers,  vi,  p.  48;  also  Ann  Pellew  and  R.  V.  Southwell,  Proc.  R.S.  (A), 
CLXXVi,  pp.  312-343,  1940.  The  hexagonal  pattern  is  a  particular  case  of  stability, 
but  not  necessarily  the  simplest;  it  is  only  by  experiment  that  we  know  it  to  be 
the  permanent  condition  in  an  unlimited  field. 


VII]  OF  BENARD'S  EXPERIMENT  503 

are  similar  ^to  but  more  elegant  than  James  Thomson's  tesselated 
patterns,  and  both  of  them  are  in  their  own  way  still  more  curious 
than  M.  Leduc's;  for  the  latter  depend  on  centres  of  diffusion 
artificially  inserted  into  the  system  and  determining  the  number 
and  position  of  the  "cells,"  while  in  the  others  the  cells  make 
themselves.  In  Benard's  experiment  a  thin  layer  of  Hquid  is 
warmed  in  a  copper  dish.  The  hquid  is  under  peculiar  conditions  of 
instability,  for  the  least  fortuitous  excess  of  heat  here  or  there  would 
suffice  to  start  a  current,  and  we  should  expect  the  whole  system  to 
be  highly  unstable  and  unsymmetrical.  But  if  all  be  kept  carefully 
uniform,  small  disturbances  appear  at  random  all  over  the  system; 
a  current  ascends  in  the  centre  of  each;  and  a  "steady  state,"  if 
not  a  stable  equilibrium,  is  reached  in  time,  when  the  descending 
currents,  impinging  on  one  another,  mark  out  a  "cellular  system." 
If  we  set  the  fluid  gently  in  motion  to  begin  with,  the  first  "cell- 
divisions"  will  be  in  the  direction  of  the  flow;  long  tubes  appear, 
or  "vessels,"  as  the  botanist  would  be  apt  to  call  them.  As  the 
flow  slows  down  new  cell-boundaries  appear,  at  right  angles  to  the 
first  and  at  even  distances  from  one  another;  parallel  rows  of  cells 
arise,  and  this  transitory  stage  of  partial  equilibrium  or  imperfect 
symmetry  is  such  as  to  remind  the  botanist  of  his  cambium  tissues, 
which  are,  so  to  speak,  a  temporary  phase  of  histological  equilibrium. 
If  the  impressed  motion  be  not  longitudinal  but  rotary,  the  first  fines 
of  demarcation  are  spiral  curves,  followed  by  orthogonal  inter- 
sections. 

Whether  we  start  with  liquid  in  motion  or  at  rest,  symmetry  and 
uniformity  are  ultimately  p,ttained.  The  cells  draw  towards  uni- 
formity, but  four,  five  or  seven-sided  cells  are  still  to  be  found 
among  the  prevailing  hexagons.  The  larger  cells  grow  less,,  the 
smaller  enlarge  or  disappear;  where  four  partition- walls  happen  to 
meet,  they  shift  till  only  three  converge;  the  sides  adjust  themselves 
to  equal  lengths,  the  angles  also  to  equahty.  In  the  final  stage  the 
cells  are  hexagonal  prisms  of  definite  dimensions,  which  depend  on 
temperature  and  on  the  nature  and  thickness  of  the  liquid  layer; 
molecular  forces  have  not  only  given  us  a  definite  cellular  pattern, 
but  also  a  "fixed  cell-size." 

Solid  particles  in  the  fluid  come  to  rest  in  svmmetrical  positions. 
If  they  be  heavier  they  accumulate  in  little  isolated  heaps,  each  in 


504 


THE  FORMS  OF  TISSUES 


[CH. 


the  focus  or  axis  of  a  cell;  if  they  be  hghter  they  drift  to  the 
boundaries,  then  towards  the  nodes,  where  they  tend  to  form 
tri-radiate  figures  like  so  many  'Hri-radiate  spicules".  But  if  they 
be  in  very  fine  suspension  a  curious  thing  happens:  for  as  they  are 
carried  round  in  the  vortex,  the  lowermost  layer  of  hquid,  next  to 
the  solid  floor,  keeps  free  of  particles;  and  this  "dust-free  coat*", 
rising  in  the  axis  of  the  cell  and  descending  at  its  boundary-walls, 
surrounds  an  inner  vortex  to  which  the  suspended  particles  are 
confined.  The  cell-contents  have,  so  to  speak,  become  differentiated 
into  an  "ectoplasm"  and  an  "endoplasm";  and  an  analogy  appears 


A  B 

Fig.  182.    Benard  patterns  in  smoke :  ^,  at  rest;  B,  under  shear.    After  K.  Chandra. 

with  the  phenomenon  of  protoplasmic  "rotation,"  where  the  outer 
layer  of  a  cell  tends  to  be  free  from  granules.  When  bright  glittering 
particles  are  used  for  the  suspension  (such  as  graphite  or  butterfly- 
scales)  beautiful  optical  effects  are  obtained,  deep  shadows  marking 
the  outlines  and  the  centres  of  the  cells.  Lastly,  and  this  is  by  no 
means  the  least  curious  part  of  the  phenomenon,  the  free  surface 
of  the  Hquid  is  not  plane;  but  each  little  cell  is  found  to  be  dimpled 
in  the  centre  and  raised  at  the  edges,  in  a  surface  of  very  complex 
curvature!,  and  there  is  a  curious  pulsation  in  the  flow,  especially 
when  waxes' are  used. 

*  Cf.  Tyndall,  Proc.  Roy.  Inst,  vi,  p.  3,  1870. 

t  The  differences  of  level  are  of  a  very  small  order  of  magnitude,  say  1  /x  in 
a  layer  of  spermaceti  1  mm.  thick. 


VII 


OF  BENARD'S  EXPERIMENT 


505 


Ringing  the  changes  on  Benard's  experiment,  we  may  use  unstable 
layers  of  various  liquids  or  gases,  or  even  a  thin  layer  of  smoke 
between  a  hot  plate  and  a  cold  (Fig.  182).  The  smoke  will  form  waves 
and  folds  and  rolHng  clouds;  then,  with  increasing  and  more  and 
more  symmetrical  instabihty,  polygonal  or  hexagonal  prisms;  and 
all  these  configurations  we  may  deform  or  "shear"  by  sliding  one 
plate  over  the  other.  Famihar  cloud-patterns,  as  of  a  dappled  or 
mackerel  sky,  can  be  imitated  in  this  way.  When  the  hexagonal 
prisms  have  been  developed  it  is  found  that  a  steady  shear  deforms 
them  into  a  well-known  curvihnear  tesselated  pattern  (Fig.   183) 


Direction  of 
motion 


Fig. 


183. 

of  air  or 


Shear-patterns  in  an  unstable  layer 
raoke.     After  Graham. 


Fig.  184.  Ambulaeral  plates 
of  a  sea-urchin  (Lepides- 
thes),  to  illustrate  the 
"shearing"  of  a  pattern 
normally  hexagonal. 
After  Hawkins*. 


and  this  may  be  sheared  again  into  hexagons,  oriented  in  an  opposite 
direction  to  the  first  t.  The  very  same  pattern  occurs  now  and  then 
in  organisms,  as  a  deformation  of  what  is  normally  a  pattern  of 
hexagons  (Fig.  184). 


*  From  H.  L.  Hawkins,  Phil.  Trans.  (B),  ccix,  p.  383,  1920. 

t  Cf.  A.  Graham,  Shear  patterns  in  an  unstable  layer  of  air,  Phil.  Trans.  (A), 
No.  714,  1933;  Gilbert  Walker  and  Phillips,  Q.J.R.  Met.  ;Soc.  Lvm,  p.  23,  1932; 
Mais,  Beitr.  Phys.  frei.  Atmosph.  xvii,  p.  \o,  1930;  H.  Jeffreys,  Proc.  R.S.  (A), 
cxvm,  p.  195,  1928;  Krishna  Chandra,  ibid,  clxiv,  pp.  231-242,  1938.  After  a 
steady  state  is  reached  it  is  found  that  in  air  or  smoke  the  centre  of  each 
polygon  is  a  funnel  of  descent,  but  it  is  an  ascending  column  if  the  layer  be 
liquid.  Now  the  viscosity  of  a  gas  increases,  and  that  of  a  hquid  decreases,  with 
rise  of  temperature,  and  the  greater  the  viscosity  the  more  stable  is  the  layer. 
Accordingly,  for  a  gas  the  upper,  and  for  a  liquid  the  lower  layer  is  the  more 
unstable,  and  it  is  there  that  in  each  case  the  flow  begins. 


506  THE  FORMS  OF  TISSUES  [ch. 

We  learn  from  these  experiments  of  Benard  and  others  how 
similar  distributions  of  force,  and  identical  figures  of  equihbrium, 
may  arise  through  different  physical  agencies.  We  see  that  patterns 
closely  analogous  to  those  of.  living  cells  and  tissues  may  be  due 
to  very  different  causes ;  and  we  may  be  led  to  scrutinise  anew,  with 
an  open  mind,  various  histological  configurations  whose  origin  is 
doubtful  or  obscure.  The  chitinous  shells  of  certain  water-fleas 
(Cladocera)  are  beset  with  a  roughly  hexagonal  pattern,  and  each 
little  chitinous  polygon  is  supposed  to  correspond  to,  and  to  be 
formed  by,  an  underlying  "hypodermis"  cell*.  But  we  presently 
discover  that  the  existence  of  these  hypodermis-cells  is  merely 
deduced   from   the   polygons   themselves   and   from   a   coincident 


W^' 

w° 

Fig.  185.     Soap-froth  under  pres.sure.     After  Rhumbler. 

distriburion  of  pigment;  it  might  not  be  amiss  to  look  again  into 
the  development  of  the  pattern,  with  an  open  mind  as  to  the 
possibihty  of  its  being  a  purely  physical  phenomenon.  Nor  need 
we  by  any  means  assume  that  the  calcareous  prisms  of  a  molluscan 
shell  are  necessarily  derived  from,  or  associated  with,  a  hke  number 
of  histological  elements. 

In  a  soap-froth  imprisoned  between  two  glass  plates  w^e  have  a 
symmetrical  system  of  cells  which  appear  in  optical  section  (Fig. 
185,  B)  as  regular  hexagons;  but  if  we  press  the  plates  a  little 
closer  together  the  hexagons  become  deformed  and  flattened.    The 

*  Cf.  F.  Claus,  Zur  Kenntniss. .  .des  feineren  Baues  der  Daphniden,  Ztschr.  f. 
loiss.  Zool.  XXIII,  XXVII,  pp.  3(i2-402,  1876;  Ernest  Warren,  Relationship  between 
size  of  cell  and  size  of  body  in  Dapknia  magna,  Biometrika,  (3)  ii,  pp.  255-259, 
1902;  Fritz  Werner.  Die  V'eranderung  der  Schalenform  und  der  Zellenaufbau  bei 
Scaphohberis.  Int.  Berne  der  yes.  Hydrobioloyie,  pp.  1-20,  1923. 


VII]  OF  HEXAGONAL  SYMMETRY  507 

change  is  from  a  more  to  a  less  probable  configuration — the 
entropy  is  diminished;  and  if  we  apply  no  further  pressure  the 
tension  of  the  films  adjusts  itself  again,  and  the  system  recovers 
its  former  symmetry*. 

The  epithelial  hning  of  the  blood-vessels  shews  a  curious  and 
beautiful  pattern.  The  cells  seem  diamond-shaped,  but  looking 
closer  we  see  that  each  is  in  contact  (usually)  with  six  others;  they 
are  not  rhombs,  or  diamonds,  but  elongated  hexagons,  pulled  out 
long  by  the  growth  of  the  vessel  and  the  elastic  traction  of  its  walls. 
The  sides  of  each  cell  are  curiously  waved,  and  a  simple  experiment 
explains  this  phenomenon.  If  we  make  a  froth  of  white-of-egg 
upon  a  stretched  sheet  of  rubber,  the  cells  of  the  froth  will  tend  to 


Fig.  186.     Sinuous  outlines  of  epithelial  cells,     a,  endothelium  of  a  blood-vessel; 
h,  epidermis  of  hnpatiens;    r,  epidermal  cells  of  a  grass  (Festuca). 

assume  their  normal  hexagonal  pattern ;  but  relax  the  elastic  mem- 
brane, and  the  cell-walls  are  thrown  into  beautiful  sinuous  or  wavy 
folds.  The  froth-cells  cannot  contract  as  the  rubber  does  which 
carries  them,  nor  can  the  epithelial  cells  contract  as  does  the 
muscular  coat  of  the  blood-vessel ;  in  both  cases  alike  the  cell- walls 
are  obliged  to  fold  or  wrinkle  up,  from  lack  of  power  to  shorten. 
The  epithelial  cells  on  the  gills  of  a  mussel  I  are  wrinkled  after 
the  same  fashion;  but  the  more  coarsely  sinuous  outlines  of  the 
epithelium  in  many  plants  is  another  story,  and  not  so  easily 
accounted  for. 

The  hexagonal  pattern  is  illustrated  among  organisms  in  count- 
less cases,  but  those  in  which  the  pattern  is  perfectly  regular,  by 

*  That  everything  is  passing  all  the  while  towards  a  ''more  probable  state''  is 
known  as  the  "principle  of  Carnot,"  and  is  the  most  general  of  all  physical  laws 
or  aphorisms. 

t  Cf.  James  Gray's  Experimental  Cytology,  p.  252. 


508  THE  FORMS  OF  TISSUES  [ch. 

reason  of  perfect  uniformity  of  force  and  perfect  equality  of  the 
individual  cells,  are  not  so  numerous.  The  hexagonal  cells  of  the 
pigmented  epithelium  of  the  retina  are  a  good  example.  Here  we 
have  a  single  layer  of  uniform  cells,  reposing  on  the  one  hand  upon 
a  basement  membrane,  supported  behind  by  the  sohd.  wall  of  the 
sclerotic,  and  exposed  on  the  other  hand  to  the  uniform  fluid 
pressure  of  the  vitreous  humour.  The  conditions  all  point,  and 
lead,  to  a  symmetrical  result:  the  cells,  uniform  in  size,  are  flattened 
out  to  a  uniform  thickness  by  uniform  pressure,  and  their  reaction 
one  upon  another  converts  each  flattened  disc  into  a  regular 
hexagon.  An  equally  symmetrical  case,  one  of  the  first-known 
examples  of  an  "epithehum,"  is  to  be  found  on  the  inner  wall  of 
the  amnion,  where,  as  Theodor  Schwann  remarked,  "die  sechs- 
eckige  Plattchen  sind  sehr  schon  und  gross*." 


Fig.  187.     Epidermis  of  Girardia.     After  GoebeJ. 

In  an  ordinary  columnar  epithelium,  such  as  that  of  the  intestine, 
again  the  columnar  cells  are  compressed  into  hexagonal  prisms; 
but  here  the  cells  are  less  uniform  in  size,  small  cells  are  apt  to 
be  intercalated  among  the  larger,  and  the  perfect  symmetry  is 
lost  accordingly.  But  obviously,  wherever  we  have,  in-  addition 
to  the  forces  which  tend  to  produce  the  regular  hexagonal  sym- 
metry, some  other  component  arising  asymmetrically  from  growth 
or  traction,  then  our  regular  hexagons  will  be  distorted  in  various 
simple  ways.  Thus  in  the  delicate  epidermis  of  a  leaf  or  young 
shoot  we  begin  with  hexagonal  cells  of  exquisite  regularity:  on 
which,  however,  subsequent  longitudinal  growth  may  impose  an 
equally  simple  and  symmetrical  deformation  or  polarity  (Fig.  187). 

In  the  growth  of  an  ordinary  dicotyledonous  leaf,  we  see  reflected 
in  the  form  of  its  cells  the  tractions,  irregular  but  on  the  whole 
longitudinal,  which  growth  has  superposed  on  the  tensions  of  the 

*   Untersuchungen,  p.  84;    cf.  Sydenham  Society's  translation,  p.  75. 


VII]  OF  EPIDERMAL  CELLS  509 

partition  walls  (Fig.  188).  In  the  narrow  elongated  leaf  of  a  mono- 
cotyledon, such  as  a  hyacinth,  the  elongated,  apparently  quad- 
rangular cells  of  the  epidermis  appear  as  a  necessary  consequence 
of  the  simpler  laws  of  growth  which  gave  its 
simple  form  to  the  leaf  as  a  whole.  In  all 
these  cases  ahke,  however,  the  rule  still 
holds  that  only  three  partitions  (in  surface 
view  or  plane  projection)  meet  in  a  point; 
and  near  their  point  of  meeting  the  walls  are 
manifestly  curved  for  a  little  way,  so  as  to 
permit  the  triple  conjunction  to  take  place 
at  or  near  the  co-equal  angles  of  120°,  after 
the  fashion  described  above. 

Briefly  speaking,  wherever  we  have  a  system  j^^jg  ^g^  Epidermal  cells 
of  cylinders  or  spheres,  associated  together         from   leaf  of  Elodea 

with  sufficient  mutual  interaction  to   bring         canadensis.         After 

,       ^  Berthold. 

them  into  complete  suriace  contact,  there, 

in  section  or  in  surface  view,  we  tend  to  get  a  pattern  of  hexagons. 
In  thickened  cells  or  fibres  of  bast  or  wood,  the  "  sclerenchyma " 
of  vegetable  histology,  the  hexagonal  pattern  is  all  but  lost,  and  we 
see  in  cross-section  the  more  or  less  circular  transverse  outlines  of 
elongated  and  tapering  cells.  Looking  closer  we  see  that  the 
primitive  cell-walls  preserve  their  angular  contours,  and  shew  much 
as  usual  an  hexagonal  pattern,  with  only  such  irregularities  as 
follow  from  the  unequal  sizes  of  the  associated  cells.  But  when 
these  primary  walls  are  once  laid  down,  the  secondary  deposits  which 
follow  them  are  under  different  conditions;  and  these  obey  the  law 
of  minimal  areas  in  their  own  way,  by  filling  up  the  angles  of  the 
primary  cell  and  by  continuing  to  grow  inwards  in  concentric  and 
more  and  more  nearly  circular  rings. 

While  the  formation  of  an  hexagonal  pattern  on  the  basis  of  ready-formed 
and  symmetrically  arranged  material  units  is  a  very  common,  and  indeed  the 
general  way,  it  does  not  follow  that  there  are  not  others  by  which  such  a 
pattern  can  be  obtained.  For  instance,  if  we  take  a  little  triangular  dish  of 
mercury  and  set  it  vibrating  (either  by  help  of  a  tuning-fork,  or  by  simply 
tapping  on  the  sides)  we  shall  have  a  series  of  little  waves  or  ripples  starting 
inwards  from  each  of  the  three  faces;  and  the  intercrossing,  or  interference 
of  these  three  sets  of  waves  produces  crests  and  hollows,  and  intermediate 
points  of  no  disturbance,  whose  loci  are  seen  as  a  beautiful  pattern  of  minute 


510.  THE  FORMS  OF  TISSUES  [ch. 

hexagons.  It  is  possible  that  the  very  minute  and  astonishingly  regular 
pattern  of  hexagons  which  we  see  on  the  surface  of  many  diatoms  (Fig.  189) 
may  be  a  phenomenon  of  this  order*.  The  same  may  be  the  case  also  in  Arcella, 
where  an  apparently  hexagonal  pattern  is  found  not  to  consist  of  simple 
hexagons,  but  of  "straight  lines  in  three  sets  of  parallels,  the  lines  of  each 
set  making  an  angle  of  sixty  degrees  with  those  of  the  other  two  setst-"  We 
must  also  bear  in  mind,  in  the  case  of  the  minuter  forms,  the  large  possibilities 
of  optical  illusion.  For  instance,  in  one  of  Abbe's  "diffraction-plates,"  a 
pattern  of  dots,  set  at  equal  interspaces,  is  reproduced  on  a  very  minute  scale 
by  photography;  but  under  certain  conditions  of  microscopic  illumination 
and  focusing,  these  isolated  dots  appear  as  a  pattern  of  hexagons. 

A  symmetrical  arrangement  of  hexagons,  such  as  we  have  just  been  studying, 
suggests  various  geometrical  corollaries,  of  which  the  following  may  be  a 
useful  one.  We  sometimes  desire  to  estimate  the  number  of  hexagonal  areas  or 
facets  in  some  structure  where  these  are  numerous,  such  for  instance  as  the 
cornea  of  an  insect's  eye,  or  in  the  minute  pattern  of  hexagons  on  many 
diatoms.     An  approxipiate  enumeration  is  easily  made  as  follows. 

For  the  area  of  a  hexagon  (if  we  call  8  the  short  diameter,  that  namely 
which  bisects  two  of  the  opposite  sides)  is  8^  x  v  3/2,  the  area  of  a  circle 
being  ci^.7r/4.  Then,  if  the  diameter  {d)  of  a  circular  area  include  n  hexagons, 
the  area  of  that  circle  equals  {n.Sfxn/i:.  And,  dividing  this  number  by 
the  area  of  a  single  hexagon,  we  obtain  for  the  number  of  areas  in  the  circle, 
each  equal  to  a  hexagonal  facet,  the  expression  w^  x7r/4  x  2/\/3  =  0'9077i^  or 
9/lO.w^  nearly. 

This  calculation  deals,  not  only  with  the  complete  facets,  but  with  the 
areas  of  the  broken  hexagons  at  the  periphery  of  the  circle.  If  we  neglect 
these  latter,  and  consider  our  whole  field  as  consisting  of  successive  rings  of 
hexagons  about  a  central  one,  we  obtain  a  simpler  rule.  For  obviously, 
around  our  central  hexagon  there  stands  a  zone  of  six,  and  around  these 


*  Cf.  some  of  J.  H.  Vincent's  photographs  of  ripples,  in  Phil.  Mag.  1897-99; 
or  those  of  F.  R.  Watson,  in  Phys.  Review,  1897,  1901,  1916.  The  appearance  will 
depend  on  the  rate  of  the  wave,  and  in  turn  on  the  surface-tension;  with  a  low 
tension  one  would  probably  see  only  a  moving  "jabble."  Cf.  also  Faraday,  On 
the  crispations  of  fluids  resting  upon  a  vibrating  support,  Phil.  Mag.  1831,  p.  299; 
and  Rayleigh,  Sound,  u,  p.  346,  1896.  FitzGerald  thought  diatom -patterns  might 
be  due  to  electromagnetic  vibrations  (Works,  p.  503,  1902);  with  which  of.  W.  D. 
Dye,  Vibration-patterns  of  quartz  plates,  Proc.  R.S.  (A),  cxxxviii,  p.  1,  1932. 
Dye's  Fig.  17,  which  he  calls  "one  of  the  most  beautiful  types  of  minor  vibration 
met  with  in  discs",  is  closely  akin  to  the  diatom  Orthoneis  splendida.  In  both  cases 
two  nodal  systems,  conjugate  to  one  another,  are  based  on  two  foci  near  the  ends  of 
an  elliptical  plate;  .  but  bands  in  the  experimental  plate  are  further  broken  up  into 
rows  of  dots  in  the  diatom.  See  also  Max  Schultze,  Die  Struktur  der  Diatomeenschale 
verglichen  mit  gewi.ssen  aus  Fluorkiesel  kiinstlich  darstellbaren  Kieselhauten,  Verh. 
nnturh.  Ver.  Bonn,  xx,  pp.  1-42,  1863;  Trans.  Microsc.  Soc.  (N.S.),  xi,  pp.  120-136, 
1863;    H.  J.  Slack,  Monthly  Microsc.  Journ.  1870,  p.  183. 

t  J.  A.  Cushman  and  W.  P.  Henderson,  Amer.  Nat.  XL,  pp.  797-802,  1906. 


VIlJ 


OF  HKXAIJONAL  PATTERNS 


511 


Fig.  189. 


A  diatom.  Triccrniiinn  sj)..  shewing  j)attern  of  hrxagoiis; 
().  Prochnow,  foDnenkunst  der  Xatur. 


y'.HH).      From 


a  zone  of  twelve,  and  around  these  a  zone  of  eighteen,  and  so  on. 
total  number,  excluding  the  central  hexagon,  is  accordingly: 


And  the 


For  one  zone 
„    two  zones 
,,    three  zones 
„    four  zones 
„    five  zones 


6=3x1x2=6x1 
18  2x3  3 

36  3x4  6 

60  4x5         10 

90  5x6         15 


and  so  forth.  If  N  be  the  number  of  zones,  and  if  we  add  one  to  the  above 
numbers  for  the  odd  central  hexagon,  then  the  rule  is  that  the  total  number 
H  =  3N  (A'  +  l)  +  l.     Thus,  if  in  a  preparation  of  a  fly's  cornea  I  can  count 


512  THE  FORMS  OF  TISSUES  [ch. 

twenty-five  facets  in  a  line  from  a  central  one,  the  total  number  in  the  entire 
field  is  (3  X  25  X  26)  + 1  =  1951  *. 

The  electrical  engineer  is  dealing  with  the  selfsame  problem  when  he  finds 
he  can  pack  6  +  18  +  36+  ...  wires  around  a  central  wire,  to  form  a  multiple 
cable  of  1,  2  and  3  concentric  strands.  He  coimts  them  by  the  same  formula, 
in  the  simpler  form  of  6<+ 1 :  where  t  is  a  "triangular  number,"  1,  3,  6,  10,  etc., 
corresponding  to  the  number  of  strands.  Thus  1951=6x325  +  1;  325  being 
the  triangular  number  of  25,  1  +  2  +  3  + . . .  +  25. 

We  have  many  varied  examples  of  this  principle  among  corals, 
wherever  the  polypes  are  in  close  juxtaposition,  with  neither  empty 
space  nor  accumulations  of  matrix  between  their  adjacent  walls. 
Favosites  gothlandica,  for  instance,  furnishes  us  with  an  excellent 
example.  In  the  great  genus  Lithostrotion  we  have  some  species 
which  are  "massive"  and  others  which  are  "fasciculate."  In  other 
words,  in  some  the  long  cylindrical  corallites  are  closely  packed 
together,  and  in  others  they  are  separate  and  loosely  bundled  (Fig. 
190);  in  the  former  the  corallites  are  squeezed  into  hexagonal 
prisms,  while  in  the  latter  they  retain  their  cylindrical  form.  Where 
the  polypes  are  comparatively  few,  and  so  have  room  to  spread, 
the  mutual  pressure  ceases  to  work  or  only  tends  to  push  them 
asunder,  letting  them  remain  circular  in  outhne  (e.g.  Thecosmilia). 
Where  they  vary  gradually  in  size,  as  for  instance  in  Cyathophyllum 
hexagonum,  they  are  more  or  less  hexagonal  but  are  not  regular 
hexagons;  and  where  there  is  greater  and  more  irregular  variation 
in  size,  the  cells  will  be  on  the  average  hexagonal,  but  some  will 
have  fewer  and  some  more  sides  than  six,  as  in  the  annexed  figure 
of  Aracknophyllum  (Fig.  192).  Where  larger  and  smaller  cells, 
corresponding  to  two  different  kinds  of  zooids,  are  mixed  together, 
we  may  get  various  results.    If  the  larger  cells  are  numerous  enough 

*  This  estimate  neglects  not  merely  the  broken  hexagons,  but  all  those  whose 
centres  lie  between  the  circle  and  a  hexagon  inscribed  in  it.  The  discrepancy 
is  considerable,  but  a  correction  is  easily  made.  It  will  be  found  that  the  numbers 
arrived  at  by  the  two  methods  are  approximately  as  6:5.  For  more  detaOed 
calculations  see  a  paper  by  H.M.  (?  H.  Munro)  in  Q.J. M.S.  vi,  p.  83,  1858.  The 
methods  of  enumeration  used  by  older  writers,  especially  by  Leeuwenhoek,  are 
sometimes  curious  and  interesting;  cf.  Hooke,  Micrographia,  1665,  p.  176; 
Leeuwenhoek,  Arcana  naturae,  1695,  p.  477;  Phil.  Trans.  1698,  p.  169;  Epist. 
physiolog.  1719,  p.  342;  Swammerdam,  Biblia  Naturae,  1737,  p.  490.  Leeuwenhoek 
found,  or  estimated,  3181  facets  on  the  cornea  of  a  scarab,  and  8000  on  that  of 
a  fly;  M.  Puget,  about  the  same  time,  found  17,325  in  that  of  a  butterfly.  See  also 
Karl  Leinemann,  Die  Zahl  der  Facetten  in  den.  .  .Coleopteren,  Hildesheim,  1904. 


VIl] 


OF  HEXAGONAL  SYMMETRY 


513 


to  be  more  or  less  in  contact  with  one  another  (e.g.  various  Monti- 
cuHporae)  they  will  be  irregular  hexagons,  while  the  smaller  cells 
between  them  will  be  crushed  into  all  manner  of  irregular  angular 


Fig.  190.     Lithostrotion  Martini. 
After  Nicholson. 


Fig.  191.     Cyathophyllum  hexagonum. 
From  Nicholson,  after  Zittel. 


Fig.  192.     Arachnophyllujn  pentagonum. 
After  Nicholson. 


m 

m 

Fig.  193.    Heliolites.    After  Woods. 


forms.  If  on  the  other  hand  the  large  cells  are  comparatively  few 
and  are  large  and  strong-walled  compared  with  their  smaller  neigh- 
bours, then  the  latter  alone  will  be  squeezed  into  hexagons  while 
the  larger  ones  will  tend  to  retain  their  circular  outhne  undisturbed 
(e.g.  Heliopora,  Heliolites,  etc.  (Fig.  193)). 


514 


THE  FORMS  OF  TISSUES 


[Un. 


When,  as  happens  in  certain  corals,  the  peripheral  walls  or 
"thecae"  of  the  individual  polypes  remain  undeveloped  but  the 
radiating  septa  are  formed  and  calcified,  then  we  obtain  new  and 
beautiful  mathematical  configurations  (Fig.  194).  For  the  radiating 
septa  are  no  longer  confined  to  the  circular  or  hexagonal  bounds  of 
a  polypite,  but  tend  to  meet  and  become  confluent  with  their 
neighbours  on  every  side;  and,  tending  to  assume  positions  of 
equihbrium,  or  of  minimum,  under  the  restraints  to  which  they 
are  subject,  they  fall  into  congruent  curves,  which  correspond  in 
a  striking  manner  to  lines  running  in  a  common  field  of  force 
between  a  number  of  secondary  centres.  Similar  patterns  may  be 
produced  in  various  ways  by  the  play  of  osmotic  or  magnetic  forces ; 


Fig.  194.     Surface-views  of  corals  with  undeveloped  thecae  and  confluent  septa. 
A,  Thamnastraea ;  B,  Comoseris.     From  Nicholson,  after  Zittel. 

and  a  very  curious  case  is  to  be  found  in  those  compHcated  forms 
of  nuclear  division  known  as  triasters,  polyasters,  etc.,  whose 
relation  to  a  field  of  force  Hartog  in  part  explained*.  It  is  obvious 
that  in  our  corals  these  curving  septa  are  all  orthogonal  to  the 
non-existent  hexagonal  boundaries ;  and,  as  the  phenomenon  is  due 
to  the  imperfect  development,  or  non-existence,  of  a  thecal  wall,  it 
is  not  surprising  that  we  find  identical  configurations  among  various 
corals,  or  families  of  corals,  not  otherwise  related  to  one  another. 
We  find  the  same  or  very  similar  patterns  displayed,  for  instance, 
inSynhelia  (Oculinidae),  in  Philhpsastraea  (Rugosa),  in  Thamnastraea 
(Fungida),  and  in  many  more. 

*  Cf.  M.  Hartog,  The  dual  force  of  the  dividing  cell,  Science  Progress  (N.S.), 
I,  Oct.  1907,  and  other  papers.  Also  Baltzer,  Mehrpolige  Mitosen  bet  Seeeigeleiern, 
Diss.,  1908. ' 


VII 


OF  HEXAGONAL  SYMMETRY 


515 


Fig.    195. 

Female  cone 

of  Zamia. 


A  beautiful  hexagonal  pattern  is  seen  in  the  male  and  female 
cones  of  Zamia,  where  the  scales  which  bear  the  pollen-sacs  or  the 
ovules  are  crowded  together,  and  are  so  formed  and  circumstanced 
that  they  cannot  protrude  and  overlap.  They  become 
compressed  accordingly  into  regular  hexagons*,  smaller 
and  more  regular  in  the  male  cone  than  in  the  female, 
in  which  latter  the  cone  as  a  whole  has  tended  to  grow 
more  in  breadth  than  in  length,  and  the  hexagons  are 
somewhat  broader  than  they  are  long.  In  a  cob  of 
maize  the  hexagonal  form  of  the  grains,  such  as  should 
result  from  close-packing  and  mutual  compression,  is 
exhibited  faintly  if  at  all;  for  growth  and  elongation 
of  the  spike  itself  has  reheved,  or  helped  to  reheve,  the 
mutual  pressure  of  the  grains. 

The  pine-cone  shews  a  simple,  but  unusual  mode  of 
close-packing.  The  spiral  arrangement  causes  each 
scale  to  lie,  to  right  and  to  left,  on  two  prmcipal  spirals; 
it  has  close  neighbours  on  four  sides,  and  mutual 
compression  leads  to  a  square  or  rhomboidal,  instead  of  an  hexa- 
gonal, configuration*.  On  the  other  hand,  the  scales  of  the  larch - 
cone  overlap:  therefore  they  are  not  subject  to  compression,  but 
grow  more  freely  into  leaf- like  curves. 

The  story  of  the  hexagon  leads  us  far  afield,  and  in  many  directions, 
but  it  begins  with  something  simpler  even  than  the  hexagon.  We 
have  seen  that  in  a  soapy  froth  three  films,  and  three  only,  meet 
in  an  edge,  a  phenomenon  capable  of  explanation  by  the  law  of 
areae  minirnae.  But  the  conjunction,  three  by  three,  of  almost  any 
assemblage  of  partitions,  of  cracks  in  drying  mud,  of  varnish  on  an 
old  picture,  of  the  various  cellular  systems  we  have  described,  is  a 
general  tendency,  to  be  explained  more  simply  still.  It  would  be 
a  complex  pattern  indeed,  and  highly  improbable,  were  all  the  cracks 
(for  instance)  to  meet  one  another  six  by  six;  four  by  four  would 
be  less  so,  but  still  too  much;  and  three  by  three  is  nature's  way, 
simply  because  it  is  the  simplest  and  the  least.  When  the  partitions 
meet  three  by  three,  the  angles  by  which  they  do  so  may  vary 
indefinitely,  but  their  average  will  be  120°;    and  if  all  be  on  the 

*  In  some  small,  few-scaled  cones  the  packing  remains  incomplete,  and  the  scales 
are  four-,  five-,  or  six-sided,  as  the  case  may  be. 


516  THE  FORMS  OF  TISSUES  [ch. 

average  angles  of  120°,  the  polygonal  areas  must,  on  the  average, 
be  hexagonal.  This,  then,  is  the  simple  geometrical  explanation, 
apart  from  any  physical  one,  of  the  widespread  appearance  of  the 
pattern  of  hexagons. 

If  the  law  of  minimal  areas  holds  good  in  a  "cellular"  structure, 
as  in  a  froth  of  soap-bubbles  or  in  a  vegetable  parenchyma,  then 
not  merely  on  the  average,  but  actually  at  every  node,  three  partition- 
walls  (in  plane  projection)  meet  together.  Under  perfect  symmetry 
they  do  so  at  co-equal  angles  of  120°,  and  the  assemblage  consists 


Fig.  196.     Cracks  in  drying  mud;   a  thread  encircles  and  marks  out  a  "'polar 
furrow";    cf.  p.  487.     From  R.  H.  Wodehouse. 

(in  plane  projection)  of  co-equal  hexagons;  but  the  angles  may  vary, 
the  cells  be  unequal,  and  the  hexagons  interspersed  with  other 
polygonal  figures.  Nevertheless,  so  long  as  three  partition- walls  and 
only  three  meet  together,  the  cells  are,  ipso  facto,  on  the  average 
hexagonal*. 

We  may  count  the  cells  if  we  please.     A  section  of  Cycas-petiole 
gave  the  following  numbers:  ^ 

Number  of  sides  3     4      5         6       7     8     9 

„  instances        0    8     97     207     96     9    0 
Mean  number  of  sides:   6-00. 

The  fine  emulsion  of  an  Agfa  plate  shews  a  beautiful  polygonal 
pattern  which  obeys  the  law  of  the  triple  node;    and  a  patch  of  a 

*  A  more  elaborate  proof  is  given  by  W.  C.  Goldstein,  On  the  average  number 
of  sides  of  polygons  of  a  net,  Ann.  Math.  (2),  xxxii,  pp.  149-153,  1931. 


VII 


OF  HEXAGONAL  SYMMETRY 


517 


thousand  cells  in  such  a  plate  has  been  found,  Uke  the  Cycas-petiole, 
to  average  out  at  six  sides  each,  precisely* 

The  cracking  of  a  fine  varnish  may  illustrate  (as  in  Fig.  19.7)  the 
same  phenomenon.  A  little  water  in  the  varnish  tends  to  accumu- 
late between  the  cells,  interferes  with  their  close-packing,  and 
complicates  the  arrangement  of  their  partitions. 

The  horny  plates  which  form  the  carapace  of  a  tortoise  (different 
as  the  case  may  seem)  still  obey  the  two  guiding  principles  (1)  that 
the  polygonal  boundaries  meet  in  three-way  nodes,  and  (2)  that  the 
three  angles  tend  towards  equality,  always  provided  that  no  alien 
influences  interfere.  These  principles  are  of  the  widest  apphcation; 
the  carapace  of  a  Eurypterid,  the  dermal  armour  of  an  Old  Red 


Fig.  197.     Cellular  patterns  in  varnish,   a,  dissolv'ed  in  dry  acetone; 
b,  containing  a  little  water. 

Sandstone  fish  like  Hugh  Miller's  Asterolepis,  exhibit  them  at  a 
glancef.  The  carapace  of  our  tortoise  is  formed  of  a  bony  framework 
of  ribs  and  vertebrae,  overlaid  by  superficial  plates  of  horn  or 
tortoiseshell ;  it  is  these  latter  with  which  we  are  about  to  deal. 
They  are  arranged  in  three  rows  down  the  back,  and  a  marginal  row 
of  smaller  plates  surrounds  the  others ;  there  are  (normally)  twenty- 
four  plates  in  the  marginal  series,  and  five  large  ones  in  the  median 
longitudinal  row.     With  these  few^  facts,  and  our  general  principles 

*  F.  T.  Lewis,  Polygons  in  a  film... and  the  pattern  of  simple  epithelium, 
Anat.  Record,  l,  pp.  235-265,  1931. 

t  The  all  but  universal  law  of  the  triple  corner,  or  triradiate  suture,  is  now 
and  then  enough  to  give  a  deceptive  likeness  to  very  different  things.  When 
Cope  marred  his  brilliant  classification  of  the  Ostracoderm  fishes  by  seeing  in  the 
carapace  of  Bothriolepis  a  likeness  to  the  dorsal  plates  of  the  tunicate  Chelyosoma, 
it  was  this  and  this  alone  which  led  him  astray. 


518  THE  FORMS  OF  TISSUES  [ch. 

in  hand,  how  far  can  we  go  towards  depicting  the  carapace?  The 
five  plates  of  the  median  row  must  alternate  with  their  lateral 
neighbours,  and  four  plates  in  each  lateral  row  will,  accordingly,  be 
the  simplest  case  or  most  probably  number.  If  at  each  three-way 
junction  between  median  and  lateral  plates  the  angles  tend  to 
equality,  it  follows  that  the  median  plates  become  converted  into 
more  or  less  regular,  or  at  least  symmetrical,  hexagons.  As  to  the 
twenty-four  marginal  plates,  let  us  put  one  in  front*  and  one 
behind,  leaving  eleven  for  each  side;   and  let  us  see  to  it  carefully 


Fig.  198.   Asterolepis:  an  Old  Red 
Sandstone  fish.    After  Traquair. 


Fig.  199.    Horny  carapace  of  a 
tortoise ;  diagrammatic. 


that  the  sutures  between  these  do  not  coincide  but  alternate 
with  those  of  the  lateral  row.  Here  we  begin  to  meet  with  con- 
ditions of  restraint  analogous  to  those  of  the  surface  layer  of  a 
froth,  for  the  long  marginal  cells  must  remain  marginal,  and  their 
sides  must  continue  to  run  more  or  less  parallel,  or  more  or  less 
perpendicular,  to  the  edge   of   the  shell;  »only  in  the  immediate 

*  The  Old  World  tortoises  have  twenty-five  marginal  plates,  those  of  the 
New  World  lack  the  anterior  median,  or  "nuchal"  plate.  This  difference  is  a 
biological  accident,  it  has  neither  mathematical  interest  nor  functional  significance; 
it  exemplifies  the  aphorism  that  whatsoever  is  possible  Nature  will  sooner  or 
later  do. 


VII]  OF  HEXAGONAL  SYMMETRY  519 

neighbourhood  of  each  corner  will  the  sides  tend  to  curve  in,  so 
forming  a  notch  whose  curved  sides  have  tangents  approximately 
120°  apart,  again  just  as  in  our  projection  of  the  surface  of  a  froth 
(p.  494).  Already  these  considerations  lead  us  to  a  fair  sketch  of, 
or  first  approximation  to,  the  carapace  of  a  tortoise;  but  we  may 
go  on  a  httle  further.  The  horny  plates  and  the  bony  carapace 
below  must  grow  at  such  a  rate  as  to  keep  pace,  more  or  less  exactly, 
with  one  another;  but  it  does  not  follow  that  they  will  keep  time 
precisely.  If  the  horny. plates  grow  ever  so  httle  faster  than  the 
bones  below,  they  will  fail  to  fit,  will  overcrowd  one  another,  and 
will  be  forced  to  bulge  or  wrinkle.  Both  of  these  things  they  often, 
and  even  characteristically,  do;  the  wrinkles  appear  in  orderly, 
parallel  folds,  pointing  to  alternate  periods,  or  spurts,  of  faster  and 
slower  growth;  and  the  characteristic  patterns  which. ensue  are 
the  visible  expression  of  these  differential  growth-rates. 

In  all  this  we  assume  tHat  the  plates  are  lying  in  one  and  the 
same  plane  or  even  surface,  abutting  against  one  another  as  they 
grow,  and  so  crowding  and  squeezing  one  another  into  the  fornj  of 
straight-edged  polygons.  The  result  will  be  very  different  if  they 
overlap,  after  the  manner  of  slates  on  a  roof:  the  difference  is  what 
we  have  seen  to  exist  between  the  cones  of  Pinus  and  of  Larix. 
The  overlapping  edges  will  be*  free  to  grow  into  natural,  rounded 
curves ;  each  plate,  uncrowded  and  unconstrained,  will  stay  smooth 
and  un wrinkled;  the  number  and  order  of  the  plates  will  be  the 
same  as  before — but  the  shell  will  be  no  longer  that  of  a  tortoise, 
but  of  the  turtle  from  which  "  tortoise-shell"  is  obtained. 

A  snow-crystal  is  a  very  beautiful  example  of  hexagonal  sym- 
metry. It  belongs  to  another  order  of  things  to  those  we  have  been 
speaking  of:  for  in  substance  it  is  a  solid,  and  in  form  it  is  a  crystal, 
and  its  own  intrinsic  molecular  forces  build  it  up  in  its  own  way. 
But  (as  we  have  mentioned  once  before)  it  is  an  exquisite  illus- 
tration of  Nature's  way  of  producing  infinite  variety  from  the 
permutations  and  combinations  of  a  single  type.  The  snowflake  is 
a  crystal  formed  by  sublimation,  that  is  to  say  by  precipitation 
from  a  vapour  without  passing  through  a  liquid  phase.  It  begins 
as  a  tiny  hexagon,  the  making  of  which  tends  to  use  up  the  vapour 
near  by;  the  angles  of  the  hexagon  jut  out,  so  to  speak,  into  regions 
of  greater,  or  less  depleted,  vapour-pressure,  and  at  these  corners 


520  THE  FORMS  OF  TISSUES  [ch. 

further  crystallisation  will  next  set  in.  The  hexagon  will  tend  to 
grow  out  into  a  six-rayed  star;  and  later  and  more  slowly  the 
material  for  further  crystallisation  will  make  its  way  between  the 
rays,  and  begin  to  build  side-growths  on  them*. 

The  basaltic  column 

Hexagonal  patterns  are  by  no  means  confined  to  the  organic 
world.  The  basalt  of  Staffa  and  the  Giant's  Causeway  shews  a 
wonderful  array  of  prismatic  columns  of  irregular  size  and  form, 


Fig.  200.     Basalt  at  Giant's  Causeway.     B}^  Mr  R.  Welch,  Belfast. 

but  mostly  hexagonal;  so  also  does  the  frozen  soil  of  Spitzbergen; 
starch  sets  on  coohng  into  analogous  prisms,  but  in  a  ruder  fashion 
as  on  a  smaller  scale;  and  all  these  are  due  to  simple  forces  in  a 
simple  field,  namely  to  tension,  or  shrinkage,  in  a  horizontal  mass 
or  layer.  Imagine  a  sheet  or  "sill"  of  intrusive  basalt,  thrust 
in  as  a  molten  mass  between  older  rocks.  It  is  gradually  chilled 
by  the  cold  air  above  or  by  the  rocks  on  either  side,  and  its  inner 
mass,  cooling  slower  than  the  outer  layer,  contracts  slowly.  Nothing 
hinders  its  vertical  contraction,  rather  is  this  helped  by  its  own  weight 
and  by  the  load  above;  but  no  further  lateral  contraction  can  take 
place  without  splitting  the  mass,  once  the  basalt  sets  hard.     Con- 

♦  Cf.  Gerald  Seligmann,  Nature,  26  June,  1937,  p.  1090. 


VII]  OF  THE  GIANT'S  CAUSEWAY  521 

traction,  however,  does  take  place,  irresistibly,  and  it  may  be  that 
long  cracks  appear;  the  strain  being  so  far  relieved,  the  next  cracks 
will  tend  to  take  place  at  right  angles  to  the  first.  But  more 
commonly  rupture  is  delayed  until  considerable  strain-energy  has 
been  stored  up ;  once  started,  it  proceeds  explosively  from  a  number 
of  centres,  and  shatters  the  whole  mass  into  prismatic  fragments. 
However  quickly  and  explosively  the  cracks  succeed  one  another 
each  reheves  an  existing  tension,  and  the  next  crack  will  give  rehef 
in  a  different  direction  to  the  first.  When  one  crack  meets  another 
it  will  seldom  cross  it,  for  the  strain  which  led  to  the  former  fracture 
does  not  extend  into  the  new  field.  In  short  the  cracks  will  be 
found  to  meet  one  another  three,  by  three,  and  therefore  at  angles 
071  the  average  of  120°,  and  the  columns  will  be  on  the  average 
hexagonal.  For  the  making  of  a  prismatic  structure  all  that  is 
required  is  more  or  less  uniform  tensile  strain  in  the  two  dimensions 
of  a  horizontal  plane;  uniform  tension  in  three  dimensions  would 
have  given  rise  to  a  cellular  structure,  of  which  the  hexagonal 
"causeway"'  is  the  two-dimensional  analogue. 

The  columnar  structure  is  accompanied  by  sundry  secondary 
phenomena.  The  vertical  columns  tend  to  break  across  on  further 
contraction,  and  exhibit  rounded  or  basin-shaped  ends,  fitting 
together  in  a  shallow  ball-and-socket;  this  beautiful  configuration 
has  only  lately  been  explained*.  When  cooling  has  caused  the 
mass  to  split  into  vertical  columns,  air  or  it  may  be  water  enters 
the  rifts  and  further  cools  or  quenches  the  now  sohd  but  still  glowing 
basalt.  Each  column  tends  to  be  chilled  all  round  while  still  hot 
within;  but  the  hot  unshrunken  mass  within  checks  or  hinders  the 
contraction  of  the  cooler  outer  layers.  Thus  unequal  coohng  causes 
vertical  as  well  as  horizontal  tensions;  and  just  as  these  last  are 
relieved  by  the  existing  cracks,  so  new  rifts  appear  crosswise  to  the 
column,  and  relieve  the  vertical  tensions. 

If  the  coohng  come  downward  from  above,  then,  at  any  given 
level,  the  column  will  always  be  cooler  above  it  than  below;   the 

*  F  \V  Preston  0  ball-and-socket  jointing  in  basalt  prisms,  Proc  R  S  (B), 
c  VI,  pp.  87-92,  1930  A  study  of  the  rupture  of  glass  Trans  Soc  of  Glass  Technology ^ 
1926  p  263  On  the  general  subject  of  prismatic  structure  m  igneous  rocks,  see 
also  Robert  Mallet,  Phil.  Mag.  (4),  l.  pp.  122-135,  201-226,  1875,  James  Thomson, 
Trans.  Geol.  Soc.  Glasgow,  March,  1877;  Coll.  Works,  p.  422;  R.  B.  Sosman,  Journ. 
Geology,  xxiv,  p.  215,  1916. 


522 


THE  FORMS  OF  TISSUES 


[CH. 


part  above  will  shrink  the  more;  and  this  will  set  up  a  horizontal 
shear  in  the  interior  of  the  column,  in  addition  to  the  existing 
vertical  tension.  The  principal  stress,  compounded  of  shear  and 
tension,  will  be  neither  vertical  nor  horizontal,  but  inclined 
obhquely  between  the  two.  Now  in  a  brittle  substance,  such  as 
glass  or  basalt,  an  advancing  fracture  tends  to  advance  at  right 
angles  to  the  principal  tension.  Where  the  surface  of  the  column 
meets  the  cool  air  the  tension  is  parallel  to  the  face,  and  the  fissure 
enters  at  right  angles  to  the  face,  that  is  to  say,  horizontally;  it  is 
for  this  reason  that  the  ball-and-socket  joint  is  found  to  have 
a  square  lip.  Once  inside  the  boundary,  however,  the  advancing 
rift  finds  itself  in  a  region  where  the  principal  stress  is  inclined, 


4 


/. 


t       P 


3 


Fig.  201.  (1)  Diagram  of  the  vertical  and  shearing  stresses  in  a  shrinking  column 
of  basalt,  (2)  The  same  in  the  neighbourhood  of  the  point  P.  (3)  JSS'  re- 
sultant stress,  and  TT'  direction  of  rupture.     After  F.  W.  Preston. 


slightly,  to  the  vertical;  the  crack  consequently  bends  down,  the 
downward  tilt  increases  for  a  short  distance  and  then  approaches 
the  horizontal  again,  and  the  opposing  surfaces  of  "ball  and  socket" 
are  thus  defined.  The  crack  often  fails  to  complete  its  journey,  and 
leaves  a  core  of  rock  unbroken  in  the  middle  of  the  bowl. 

The  curved  sides  of  the  basin,  its  square  lip,  its  flattened  centre, 
often  incomplete,  are  thus  all  explained;  and  whether  it  be  convex 
or  concave,  dome  or  basin,  merely  depends  on  whether  the  cooling 
or  quenching  came  from  above  or  from  below. 

The  basin,  or  bowl,  will  always  be  a  shallow  one.  For  if  at 
a  point  P,  within  the  column,  the  vertical  tension  be  /,  and  the 
horizontal  shear-stress  be  /,.,  then  the  direction  of  the  principal 
planes  will  be  2(/.  ==  tan-^  (-  2fjf) ;  so  that,  since  /  and  /^  are  both 
positive,  <f>j^  hes  between  45°  and  90°,  while  ^2  ^i^s  between  135° 


VII]  OF  RIFTS  AND  CRACKLES  523 

and  180°.  Hence  the  rift,  dipping  down  at  the  angle  TT\  will 
never  dip  at  a  greater  angle  than  45°,  and  generally  much  less. 
Now  a  spherical  bowl  whose  hp  is  at  45°  to  the  horizontal  has  a 

depth  of times  its  lip  diameter,  or  approximately  one-fifth. 

And  one-fifth  of  the  diameter  of  the  column  is,  approximately,  the 
depth  of  the  deepest  bowl. 

The  hexagonal  pattern  and  the  three-way  corners  on  which  it 
is  based  are  characteristic  (as  we  have  already  explained)  of  a 
condition  of  symmetry  or  uniformity  .under  which  a  partition  is 
as  hkely  to  arise  in  any  one  direction  as  in  any  other,  and  no 
series  of  partitions  has  precedence  over  the  rest.     We  have  seen, 


Fig.  2U2.     Crackles  on  a  porcelain  bowl.     From  H.  Hukusima. 

in  the  dragon-fly's  wing  and  in  cambium-tissue,  how  different  is 
the  result  when  primary  partitions  are  first  established  and  con- 
solidated, to  be  followed  by  a  secondary  and  a  weaker  set.  The 
"crackles"  on  a  porcelain  bowl  look  somewhat  like  a  cellular 
epithelium;  but  the  porcelain  has  been  under  strain  in  more  ways 
than  one.  The  plastic  clay  was  first  shaped  upon  a  wheel,  and 
potential  stress-energy  so  acquired  is  stored  up  even  in  the  finished 
ware ;  again,  as  the  ware  cools  and  shrinks  after  it  is  drawn  from  the 
kiln  the  glaze  is  apt  to  cool  quicker  and  shrink  more  than  the  paste 
below,  and  tension -energy  is  stored  up  till  the  glaze  ruptures  and 
the  cracks  appear.'  Various  rates  of  cooling  and  of  contraction,  the 
nature  of  paste  and  glaze,  the  shape  of  the  ware,  and  even  the  way 
in  which  the  potter  worked  the  clay,  may  all  influence  the  pattern 
of  the  crackle.     The  primary  crack  will  be  perpendicular  to  the 


524  THE  FORMS  OF  TISSUES  [ch. 

main  tension;  secondaries  will  tend  to  be  more  or  less  orthogonal 
to  the  primary  cracks;  two  secondaries  on  opposite  sides  of  a 
primary  will  tend  to  be  near  togetiier,  though  not  opposite;  and 
spiral  cracks  are  often  to  be  seen,  the  remote  effect  of  plasticity 
under  the  potter's  wheel.  The  net  result  is  that  certain  primary 
cracks  appear,  related  (more  or  less  clearly)  to  the  circular  or  spiral 
shaping  of  the  clay  upon  the  wheel;  and  each  primary  crack  so 
reheves  the  tension  in  one  direction  that  the  secondaries  tend  to 
follow  in  a  direction  at  right  angles  to  the  first.  While  co-equal 
angles  of  120°  were  likely  to  occur  in  certain  symmetrical  cases, 
leading  to  a  simultaneous  pattern  of  hexagons,  in  other  cases  suc- 


Fig.    203.     Colour-patterns    of    kidney-beans    with    diagrammatic    contour-lines 
added;    a,  b,  Japanese  "quail- beans";    c,  scarlet-runner.     After  M.  Hirata. 

cessive  partitions  or  cracks  tend  to  be  at  right  angles  to  one  another ; 
and  Sachs's  Law  becomes  truer  of  the  porcelain  than  of  the  plant*. 
In  this  latter  case,  and  doubtless  in  many  more,  we  are  deahng 
not 'with  a  random  pattern,  but  with  one  based  on  systematic  and 
predetermined  lines.  The  apparently  confused  or  random  pattern 
of  a  kidney-bean  comes  under  the  same  class  of  configurations, 
inasmuch  as  it  also  is  based  on  an  underlying  polarity,  whose 
centre  of  symmetry  is  in  the  stalk  or  "hilus."  For  simphcity's 
sake,  imagine  the  bean  round  hke  a  pea,  and  its  surface  mapped 
out,  orthogonally,  by  two  sets  of  boundary-hnes,  radial  and  con- 
centric. Then  suppose  an  asymmetry  of  growth  to  be  introduced 
so  that  the  round  pea  grows  into  the  elhpsoid  of  a  bean;  and 
suppose  that  the  whole  system  of  boundary-lines  is  subject  to  the 
same  conformal  transformation — which  elliptic  functions  might  help 

*  See  H.  Hukueima,  Cracks  upon  the  glazed  surface  of  ceramic  wares,  Sci. 
Papers,  Inst,  of  Chem.  and  Phys.  Research,  Tokyo,  xxvii,  pp.  235-243.  1935. 


VII]  OF  THE  BEE'S  CELL  525 

us  to  define.  The  colour-pattern  of  the  bean  will  then  be  found 
following  the  direction  of  the  boundary-lines,  and  occupying  areas 
or  patches  corresponding  to  parts  of  the  orthogonal  system.  The 
lines  are  equipotential  hnes,  or  akin  thereto.  If  we  varnish  an 
elastic  bag,  dry  it  and  expand  it,  the  varnish  will  tend  to  crack  along 
the  same  orthogonal  boundaries*. 

The  bee^s  cell 

The  most  famous  of  all  hexagonal  conformations,  and  one  of  the 
most  beautiful,  is  the  bee's  cell.  As  in  the  basalt  or  the  coral,  we 
have  to  deal  with  an  assemblage  of  co-equal  cylinders,  of  circular 
section,  compressed  into  regular  hexagonal  prisms;  but  in  this  case 
we  have  two  layers  of  such  cyhnders  or  prisms,  one  facing  one 
way  and  one  the  other,  and  a  new  problem  arises  in  connection 
with  their  inner  ends.  We  may  suppose  the  original  cyhnders 
to  have  spherical  endsf,  which  is  their  normal  and  symmetrical  way 
of  terminating;  then,  for  closest  packing,  it  is  obvious  that  the  end 
of  any  one  cylinder  in  the  one  layer  will  touch,  and  fit  in  between, 
the  ends  of  three  cylinders  in  the  other.  It  is  just  as  when  we 
pile  round-shot  in  a  heap;  we  begin  with  three,  a  fourth  fits  into 
its  nest  iDetween  the  three  others,  and  the  four  form  a  ''tetrad," 
or  regular  tetragonal  arrangement. 

Just  as  it  was  obvious,  then,  that  by  mutual  pressure  from  the 
sides  of  six  adjacent  cells  any  one  cell  would  be  squeezed  into  a 
hexagonal  prism,  so  is  it  also  obvious  that,  by  mutual  pressure 
against  the  ends  of  three  opposite  neighbours,  the  end  of  each  and 
every  cell  will  be  compressed  into  a  trihedral  pyramid.  The  three 
sides  of  this  pyramid  are  set,  in.  plane  projection,  at  co-equal  angles 
of  120°  to  one  another;  but  the  three  apical  angles  (as  in  the 
analogous  case  already  described  of  a  system  of  soap-bubbles)  are, 

*  M.  Hirata,  Coloured  patches  in  kidney-beans,  Set.  Papers,  Inst.  Chem.  Researchy 
Tokyo,  XXVI,  pp.  122-135,  1936. 

f  In  the  combs  of  certain  tropical  bees  the  hexagona  structure  is  imperfect  and 
the  cells  are  not  far  removed  from  cylinders.  They  are  set  in  tiers,  not  contiguous 
but  separated  by  little  pillars  of  wax,  and  the  base  of  each  cell  is  a  portion  of 
a  sphere.  They  differ  from  the  ordinary  honeycomb  in  the  same  sort  of  way  as 
the  fasciculate  from  the  massive  corals,  of  which  we  spoke  on  p.  512.  Cf.  Leonard 
Martin,  Sur  les  Melipones  de  Bresil,  La  Nature,  1930,  pp.  97-100. 


526  THE  FORMS  OF  TISSUES  [ch. 

by  the  geometry  of  the  case*,  co-equal  angles  of  109°  and  so  many 
minutes  and  seconds. 

If  we  experiment,  not  with  cyHnders  but  with  spheres,  if  for 
instance  we  pile  bread-pills  together  and  then  submit  the  whole  to 
a  uniform  pressure,  as  we  shall  presently  find  that  Buffon  did :  each 
ball  (hke  the  seeds  in  a  pomegranate,  as  Kepler  said)  will  be  in  contact 
with  twelve  others — six  in  its  own  plane,  three  below  and  three  above, 
and  under  compression  it  will  develop  twelve  plane  surfaces. 
It  will  repeat,  above  and  below,  the  conditions  to  which  the  bee's 
cell  is  subject  at  one  end  only;  and,  since  the  sphere  is  symmetrically 
situated  towards  its  neighbours  on  all  sides,  it  follows  that  the 
twelve  plane  sides  to  which  its  surface  has  been  reduced  will  be  all 
similar,  equal  and  similarly  situated.  Moreover,  since  we  have 
produced  this  result  by  squeezing  our  original  spheres  close  together,  it 
is  evident  that  the  bodies  so  formed  completely  fill  space.  The  regular 
solid  which  fulfils  all  these  conditions  is  the  rhombic  dodecahedron. 
The  bee's  cell  is  this  figure  incompletely  formed;  it  represents,  so 
to  speak,  one-half  of  that  figure,  with  its  apex  and  the  six 
adjacent  corners  proper  to  the  rhombic  dodecahedron,  but  six  sides 
continued,  as  a  hexagonal  prism,  to  an  open  or  unfinished  endf. 

The  bee's  comb  is  vertical  and  the  cells  nearly  horizontal,  but 
sloping  slightly  downwards  from  mouth  to  floor;  in  each  prismatic 
cell  two  sides  stand  vertically,  and  two  corners  lie  above  and  below. 
Thus  for  every  honeycomb  or  "section"  of  honey,  there  is  one  and 
only  one  "right  way  up";  and  the  work  of  the  hive  is  so  far  con- 
trolled by  gravity.  Wasps  build  the  other  way,  with  the  cells  upright 
and  the  combs  horizontal;  in  a  hornet's  nest,  or  in  that  of  Polistes, 
the  cells  stand  upright  like  the  wasp's,  but  their  mouths  look  down- 
wards in  the  hornet's  nest  arid  upwards  in  the  wasp's. 

What  Jeremy  Taylor  called  "the  discipline  of  bees  and  the  rare 
fabric  of  honeycombs  "  must  have  attracted  the  attention  and  excited 
the  admiration  of  mathematicians  from  time  immemorial.  "Ma 
maison  est  construite,"  says  the  bee  in  the  Arabian  Nights,  "selon 

*  The  dihedral  angle  of  120°  is,  physically  speaking,  the  essential  thing;  the 
Maraldi  angle,  of  109°,  etc.,  is  a  geometrical  consequence.  Cf.  G.  Cesaro,  Sur  la 
forme  de  Falveole  de  I'abeille,  Bull.  Acad.  R.  Belgique  (Sci.),  1920,  p.  100. 

t  See  especially  Haiiy,  the  crystallographer;  Sur  le  rapport  des  figures  qui 
existe  entre  Falveole  des  abeilles  et  le  grenat  dodecaedre,  Journ.  d'hist.  naturelle, 
n,  p.  47,  1792. 


VII]  OF  THE  BEE'S  CELL  527 

lesloisd'une  severe  architecture;  et  Euclidos  lui-meme  s'instruirait 
en  admirant  la  geometrie  de  ses  alveoles*."  Ausonius  speaks  of 
the  geometrica  forma  favorum,,  and  Pliny  tells  of  men  who  gave  a 
lifetime  to  its  study. 

Pappus  the  Alexandrine  has  left  us  an  account  of  its  hexagonal 
plan,  and  drew  from  it  the  conclusion  that  the  bees  were  endowed 
Vith  "a  certain  geometrical  forethought "f-  "There  being,  then, 
three  figures  which  of  themselves  can  fill  up  the  space  round  a  point, 
viz.  the  triangle,  the  square  and  the  hexagon,  the  bees  have  wisely 
selected  for  their  structure  that  which  contains  most  angles,  sus- 
pecting indeed  that  it  could  hold  more  honev  than  either  of  the 


•Y^^v;v^<"r 


Fig.  204.     Portion  of  a  honeycomb.     After  WiJlem. 

Other  two  J."  Erasmus  Bartholin  was  apparently  the  first  to 
suggest  that  the  hypothesis  of  '"economy"  was  not  warranted,  and 
that  the  hexagonal  cell  was  no  more  than  the  necessary  result  of 
equal  pressures,  each  bee  striving  to  make  its  own  little  circle  as 
large  as  possible. 

The  investigation  of  the  ends  of  the  cell  was  a  more  difficult 
matter  than  that  of  its  sides,  and  came  later,  in  general  terms  the 
arrangement  was  doubtless  often  studied  and  described:    as  for 

*  Ed.  Mardrus,  xv,  p.  173. 

t  (pvaLK7]v  ye(i}/j.eTpiKi]u  Trpbvoiav.  Pappus,  Bk.  V;  cf.  Heath,  Hist,  of  Gk.  Math,  ii, 
p.  589.  St  Basil  discusses  ttiv  yewixerplav  t^s  cro<pu}Ta.TTj<>  fxeXlaoTjs:  Hexaem.  vni, 
p.  172  (Migne);  Virgil  speaks  of  the  pars  divinae  mentis  of  the  bee,  and  Kepler 
found  the  bees  anxmd  praedifas  et  geomet.  riae  suo  modo  capaces. 

X  This  was  according  to  the  "theorem  of  Zenodorus."  The  use  by  Pappus  of 
"economy"  as  a  guiding  principle  is  remarkable.  For  it  means  that,  like  Hero 
with  his  mirrors,  he  had  a  pretty  clear  adumbration  of  that  principle  of  minima, 
which  culminated  in  the  principle  of  least  action,  which  guided  eighteenth-century 
physics,  was  generalised  (after  Fermat)  by  Lagrange,  inspired  Hamilton  and 
Maxwell,  and  reappears  in  the  latest  developments  of  wave-mechanics. 


528  THE  FORMS  OF  TISSUES  [ch. 

instance,  in  the  Garden  of  Cyrus:  "And  the  Combes  themselves 
so  regularly  contrived  that  their  mutual  intersections  make  three 
Lozenges  at  the  bottom  of  every  Cell*  which  severally  regarded 
make  three  Rows  of  neat  Rhomboidall  Figures,  connected  at  the 
angles,  and  so  continue  three  several  chains  throughout  the  whole 
comb  "  Or  as  Reaumur  put  it,  a  little  later  on:  "trois  cellules 
accolees  laissent  un  vuide  pyramidal,  precisement  semblable  a  celui 
de  la  base  d'une  autre  cellule  tourn^e  en  sens  contraire." 

Kepler  had  deduced  from  the  space-filling  symmetry  of  the  honey- 
comb that  its  angles  must  be  those  of  the  rhombic  dodecahedron; 
and  Swammerdam  also  recognised  the  same  geometrical  figure  in 
the  base  of  the  cell*.  But  Kepler's  discovery  passed  unnoticed, 
and  Maraldi  the  astronomer,  Cassini's  nephew,  has  the  credit  of 
ascertaining  for  the  first  time  the  shape  of  the  rhombs  and  of  the 
sohd  angle  which  they  bound,  while  watching  the  bees  in  "les  ruches 
vitrees  dans  le  jardin  de  M.  Cassini  attenant  I'Observatoire  de 
Paris -f."  The  angles  of  the  rhomb,  he  tells  us,  are  110°  and  70°: 
"Chaque  base  d'alveole  est  formee  de  trois  rombes  presque  toujours 
egaux  et  semblables,  qui,  suivant  les  mesures  que  nous  avons  2^rises, 
ont  les  deux  angles  obtus  chacun  de  110  degres,  et  par  consequent 
les  deux  aigus  chacun  de  70  degres."  Further  on  (p.  312),  he 
observes  that  on  the  magnitude  of  the  angles  of  the  three  rhombs 
at  the  base  of  the  cell  depends  that  of  the  basal  angles  of  the  six 
trapezia  which  form  its  sides;  and  it  occurs  to  him  to  ask  what 
must  these  angles  be,  if  those  of  the  floor  and  those  of  the  sides  be 
equal  one  to  another.  The  solution  of  this  problem  is  that  "les 
angles  aigus  des  rombes  etant  de  70  degres  32  minutes,  et  les  obtus 
de  109  degres  28  minutes,  ceux  des  trapezes  qui  leur  sont  contigus 
doivent  etre  aussi  de  la  meme  grandeur."  And  lastly:  "II  resulte 
de  cette  grandeur  d' angle  non  seulement  une  plus  grande  facilite  et 
simpHcite  dans  la  construction,  a  cause  que  par  cette  maniere  les 
abeilles  n'employent  que  deux  sortes  d'angles,  mais  il  en  resulte 
encore  une  plus  belle  simetrie  dans  la  disposition  et  dans  la  figure 

*  Kepleri  Opera  omnia,  ed.  Fritsch,  v,  pp.  115,  122,  178,  vii,  p.  719,  1864; 
Swammerdam,  Tractatus  de  apibus  (observations  made  in  1673). 

t  Obs.  sur  les  abeilles,  Mem.  Acad.  R.  Sciences  (1712),  1731,  pp.  297-331. 
Sir C.  Wren  had  used  "transparent  bee-hives"  long  before;  see  his  Letter  concerning 
that  pleasant  and  profitable  invention,  etc.,  in  S.  Hartlib's  Reformed  Common- 
Wealth  of  Bees,  1655. 


VII]  OF  THE  BEE'S  CELL  529 

de  r Alveole."     In  short,  Maraldi  takes  the  two  principles  of  sim- 
phcity  and  mathematical  beauty  as  his  sure  and  sufficient  guides. 

The  next  step  was  that  which  had  been  foreshadowed  long  before 
by  Pappus.  Though  Euler  had  not  yet  published  his  famous 
dissertation  on  curves  maximi  minimive  proprietate  gaudentes,  the 
idea  of  maxima  and  minima  was  in  the  air  as  a  guiding  postulate, 
an  heuristic  method,  to  be  used  as  Maraldi  had  used  his  principle 
of  simphcity.  So  it  occurred  to  Reaumur,  as  apparently  it  had  not 
done  to  Maraldi,  that  a  minimal  configuration,  and  consequent 
economy  of  material  in  the  waxen  walls  of  the  cell,  might  be  at 
the  root  of  the  matter:  and  that,  just  as  the  close-packed  hexagons 
gave  the  minimal  extent  of  boundary  in  a  plane,  so  the  figure  deter- 
mined by  Maraldi,  namely  the  rhombic  dodecahedron,  might  be 
that  which  employs  the  minimum  of  surface  for  a  given  content: 
or  which,  in  other  words,  should  hold  the  most  honey  for'  the  least 
wax.  "Convaincu  que  les  abeilles  employent  le  fond  pyramidal  qui 
merite  d'etre  prefere,  j'ai  soup9onne  que  la  raison,  ou  une  des 
raisons,  qui  les  avoit  decidees  etait  I'epargne  de  la  cire;  qu'entre 
les  cellules  de  meme  capacite  et  a  fond  pyramidal,  celle  qui  pouvait 
etre  faite  avec  moins  de  matiere  ou  de  cire  etoit  celle  dont  chaque 
rhombe  avoit  deux  angles  chacun  d'environ  110  degres,  et  deux 
chacun  d'environ  70°."  He  set  the  problem  to  Samuel  Koenig, 
a  young  Swiss  mathematician:  Given  an  hexagonal  cell  terminated 
by  three  similar  and  equal  rhombs,  what  is  the  configuration  which 
requires  the  least  quantity  of  material  for  its  construction?  Koenig 
confirmed  Reaumur's  conjecture,  and  gave  109°  26'  and  70°  34'  as 
the  angles  which  should  fulfil  the  condition;  and  Reaumur  then 
sent  him  the  Memoires  de  I'Academie  for  1712,  where  Koenig  was 
"agreeably  surprised"  to  find:  "que  les  rombes  que  sa  solution 
avait  determine,  avait  a  deux  minutes  pres*  les  angles  que 
M.  Maraldi  avait  trouves  par  des  mesures  actuelles  a  chaque  rhombe 
des  cellules  d'abeilles. . . .  Un  tel  accord  entre  la  solution  et  les 
mesures  actuelles  a  assurement  de  quoi  surprendre."  Koenig 
asserted  that  the  bees  had  solved  a  problem  beyond  the  reach  of 

*  The  discrepancy  was  due  to  a  mistake  of  Koenig's,  doubtless  misled  by  his 
tables,  in  the  determination  of  V2;  but  Koenig's  own  paper,  sent  to  Reaumur, 
remained  unpublished  and  his  method  of  working  is  unknown.  An  abridged 
notice  appears  in  the  Mdm.  de  VAcad.  1739,  pp.  30-35. 


530  THE  FORMS  OF  TISSUES  [ch. 

the  old  geometry  and  requiring  the  methods  of  Newton  and  Leibniz. 
Whereupon  Fontenelle,  as  Secretaire  Perpetuel,  summed  up  the 
case  in  a  famous  judgment,  in  which  he  denied  intelhgence  to  the 
bees  but  nevertheless  found  them  blindly  using  the  highest  mathe- 
matics by  divine  guidance  and  command*. 

When  CoHn  Maclaurin  studied  the  honeycomb  in  Edinburgh,  a 
few  years  after  Maraldi  in  Paris,  he  proceeded  to  solve  the  problem 
without  using  "any  higher  Geometry  than  was  known  to  the 
Antients,"  and  he  began  his  account  by  saying:  "These  bases  are 
fornled  from  Three  equal  Rhombus's,  the  obtuse  angles  of  which 
are  found  to  be  the  doubles  of  an  Angle  that  often  offers  itself  to 
mathematicians  in  Questions  relating  to  Maxima  and  Minima. f"  It 
was  an  angle  of  109°  28'  16",  with  its  supplement  of  70°  31'  44". 
And  this  angle  of  the  bee's  cell  determined  by  Maraldi,  Koenig  and 
Maclaurin  in  their  several  ways,  this  angle  which  has  for  its  cosine  1/3 
and  is  double  of  the  angle  which  has  for  its  tangent  V2,  is  on  the 
one  hand  an  angle  of  the  rhombic  dodecahedron,  and  on  the  other 
is  that  very  angle  of  simple  tetrahedral  symmetry  which  the  soap- 
films  within  the  tetrahedral  cage  spontaneously  assume,  and  whose 
frequent  appearance  and  wide  importance  we  havo  already  touched 
upon  J. 

That  "the  true  theoretical  angles  were  109°  28'  and  70°  32', 
precisely  corresponding  ivith  the  actual  measurement  of  the  bee's  cell,'' 
and  that  the  bees  had  been  "proved  to  be  right  and  the  mathema- 
ticians   wrong,"   was    long   believed   by    many.     Lord   Brougham 

*  La  grande  merveille  est  que  la  determination  de  ces  angles  passe  de  beaucoup 
les  forces  de  la  Geometrie  commune,  et  n'appartient  qu'aux  nouvelles  Methodes 
fondees  sur  la  Theorie  de  I'lnfini.  Mais  a  la  fin  les  Abeilles  en  S9auraient  trop, 
et  I'exces  de  leur  gloire  en  est  la  ruine.  II  faut  remonter  jusqu'a  une  Intelligence 
infinie,  qui  les  fait  agir  aveuglemerit  sous  ses  ordres,  sans  leur  accorder  de  ces 
lumieres  capables  de  s'accroitre  et  de  se  fortifier  par  elles-memes,  qui  font  I'honneur 
de  notre  Raison."     Histolre  de  VAcademie  Royale,  1739,  p.  3.j. 

t  Colin  Maclaurin,  On  the  bases  of  the  cells  wherein  the  bees  deposit  their 
honey,  Phil.  Trans,  xlii,  pp.  561-571,  1743;  also  in  the  Abridgement,  viii,  pp.  709- 
713,  1809;  it  was  characteristic  of  Maclaurin  to  use  geometrical  methods  for 
wellnigh  everything,  even  in  his  book  on  Fluxions,  or  in  his  famous  essay  on  the 
equilibrium  of  spinning  planets.  Cf.  also  Lhuiller,  Memoire  sur  le  minimum  du 
cire  des  alveoles  des  Abeilles,  et  en  particulier  sur  un  minimum  minimorum  relatif  a 
cette  matiere,  Nouv.  Mem.  de  VAcad.  de  Berlin,  1781  (1783),  pp.  277-300.  Cf. 
Castillon,  ibid,  (commenting  on  Lhuiller);  also  Ettore  Carruccio,  Notizie  storiche 
eulla  geometria  delle  api,  Periodica  di  Mathematiche,  (4)  xvi,  pp.  35-54,  1936. 

X  Supra,  p.  497.     The  faces  of  a  regular  octahedron  meet  at  the  same  angle. 


VII 


OF  THE  BEE'S  CELL  531 


helped  notably  to  spread  these  and  other  errors,  and  his  writings 
on  the  bee's  cell  contain,  according  to  Glaisher,  "as  striking 
examples  of  bad  reasoning  as  are  often  to  be  met  with  in  writings 
relating  to  mathematical  subjects."  The  fact  is  that,  were  the 
angles  and  facets  of  the  honevcomb  as  sharp  and  smooth,  and  as 
constant  and  uniform,  as  those  of  a  quartz-crystal,  it  would  still  be 
a  dehcate  matter  to  measure  the  angles  within  a  minute  or  two  of 
arc,  and  a  technique  unknown  in  Maraldi's  day  would  be  required 
to  do  it.  The  minute-hand  of  a  clock  (if  it  move  continuously) 
moves  through  one  degree  of  arc  in  ten  seconds  of  time,  and  through 
an  angle  of  two  minutes  m  one-third  of  a  second; — and  this  last  is 
the  angle  which  Maraldi  is  supposed  to  have  measured.  It  was 
eighty  years  after  Maraldi  had  told  Reaumur  what  the  angle  was 
that  Boscovich  pointed  out  for  the  first  time  that  to  ascertain  the 
angle  to  the  nearest  minute  by  direct  admeasurement  of  the  waxen 
cell  was  utterly  impossible.  Yet  Reaumur  had  certainly  beheved, 
and  apparently  had  persuaded  Koenig,  that  Maraldi's  determina- 
tions, first  and  last,  were  the  result  of  measurement ;  and  Fontenelle, 
the  historian  of  the  Academy,  epitomising  Koenig's  paper,  speaks 
of  "les  mesures  actuelles  de  M.  Maraldi,"  of  the  bees  being  in 
error  to  the  trifling  extent  of  2',  and  of  the  grande  merveille  of  their 
so  nearly  solving  a  problem  belonging  to  the  higher  geometry. 
Boscovich,  in  a  long-forgotten  note,  rediscovered  by  Glaisher,  puts 
the  case  in  a  nutshell:  "Mirum  sane  si  Maraldus  ex  observatione 
angulum  aestimasset  intra  minuta,  quod  in  tarn  exigua  mole  fieri 
utique  non  poterat.  At  is  (ut  satis  patet  ex  ipsa  ejus  determina- 
tione)  affirmat  se  invenisse  angulos  circiter  110°  et  70°,  nee  minuta 
emit  ex  observatione  sed  ex  equahtate  angulorum  pertinentium  ad 
rhombos  et  ad  trapezia ;  ad  quam  habendam  Geometria  ipsa  docuit 
requiri  ilia  minuta*."  Indeed  he  goes  on  to  say  the  wonder  is  that 
the  angles  could  be  measured  even  within  a  few  degrees,  variable 
and  irregular  as  they  are  seen  to  be,  and  as  even  Reaumur  f  knew 

*  In  his  note  De  apium  cellulis,  appended  to  the  philosophical  poem  of  Benedict 
Stay,  II,  pp.  498-504,  Romae,  1792. 

t  Op.  cit.  V,  p.  382.  Several  authors  recognised  that  the  cells  are  far  from 
identical,  and  do  no  more  than  approximate  to  an  average  or  ideal  angle:  e.g. 
Swammerdam  in  the  Biblia  Naturae,  u,  p.  379;  G.  S.  Klugel,  Grosstes  u.  Kleinstes, 
in  Mathem.  Worterb.  1803,  Castillon,  op.  cit.;  and  especially  Jeffries  Wyman, 
Notes  on  the  cells  of  the  bee,  Proc.  Amer.  Acad.  Sci.  and  Arts,  vn,  1868. 


532 


THE  FORMS  OF  TISSUES 


[CH. 


well  they  were.  The  old  misunderstanding  was  at  last  explained 
and  corrected  by  Leslie  Ellis;  and  better  still  by  Glaislier,  in  a 
Uttle-known  but  very  beautiful  paper*.  For  these  two  mathe- 
maticians shewed  that,  though  Maraldi's  account  of  his  "measure- 
ments" led  to  misunderstanding,  yet  he  had  really  done  well  and 
scientifically  when  he  eked  out  a  rough  observation  by  finer  theory, 
and  deemed  himself  entitled  thereby  to  discuss  the  cell  and  its  angles 
in  the  same  precise  terms  that  he  would  use  as  a  mathematician  in 
speaking  of  its  geometrical  prototype  f. 

y  Many   diverse    proofs  {    have    been 

given  of  the  minimal  character  of  the 
bee's  cell,  some  few,  hke  Maclaurin's, 
purely  geometrical,  others  arrived  at  by 
help  of  the  calculus.  The  following 
seems  as  simple  as  any: 

ABCDEF,  abcdef,  is  a  right  prism 
upon  a  regular  hexagonal  base.  The 
corners  B,  D,  F  are  cut  oif  by  planes 
through  the  fines  AC,  CE,  EA,  meeting 
in  a  point  V  on  the  axis  VN  of  the 
prism,  and  intersecting  Bb,  Dd,  Ff,  in 
Z,  Y,  Z.  The  volume  of  the  figure  thus 
formed  is  the  same  as  that  of  the 
L  original  prism  with  its  hexagonal  ends : 
for,  if  the  axis  cut  the  hexagon  ABCDEF 
in  N,  the  volumes  ACVN,  ACBX  are 
equal. 
pjg  205.  It  is  required  to  find  the  inchnation 

to  the  axis  of  the  faces  forming  the 


/ 


L.eslie  Ellis,  On  the  form  of  bees'  cells,  in  Mathematical  and  other  Writings, 
1863,  p.  353;  J.  W.  L.  Glaisher,  do.,  Phil.  Mag.  (4),  xlvi,  pp.  103-122,  1873. 

t  The  learned  and  original  Kieser,  in  his  Memoire  sur  Vorganisation  des  plantes, 
1812,  p.  iv,  gives  advice  to  the  same  effect:  "II  est  indispensable  de  se  former, 
avant  de  dessiner,  une  idee  de  Tobjet  dans  sa  plus  grande  perfection,  et  de  dessiner 
selon  cette  idee,  et  non  pas  I'objet  plus  ou  moins  imparfait,  plus  ou  moins  altere 
par  le  scalpel..  Voila  la  methode  qu'ont  suivi  Haller,  Albinus  et  tous  les  autres 
grands  anatomistes. .  . .  Mais  il  faut  employer  pour  cela  la  plus  grande  precaution, 
la  circonspection  la  plus  tranquille  pour  I'observation,  etc." 

I  Cf.  Koenig,  Lhuiller  and  Boscovich,  opp.  cit. ;  H.  Hennessy,  Proc.  E.S.  xxxix, 
p.  253,  1885;   XLi,  pp.  4^2,  443,  1886;   XLii.  pp.  176,  177,  1887. 


VII]  OF  THE  BEE'S  CELL  533 

trihedral  angle  at  F,  such  that  the  surface  of  the  whole  figure  may 
be  a  minimum. 

Let  the  angle  NVX,  which  is  the  inclination  of  the  plane  of  the 
rhonlbus  to  the  axis  of  the  prism,  =  d\  the  side  of  the  hexagon,  as 
AB,  =  s\    and  the  height,  as  Aa,  =  h. 

Then  AC  =  25  cos  30°,  =  s  VS.  And,  from  inspection  of  the 
triangle  LXB, 

VX 


sin  6 
Therefore  the  area  of  the  rhombus 


VAXC 


V3 


2sin^* 


And  the  area  of       AabX  =  -{2h—  ^VX  cos  6), 

2i  " 


=  '  (2A  -  \s  cot  Q). 

z 

Therefore  the  total  area  of  the  figure 

=  the  hexagonal  base  ahcdef  +  3s  (2h  —  |s  cot  6)  +  3  —-. — -, 


(^(area)      Ss^  /     1         a/3  cos  ^ 

Therefore  — j^ —  =  -^    ~^~t^ .  o  n 

ad  2  Vsm^  d       sm^  d. 


\ 


^        ,  .  .  .  ,  d  (area)      ^      ,  ^1 

But  this  expression  vanishes,  or =  0,  when  cos  U  =  -— =, 

^  dd  V3 

that  is  to  say,  when        6  =  54°  44'  8"  =  }  (109°  28'  16"). 

Such  then  are  the  conditions  under  which  the  total  area  of  the 
figure  has  its  minimal  value. 

The  following  is,  in  substance,  Maclaurin's  elementary  but  somewhat  lengthy 
proof  of  the  minimal  properties  of  the  bee's  cell,  using  "no  higher  Geometry 
than  was  known  to  the  Antients." 

Let  ABCD,  abed,  represent  one-half  of  a  right  prism  on  a  regular  hexagonal 
base;  and  let  AabE,  EbcC  be  the  trapezial  portions  of  two  adjacent  sides,  to 
which  one  of  the  three  rhombs,  AECe,  is  fitted. 

Let  0  be  the  centre  of  the  hexagon,  of  which  AB,  EC  are  adjacent  sides; 
join  ^C  and  OB,  intersecting  in  P.     Then,  because  AOC  =  ABC,  and  BE  =  Oe, 


534 


THE  FORMS  OF  TISSUES 


[CH. 


the  solid  AECB  =  AeCO;  whence  it  appears  that  the  solid  content  of  the 
whole  cell  will  be  the  same,  wherever  the  point  E  be  taken  in  Bb,  and  will 
in  fact  be  identical  with  the  content  of  the  hexagonal  prism.  We  have -then 
to  enquire  where  E  is  to  be  taken  in  Bb,  in  order  that  the  combined  surfaces 
of  the  rhomb  and  of  the  two  trapezia  may  be  a  minimum. 

Because  Ee  is  perpendicular  to  ^C  in  P,  the  area  of  the  rhombus  =  P^.^C'; 
and  the  area  of  the  two  trapezia  =  {Aa  +  Eb)  x  BC.  The  total  area  in  question, 
then,  is  PE .AC  +  2BC .Bb-BC.  BE.  But  BC .  Bb  is  constant ;  so  the  question 
remains.  When  is  PE.AC  —  BC.BE  a  minimum? 

Let  a  point  L  be  so  taken  in  Bb  that  BL :  PL  ::  BC  :  AC.  From  the 
centre  P,  in  the  plane  PBE,  describe  the  circular  arc  ER,  meeting  PL  in  R; 
and  on  PL  let  fall  the  perpendiculars  ES,  BT. 


Fig.  206. 


The  triangles  LES,  LBT,  LPB  are  all  similar.     Therefore 
LS:LE::LT:LB::LB:LP, 
and  (by  hypothesis)  : :  BC  :  AC. 

Hence  {LT -LS)  :  (LB-LE)  ::  BC  :  AC, 

i.e.  ST:  BE::  BC:  AC. 

Therefore  ST.AC  =  BE.BC 

and  consequently,      PE  .AC -BE  .BC  =  PE  .AC -ST  .AC 

=  AC{PE-8T). 
But  PE^PR;  therefore  AC  {PR  -  ST)  ^  AC  {PT  +  RS). 


VII 


OF  THE  BEE'S  CELL 


535 


But  AC  and  FT  do  not  vaiy,  while  BS  varies  with  the  position  of  E. 
Accordingly,  AC{PT  +  RS),  or  PE. AC-BE. BC,  is  a  minimum  when  RS 
vanishes:   that  is  to  say,  when  E  coincides  with  L. 

"Therefore  A  LCI  is  the  Rhombus  of  the  most  advantageous  Form  in  respect 
of  Frugality,  when  BL  is  to  PL  as  BC  to  .46'." 

Again,  since  OB  =  BC,  and  OP  =  PB,  BC^  =  4PB\  and  PC^  =  3PB^,  and 
AC  =  2PC  =  2V3.PB. 


Therefore 
and,  by  hypothesis. 


and 

Therefore 


BC:AC::2PB:2VS.PB::  1  :  Vs, 


BLiPL: 

: BC : AC 

=  1 

Vs 

PLiPB:: 

VS  :  V2, 

PBiPC 

:  1  :  V3. 

PL:  PC: 

:  1  :  V2. 

"That  is,  the  angle  CLP  is  that  whose  Tangent  is  to  the  Radius  as  V2 
is  to  1,  or  as  1-4142135  to  1-0000000;  and  therefore  is  of  54°  44'  08",  and 
consequently  the  Angle  of  the  Rhombus  of  the  Best  Form  is  that  of 
109°  28'  16".". 

When  we  have  thus  ascertained  that  the  characteristic  angles  of 
the  rhombs  are  109°  28'  16"  and  its  supplement  70°  31'  44",  the 
cosine  of  which  latter  angle  is  1/3,  the  construction  of  a  model  is 
of  the  easiest. 


Fig.  207.     Construction  of  a  model  of  the  bee's  cell 

On  AD  make  AB  =  BC  =  CD.  Let  AF  =  AD  meet  the  perpen- 
dicular BE  in  F.  Then  the  angle  BAF  (whose  cosine  =  1/3,  or 
whose  tangent  =  2\/2)  =  70°  31'  44".  Complete  the  rhomb  ADGF, 
and  repeat  three  times  as  indicated.  Make  a  developed  hexagonal 
prism  with  sides  ab,  be,  =  BF .  Cut  away  angles  bb'a,  bb'c,  etc., 
=  BAF.     Fold,  and  attach  together. 


536  THE  FORMS  OF  TISSUES  [ch. 

A   soap-bubble,  or  soap-film,  assumes  a  minimal  configuratiou 

instantaneously*,  however  small  the  saving  of  surface-area  may  be. 

But  after  learning  that  the  bee's   cell  has  undoubted   minimal 

properties,  we  should  hke  to  know  what  saving  is  actually  obtained 

by  substituting  a  rhomboidal  pyramid  for  a  plane  base  in  the 

hexagonal  prism.     It  turns  out,  after  all,  to  be  a  small  matter! 

The  calculation  was  first  made  by  Maclaurin  and  by  Lhuiller,  in 

both  cases  briefly  but  correctly.     Lhuiller  stated  that  the  whole 

amount  used  in  the  bee's  cell  was  to  that  required  for  a  flat-topped 

prismatic  cell  of  equal  volume  as  25  +  V6  (or  2745)  to  28,  the 

saving  being  thus  a  little  more  than  2  per  cent,  of  the  whole  quantity 

of  wax  required  t-    Glaisher  recalculated  the  values,  taking  the  cell 

part  by  part.     Assuming,  with  Lhuiller,  that  the  radius  of  the 

inscribed  circle  of  the  hexagon  is  to  the  depth  of  the  prismatic  cell, 

when  the  latter  has  the  same  capacity  as  the  real  cell,  as  1|  to  5, 

then,  taking  the  side  of  the  hexagon  as  unity,  we  have  for  the  same 

depth  (viz.  the  longest  side  of  the  trapezium  in  the  real  cell)  the 

25  VS 
value ;  and  then  (to  three  places  of  decimals): 

Area  of  the  three  rhombs,  f  a/2 
„      „    „    six  triangles,  f  V2 
„      „    „    six  sides  of  the  equivalent 

prismatic  cell,  -^^  V  3 
„      „    „    hexagonal  base,  f  VS 
The  whole  surface  of  the  real  cell,  accordingly, 
=  (i)-f-(iii)-(ii)  =  23-772; 

*  For  the  most  part  instantaneously;  but  sometimes,  when  there  are  two 
positions  of  nearly  equal  potential  energy,  the  film  "creeps"  from  the  less  to  the 
more  advantageous  of  the  two. 

t  We  must  take  into  account  the  depth  of  the  cell,  or  assume  a  value  for  it, 
if  we  are  to  estimate  the  percentage  saving  of  wax  on  the  whole  construction. 
But  (as  Dr  G.  T.  Bennett  says)  the  whole  saving  is  on  the  roof,  and  the  height 
of  the  house  does  not  matter;  the  question  rather  is,  what  is  saved  on  the 
rhomboidal  sloping  roof  compared  with  a  flat  one?  If  the  short  axis  of  the  rhombs 
be  2  units  (the  edge  of  the  cube),  then  3  rhombs  have  area  6  V2,  the  wall-saving 
is  2  V2,  while  the  flat  hexagonal  top  is  4  Vs.  So  the  actual  saving  is  the  diflFerence 
between  4  V2  and  4  Vs — which  looks  much  less  negligible !  But  it  is  only  on 
a  small  portion  of  the  work. 


3-182 

(i)- 

1-061 

(ii). 

21-651 

(iii). 

2-598 

(iv). 

VII]  OF  THE  BEE'S  CELL  537 

and  that  of  the  flat-bottomed  hexagonal  prism 

=  (iii)  +  (iv)  =  24-249; 

,  24-249      102  ,.^^. 

23772  ""  100 '  ^^  ^  ^^^^ ' 

so  that  the  saving  in  the  former  case  amounts  to  about  2  per  cent., 
as  Lhuiller  had  found  it  to  be. 

Glaisher  sums  up  the  matter  as  follows:  "As  the  result  of  a 
tolerably  careful  examination  of  the  whole  question,  I  may  be 
permitted  to  say  that  I  agree  with  Lhuiller  in  beheving  that  the 
economy  of  wax  has  played  a  very  subordinate  part  in  the  deter- 
mination of  the  form  of  the  cell ;  in  fact  I  should  not  be  surprised 
if  it  were  acknowledged  hereafter  that  the  form  of  the  cell  had 
been  determined  by  other  considerations,  into  which  the  saving  of 
wax  did  not  enter  (that  is  to  say  did  not  enter  sensibly;  of  course 
I  do  not  mean  that  the  amount  of  wax  required  was  a  matter  of 
absolute  indifference  to  the  bees).  The  fact  of  all  the  dihedral 
angles  being  120°  is,  it  is  not  unlikely,  the  cause  that  determined 
the  form  of  the  cell."  This  last  fact,  that  in  such  a  cell  every  plane 
cuts  every  other  plane  at  an  angle  of  120°,  was  known  both  to  Kliigel 
and  to  Boscovich;  it  is  no  mere  corollary,  but  the  root  of  the 
matter.  It  is,  as  Glaisher  indicates,  the  fundamental  physical 
principle  of  construction  from  which  the  apical  angles  of  109°  follow 
as  a  geometrical  corollary.  And  it  is  curious  indeed  to  see  how 
the  obtuse  angle  of  the  rhomb,  and  its  cosine  —J,  drew  attention 
all  the  while;  but  the  dihedral  angle  of  120°  of  the  rhombohedron, 
and  the  inclination  of  its  three  short  diagonals  at  90°  to  one  another, 
got  rare  and  scanty  notice. 

Darwin  had  hstened  too  closely  to  Brougham  and  the  rest  when 
he  spoke  of  the  bee's  architecture  as  "the  most  wonderful  of  known 
instincts";  and  when  he  declared  that  "beyond  this  stage  of  per- 
fection in  architecture  natural  selection  could  not  lead ;  for  the 
comb  of  the  hive-bee,  as  far  as  we  can  see,  is  absolutely  perfect  in 
economising  labour  and  wax." 

The  minimal  properties  of  the  cell  and  all  the  geometrical  reasoning 
in  the  case  postulate  cell -walls  of  uniform  tenuity  and  edges  which 
are  mathematically  straight.  But  the  walls,  and  still  more  their 
edges,  are  always  thickened ;  the  edges  are  never  accurately  straight, 


538 


THE  FORMS  OF  TISSUES 


[CH. 


nor  the  cells  strictly  horizontal.  The  base  is  always  thicker  than 
the  side- walls;  ,its  solid  angles  are  by  no  means  sharp,  but  filled  up 
with  curving  surfaces  of  wax,  after  the  fashion,  but  more  coarsely, 
of  Plateau's  bourrelet.  Hence  the  Maraldi  angle  is  seldom  or  never 
attained;  the  mean  value  (according  to  Vogt)  is  no  more  than 
106-7°  for  the  workers,  and  107-3°  for  the  drones.  The  hexagonal 
angles  of  the  prism  are  fairly  constant;  about  4°  is  the  limit  of 
departure,  and  about  1-8°  the  mean  error,  on  either  side. 


r^-t^ 


Fig.  208.     Brood-comb,  with  eggs. 

The  bee  makes  no  economies;  and  whatever  economies  lie  in  the 
theoretical  construction,  the  bee's  handiwork  is  not  fine  nor  accurate 
enough  to  take  advantage  of  them*. 

The  cells  vary  little  in  size,  so  little  that  Thevenot,  a  friend  of 
Swammerdam's,  suggested  using  their  dimensions  as  a  modulus 
or  standard  of  length;  but  after  all,  the  constancy  is  not  so  great 
as  has  been  supposed.  Swammerdam  gives  measurements  which 
work  out  at  5-15  mm.  for  the  mean  diameter  of  the  worker-cells, 
and  7  mm.  for  those  of  the  drones ;  Jeffries  Wyman  found  mean 
values  for  the  worker-cells  from  5-1  to  5-2  mm.;  Vogt,  after  many 
careful  measurements,  found  a  mean  of  5-37  mm.  for  the  worker- 

*  All  this  Heinrich  Vogt  has  abundantly  shewn,  in  part  by  making  casts  of  the 
interior  of  the  cells,  as  Castellan  had  done  a  hundred  years  before.  See  his 
admirable  paper  on  the  Geometric  und  Oekonomie  der  Bienenzelle,  in  Festschrift 
d.  Universitdt  Breslau,  1911,  pp.  27-274. 


VII 


OF  THE  BEE'S  CELL  539 


cells,  with  an  insignificant  difference  in  the  various  diameters,  and 
■a  mean  of  6-9  for  the  drone-cells,  with  their  horizontal  diameter  some- 
what in  excess,  and  averaging  7-1  mm.  A  curious  attempt  has  been 
made  of  late  years  by  Italian  bee-keepers  to  let  the  bees  work  on 
a  larger  foundation,  and  so  induce  them  to  build  larger  cells;  and 
some,  but  by  no  means  all,  assert  that  the  young  bees  reared  in  the 
larger  cells  are  themselves  of  larger  stature*. 

That  the  beautiful  regularity  of  the  bee's  architecture  is  due  to 
some  automatic  play  of  the  physical  forces,  and  that  it  were 
fantastic  to  assume  (with  Pappus  and  Reaumur)  that  the  bee 
intentionally  seeks  for  a  method  of  economising  wax,  is  certain; 
but  the  precise  manner  of  this  automatic  action  is  not  so  clear. 
When  the  hive-bee  builds  a  solitary  cell,  or  a  small  cluster  of  cells, 
as  it  does  for  those  eggs  which  are  to  develop  into  queens,  it  makes 
but  a  rude  construction.  The  queen-cells  are  lumps  of  coarse  wax 
hollowed  out  and  roughly  bitten  into  shape,  bearing  the  marks  of 
the  bee's  jaws  like  the  marks  of  a  blunt  adze  on  a  rough-hewn  log. 

Omitting  the  simplest  of  all  cases,  when  (among  some  humble- 
bees)  the  old  cocoons  are  used  to  hold  honey,  the  cells  built  by  the 
"solitary"  wasps  and  bees  are  of  various  kinds.  They  may  be 
formed  by  partitioning  off  little  chambers  in  a  hollow  stem;  they 
may  be  rounded  or  oval  capsules,  often  very  neatly  constructed 
out  of  mud  or  vegetable  fibre  or  httle  stones,  agglutinated  together 
with  a  salivary  glue;  but  they  shew,  except  for  their  rounded  or 
tubular  form,  no  mathematical  symmetry.  The  social  wasps  and 
many  bees  build,  usually  out  of  vegetable  matter  chewed  into  a 
paste  with  saliva,  very  beautiful  nests  of  '"combs";  and  the  close- 
set  papery  cells  which  constitute  these  combs  are  just  as  regularly 
hexagonal  as  are  the  waxen  cells  of  the  hive-bee.  But  in  these  cases 
(or  nearly  all  of  them)  the  cells  are  in  a  single  row ;  their  sides  are 
regularly  hexagonal,  but  their  ends,  for  want  of  opponent  forces, 
remain  simply  spherical. 

In  Melipona  domestica  (of  which  Darwin  epitomises  Pierre  Huber's 
description)  "the  large  waxen  honey-cells  are  nearly  spherical, 
nearly  equal  in  size,  and  are  aggregated  into  an  irregular  mass." 

*  Cf.  (int.  al.)  H.  Gontarsi,  >Sammelleistungen  von  Bienen  aus  vergrosserten 
Brutzellen,  Arch.f.  Bienenkunde,  xvi,  p.  7,  1935;  A.  Ghetti,  Celli  ed  api  piu  grandi, 
IV  Congresso  nazion.  della  S.A.I.  1935. 


540 


THE  FORMS  OF  TISSUES 


[CH. 


But  the  spherical  form  is  only  seen  on  the  outside  of  the  mass; 
for  inwardly  each  cell  is  flattened  into  "two,  three  or  m.ore  flat 
surfaces,  according  as  the  cell  adjoins  two,  three  or  more  other  cells. 
When  one  cell  rests  on  three  other  cells,  which  from  the  spheres 


Fig.  209.     An  early  stage  of  a  wasp's  nest.     Observe  the  spherical  caps,  and  the 
irregular  shape  of  the  peripheral  cells.     After  R.  Bott. 

being  nearly  of  the  same  size  is  very  frequently  and  necessarily  the 
case,  the  three  flat  surfaces  are  united  into  a  pyramid;  and  this 
pyramid,  as  Huber  has  remarked,  is  manifestly  a  gross  imitation 
of  the  three-sided  pyramidal  base  of  the  cell  of  the  hive-bee*." 


*  Origin  of  Species,  ch.  viii  (6th  ed.,  p.  221).  The  cells  of  various  bees,  humble- 
bees  and  social  wasps  have  been  described  and  mathematically  investigated  by 
K.  MiillenhofF,  Pfiuger's  Archiv,  xxxii,  p.  589,  1883;  but  his  many  interesting 
results  are  too  complex  to  .epitomise.  For  figures  of  various  nests  and  combs  see 
(e.g.)  von  Buttel-Reepen,  Biol.  Centralbl.  xxxm,  pp.. 4,  89,  129,  183,  1903. 


VII]  OF  THE  BEE'S  CELL  541 

We  had  better  be  content  to  say  that  it  depends  on  the  same 
elementary  geometry. 

The  question  is,  To  what  particular  force  are  we  to  ascribe  the 
plane  surfaces  and  definite  angles  which  define  the  sides  of  the  cell 
in  all  these  cases,  and  the  ends  of  the  cell  in  cases  where  one  row 
meets  and  opposes  another?  We  have  seen  that  Bartholin  sug- 
gested, and  it  is  still  commonly  beheved,  that  this  result  is  due  to 
mere  physical  pressure,  each  bee  enlarging  as  much  as  it  can  the 
cell  which  it  is  a-building-,  and  nudging  its  wall  outwards  till  it  fills 
every  intervening  gap,  and  presses  hard  against  the  similar  efforts 
of  its  neighbour  in  the  cell  next  door*. 

That  the  bee,  if  left  to  itself,  "works  in  segments  of  circles,"  or 
in  other  words  builds  a  rounded  and  roughly  spherical  cell,  is  ah 
old  contention!  which  some  recent  experiments  of  M.  Victor 
Willem  amplify  and  confirm  {.  M.  Willem  describes  vividly  how 
each  cell  begins  as  a  Httle  hemispherical  basin  or  "cuvette,"  how 
the  workers  proceed  at  first  with  Httle  apparent  order  and  method, 
laying  on  the  wax  roughly  like  the  mud  when  a  swallow  builds; 
how  presently  they  concentrate  their  toil,  each  burying  its  head  in 
its  own  cuvette,  and  slowly  scraping,  smoothing  and  ramming 
home;    how  those  on  the  other  side  gradually  adjust  themselves 

*  Darwin  had  a  somewhat  similar  idea,  though  he  allowed  more  play  to  the 
bee's  instinct  or  conscious  intention.  Thus,  when  he  noticed  certain  half- completed 
cell-walls  to  be  concave  on  one  side  and  convex  on  the  other,  but  to  become  perfectly 
flat  when  restored  for  a  short  time  to  the  hive,  he  says:  "It  was  absolutely  im- 
possible, from  the  extreme  thinness  of  the  little  plate,  that  they  could  have  effected 
this  by  gnawing  away  the  convex  side;  and  I  suspect  that  the  bees  in  such  cases 
stand  on  opposite  sides  and  push  and  bend  the  ductile  and  warm  wax  (which  as 
I  have  tried  is  easily  done)  into  its  proper  intermediate  plane,  and  thus  flatten  it." 
Ruber  thought  the  difference  in  form  between  the  inner  and  the  outer  cells  a  clear 
proof  of  intelhgence;  it  is  really  a  direct  proof  of  the  contrary.  And  while  cells 
differ  when  their  situations  and  circumstances  differ,  yet  over  great  stretches  of 
comb  extreme  uniformity,  unbroken  by  any  sign  of  individual  differences,  is  the 
strikingly  mechanical  characteristic  of  the  cells. 

I  It  is  so  stated  in  the  Penny  Cyclopedia,  1835,  Art.  "Bees";  and  is  expounded 
by  Mr  G.  H.  Waterhouse  [Trans.  Entom.  Soc,  London,  ii,  p.  115,  1864)  in  an 
article  of  which  Darwin  made  good  use.  Waterhouse  shewed  that  when  the 
bees  were  given  a  plate  of  wax,  the  separate  excavations  they  made  therein 
remained  hemispherical,  or  were  built  up  into  cylindrical  tubes;  but  cells  in 
juxtaposition  with  one  another  had  their  party-walls  flattened,  and  their  forms 
more  or  less  prismatic. 

X  V^ictor  Willem,  L'architecture  des  abeilles,  Bull.  Acad.  Roy.  de  Belgique  (5), 
XIV,  pp.  672-705,  1928. 


542  THE  FORMS  OF  TISSUES  [ch. 

to  their  opposite  neighbours ;  and  how  the  rounded  ends  of  the  cells 
fashion  themselves  into  the  rhomboidal* pyramids,  "a  la  suite  de 
I'amincissement  progressif  des  cloisons  communes,  et  des  pressions 
antagonistes  exercees  sur  les  deux  faces  de  ces  cloisons." 

Among  other  curious  and  instructive  observations,  M.  Willem  has 
watched  the  bees  at  work  on  the  waxen  "foundations"  now  com- 
monly used,  on  which  a  rhomboidal  pattern  is  impressed  with  a 
view  to  starting  the  work  and  saving  the  labour  of  the  bees.  The 
bees  (he  says)  disdain  these  half-laid  foundations  of  their  cells;  they 
hollow  out  the  wax,  erase  the  rhombs,  and  turn  the  pyramidal 
hollows  into  hemispherical  "cuvettes"  in  their  usual  way;  and  the 
vertical  walls  which  they  raise,  more  or  less  on  the  lines  laid  down 
for  them,  are  not  hexagonal  but  cylindrical  to  begin  with.  "La 
forme  plane,  en  facettes,  tant  de  prismes  que  des  fonds,  n'est  obtenue 
que  plus  tard,  progressivement,  comme  resultat  de  retouches, 
d'enlevements  et  de  pressions  exercees  sur  les  cloisons  qui  s'amin- 
cissent,  par  des  groupes  d'ouvrieres  operant  face  a  face,  de  maniere 
antagoniste." 

But  when  all  is  said  and  done,  it  is  doubtful  whether  such 
retouches,  enlevements  and  pressions  antagonistes,  such  mechanical 
forces  intermittently  exercised,  could  produce  the  nearly  smooth 
surfaces,  the  all  but  constant  angles  and  the  close  approach  to  a 
minimal  configuration  which  characterise  the  cell,  whether  it  be 
constructed  by  the  bee  of  wax  or  by  the  wasp  of  papery  pulp. 
We  have  the  properties  of  the  material  to  consider;  and  it  seems 
much  more  Ukely  to  me  that  we  have  to  do  with  a  true  tension 
effect:  in  other  words,  that  the  walls  assume  their  configuration 
when  in  a  semi-fluid  state,  while  the  watery  pulp  is  still  liquid  or 
the  wax  warm  under  the  high  temperature  of  the  crowded  hive. 
In  the  first  few  cells  of  a  wasp's  comb,  long  before  crowding  and 
mutual  pressure  come  into  play,  we  recognise  the  identical  con- 
figurations which  we  have  seen  exhibited  by  a  group  of  three  or 
four  soap-bubbles,  the  first  three  or  four  cells  of  a  segmenting 
egg.  The  direct  efforts  of  the  wasp  or  bee  may  be  supposed  to  be 
limited,  at  this  stage,  to  the  making  of  little  hemispherical  cups, 
as  thin  as  the  nature  of  the  material  permits,  and  packing  these 
little  round  cups  as  close  as  possible  together.  It  is  then  con- 
ceiv  ble,  and  indeed  probable,  that  the  symmetrical  tensions  of  the 


VII]  OF  THE  BEE'S  CELL  543 

semi-fluid  films  should  suffice  (however  retarded  by  viscosity)  to 
bring  the  whole  system  into  equihbrium,  that  is  to  say  into  the 
configuration  which  the  comb  actually  assumes. 

The  remarkable  passage  in  which  Bufi'on  discusses  the  bee's  cell 
and  the  hexagonal  configuration  in  general  is  of  such  historical 
importance,  and  tallies  so  closely  with  the  whole  trend  of  our 
enquiry,  that  before  we  leave  the  subject  I  wull  quote  it  in  full*: 
"Dirai-je  encore  un  mot:  ces  cellules  des  abeilles,  tant  vantees, 
tant  admirees,  me  fournissent  une  preuve  de  plus  contre  I'en- 
thousiasme  et  Tadmiration;  cette  figure,  toute  geometrique  et  toute 
reguliere  qu'elle  nous  parait,  et  qu'elle  est  en  effet  dans  la  specula- 
tion, n'est  ici  qu'un  resultat  niecanique  et  assez  imparfait  qui  se 
trouve  souvent  dans  la  nature,  et  que  Ton  remarque  meme  dans  les 
productions  les  plus  brutes;  les  cristaux  et  plusieurs  autres  pierres, 
quelques  sels,  etc.,  prennent  constamment  cette  figure 'dans  leur 
formation.  Qu'on  observe  les  petites  ecailles  de  la  peau  d'une 
roussette,  on  verra  qu'elles  sont  hexagones,  parce  que  chaque  ecaille 
croissant  en  meme  temps  se  fait  obstacle  et  tend  a  occuper  le  plus 
d'espace  qu'il  est  possible  dans  un  espace  donne:  on  voit  ces  memes 
hexagones  dans  le  second  estomac  des  animaux  ruminans,  on  les 
trouve  dans  les  graines,  dans  leurs  capsules,  dans  certaines  fleurs, 
etc.  Qu'on  remplisse  un  vaisseau  de  pois,  ou  plutot  de  quelque 
autre  graine  cylindrique,  et  qu'on  le  ferme  exactement  apres  y  avoir 
verse  autant  d'eau  que  les  intervalles  qui  restent  entre  ces  graines 
peuvent  en  recevoir;  qu'on  fasse  bouillir  cette  eau,  tons  ces  cylindres 
deviendront  de  colonnes  a  six  pans.  On  y  voit  clairement  la 
raison,  qui  est  purement  mecanique;  chaque  graine,  dont  la  figure 
est  cyhndrique,  tend  par  son  renflement  a  occuper  le  plus  d'espace 
possible  dans  un  espace  donne,  elles  deviennent  done  toutes  neces- 
sairement  hexagones  par  la  compression  reciproque.  Chaque  abeille 
cherche  a  occuper  de  meme  le  plus  d'espace  possible  dans  un  espace 
donne,  il  est  done  necessaire  aussi,  puisque  le  corps  des  abeilles  est 
cylindrique,  que  leurs  cellules  sont  hexagones — par  la  meme  raison 

*  Buffon,  Histoire  naturelle,  iv,  p.  99,  Paris,  1753.  Bonnet  criticised  BuflFon's 
explanation,  on  the  ground  that  his  description  was  incomplete;  for  Buffon  took 
no  account  of  the  Maraldi  pyramids.  Not  a  few  others  discovered  impiety  in  his 
hypotheses,  and  some  dismissed  them  with  the  remark  that  "philosophical 
absurdities  are  the  most  difficult  to  refute";  cf.  W.  Smellie,  Philosophy  of  Natural 
History,  Edinburgh,  1790,  p.  424. 


544  THE  FORMS  OF  TISSUES  [ch. 

des  obstacles  reciproques.  On  donne  plus  d'esprit  aux  mouches 
dont  les  ouvrages  sont  les  plus  reguliers;  les  abeilles  sont,  dit-on, 
plus  ingenieuses  que  les  guepes,  que  les  frelons,  etc.,  qui  savent  aussi 
r architecture,  mais  dont  les  constructions  sont  plus  grossieres  et 
plus  irregulieres  que  celles  des  abeilles:  on  ne  veut  pas  voir,  ou  Ton 
ne  se  doute  pas,  que  cette  regularite,  plus  ou  moins  grande,  depend 
uniquement  du  nombre  et  de  la  figure,  et  nuUement  de  I'intelligence 
de  ces  petites  betes ;  plus  elles  sont  nombreuses,  plus  il  y  a  des  forces 
qui  agissent  egalement  et  s'opposent  de  meme,  plus  il  y  a  par 
consequent  de  contrainte  mecanique,  de  regularite  forcee,  et  de 
perfection  apparente  dans  leurs  productions*." 

Of  parenchymatous  cells 

Just  as  Bonanni  and  other  early  writers  sought,  as  we  have  seen, 
to  explain  hexagonal  symmetry  on  mechanical  principles,  so  other 
early  naturalists,  relying  more  or  less  on  the  analogy  of  the  bee's 
cell,  endeavoured  to  explain  the  cells  of  vegetable  parenchyma ;  and 
to  refer  them  to  the  rhombic  dodecahedron  or  garnet-form,  which 
sohd  figure,  in  close-packed  association,  was  believed  in  their  time, 
and  long  afterwards,  to  enclose  space  with  a  minimal  extent  of  surface. 

*  Among  countless  papers  on  the  bee's  cell,  see  John  Barclay  and  others  in 
Ann.  of  Philosophy,  ix,  x,  1817;  Henry  Lord  Brougham,  in  Dissertations... 
connected  with  Natural  Theology,  app.  to  Paley's  Works,  i,  pp.  218-368,  1839; 
C.R.  Acad.  Sci.  Paris,  xlvi,  pp.  1024-1029,  1858;  Tracts,  Mathematical  and  Physical, 
1860,  pp.  103-121,  etc.;  E.  Carruccio,  Note  storiche  sulla  geometria  delle  api, 
Periodico  di  Matem.  (4),  xvi,  20  pp.,  1936;  G.  Cesaro,  Sur  la  forme  de  I'alveole 
des  abeilles.  Bull.  Acad.  Roy.  Belg.  (Sci.),  Avril  10,  1929;  Sam.  Haughton,  On 
the  form  of  the  cells  made  by  various  wasps  and  by  the  honey-bee,  Proc.  Nat. 
Hist.  Soc.  Dublin,  in,  pp.  128-140,  1863;  Ann.  Mdg.  Nat.  Hist.  (3),  xi,  pp.  415-429, 
1863;  A.  R.  Wallace,  Remarks  on  the  foregoing  paper,  ibid,  xii,  p.  33;  J.  0. 
Hennum,  Arch.  f.  Math.  u.  Vidensk.,  Christiania,  ix,  p.  301,  1884;  F.  Huber, 
Nouv.  obs.  sur  les  abeilles,  ii,  p.  475,  1814;  F.  W.  Hultmann,  Tidsskr.'f.  Math., 
Uppsala,  I,  p.  197,  1868;  John  Hunter,  Observations  on  bees,  Phil.  Trans.  1792, 
pp.  128-195;  Jacob,  Nouv.  Ann.  de  Math,  ii,  p.  160,  1843;  G.  S.  Kliigel,  Mathem. 
Betrachtungcn  iib.  d.  kunstreichen  Bau  d.  Bienenzellen,  Hannoversches  Mag. 
1772,  pp.  353-368;  Leon  Lalanne,  Note  sur  I'architecture  des  abeilles,  Ann.  Sc.  Nat. 
Zool.  (2),  XIII,  pp.  358-374,  1840;  B.  PoweU,  Proc.  Ashmol.  Soc.  i,  p.  10,  1844; 
K.  H  Schellbach,  Mathem  Lehrstunde  •  Lehre  v.  Grossten  u.  Kleinsten,  1860, 
pp.  35-37;  Sam  Sharpe,  Phil.  Mag  iv  pp  19-21,  1828;  J.  E.  Siegwart,  Die 
Mathematik  im  Dienste  d.  Bienenzucht,  Schw.  Bitnenzeitung,  iii,  1880;  0.  Terquem, 
Nouv.  Ann.  de  M  h  xv,  p.  176,  1856,  C.  M.  Willick,  On  the  angle  of  dock-gates  and 
the  bee's  cell,  Phtl.  Mag.  (4)  xviii,  p.  427,  1859;  C.R.  li,  p.  633,  1860;  Chauncy 
Wright,  Proc.  Amer.  Acad.  Arts  and  Sci.  iv,  p.  432,  1860. 


VII]  OF  THE  PARTITIONING  OF  SPACE  545 

We  have  mentioned  both  Hooke  and  Grew*,  and  we  have  just  heard 

Buffon  engaged  in  such  speculations;    but  the  matter  was  more 

elaborately  treated  near  the  beginning  of  last  century  by  Dieterich 

George   Kieserf,   an  ingenious  friend   and 

colleague  of  the   celebrated  Lorenz  Oken. 

Kieser  clearly  understood  that  the  cell  has 

not  a   shape   of  its   own,  but  merely  one 

impressed    on    it    by   physical   forces   and 

defined    by    mathematical    laws.      In   his 

Memoire  sur  V organisation  des  plantes,  he 

gives  an  admirable  historical  account  of  the 

work  of  Malpighi,  Hooke,  Grew,  John  Hill 

and  other  early  microscopists ;  and  then  he      Fig.  210.    A  rhombic 

says  "La  forme  des  cellules  est  variee  dans  dodecahedron. 

les  plantes  diiferentes,  mais  il  y  a  des  formes  principales,  fondees  sur 

les  lois  des  mathematiques,  que  la  nature  suit  toujours  dans  ses 

formations.. .  .La  forme  la  plus  commune  est  celle  que  prennent 

necessairement  des  globules  rondes  ou  allongees,  pressees  ensemble, 

celle  des  corps  hexagonaux  a  parois  quadrilaterales,  ou  d'une  colonne 

tres  courte  hexagone,  coupee  horizontalement  d'en  haut  et  d'en  bas." 

Here  we  have,  briefly  described  and  sufficiently  accounted  for,  the 

configuration  of  what  we  call  a  "pavement  epithelium,"  or  other 

simple  association  of  cells  in  a  single  layer. 

But  another  passage  (from  the  same  author's  Phytotomie)  is  worth 
quoting  at  length,  where  he  deals  with  cells  in  the  mass,  that  is  to 
say  with  the  three-dimensional  problem.  "Die  nach  mathematische 
Gesetzen  bestinlmte  als  nothwendige  Grundforra  der  Zelle  der  voll- 
kommenen  Zellengewebe  ist  das  langgezogene  Rhombododekaheder. 
. . .  Mathematisch  liegt  das  Beweis  dass  diese  Figur  die  Grundform 
der  vollkommenen  Zellengewebe  sei  darin,  dass  unter  alien  mathe- 
matischen  Korpern  welche  durch  Zusammensetzung  einen  sohden 
Korper  ohne  Zwischenraume  bilden,  das  Rhombododekaheder  die 
einzige  ist  welche  mit  der  wenigsteri  Masse  des  Umkreises  den 
grossten   Raum   einschliesst.     Sollte   also   aus   dem   Globus — dem 

*  R.  Hooke,  Micrographia,  1665,  pp.  115-116;  Nehemiah  Grew,  Anatomy  of 
Plants,  1682,  pp.  64,  76,  120. 

t  D.  G.  Kieser,  Memoire,  etc.,  Haarlem,  1814,  p.  89;  Phytotomie,  oder  Grundziige 
der  Anatomic  der  Pfianzen,  Jena,  1815,  p.  4. 


546  THE  FORMS  OF  TISSUES  [ch. 

urspriinglichsten  Schleimblaschen  der  Pflanzenzelle — ein  eckiger 
Korper  gebildet  werden,  so  musste  dieser  das  Rhombododekaheder 
sein,  well  dieser-  im  Hinsicht  des  Minimums  der  Masse  zu  dem 
Maximum  des  eingeschlossenen  Raumes  dem  Globus  am  nachsten 
liegt.  Als  die  Urform  der  Pflanzenzelle  ist  nicht  Globus  sondern 
Ellipsoide,  daher  muss  das  Dodekaheder,  welche  die  Grundform  der 
eckigen  Pflanzenzelle  ist,  auch  aus  dem  Ellipsoide  entstanden  sein. 
Das  Rhombododekaheder  wird  also  vom  unten  nach  oben  gestreckt, 
und  die  Grundform  der  eckigen  Pflanzenzelle  ist  das  in  perpen- 
dicularer  Richtung  langsgestreckte  Rhombododekaheder." 

These  views  and  speculations  of  Kieser's,  now  all  but  forgotten, 
were  by  no  means  neglected  in  their  day.  Oken  accepted  them, 
and  taught  them*;  Schleiden  remarks  that  "the  form  of  cells 
frequently  passes  into  that  of  the  rhombic  dodecahedron,  so  beauti- 
fully determined,  a  priori,  by  Kieserf";  and  De  CandoUe  thought 
it  necessary  to  warn  his  readers  that  cells  are  not  as  geometrically 
regular  as  pubhshed  figures  might  lead  one  to  believe  J. 

The  same  principles  apply  to  various  orders  of  magnitude,  and 
close-packing  may  be  seen  even  in  the  inner  contents  of  a  cell.  In 
vitally  stained  "goblet-cells,"  the  mucin  gathers  into  clumps  or 
droplets,  of  which  each  appears  in  optical  section  to  be  surrounded 
by  six  more.  When  fixed  they  draw  together,  appear  in  optical 
section  to  be  hexagonal,  and  we  may  take  it  that  they  have  become, 
to  a  first  approximation,  rhombic  dodecahedra§. 

These  then,  and  such  as  these,  were  the  not  unimportant  specu- 
lations on  the  forms  of  cells  by  men  who  early  grasped  the  fact  that 
form  had  a  physical  cause  and  a  mathematical  significance.  But 
their  conception  of  the  phenomenon  was  of  necessity  hmited  to  the 
play  of  the  mechanical  forces;  for  Plateau's  Statique  des  Liquides 
had  not  yet  shewn  what  the  capillary  forces  can  do,  nor  opened  a 
way  thereby  for  Berthold  and  for  Errera. 

A  very  beautiful  hexagonal  symmetry  as  seen  in  section,  or 
dodecahedral  as  viewed  in  the  soHd,  is  presented  by  the  pith  of 
certain  rushes  (e.g.  J  uncus  effusus),  and  somewhat  less  diagram- 

*  Oken,  Physiophilosophy  (Ray  Society),  1847,  p.  209. 

t  Muller's  Archiv,  1838,  p.  146. 

X  Organogenic  vegetale,  i,  p.  13,  1827. 

§  E.  S..Duthie,  in  Proc,  B,S.  (B),  cxiii,  pp.  459-463,  1933. 


VII]  OF  STELLATE  CELLS  547 

matically  by  the  pith  of  the  banana.  The  cells  are  stellate,  and 
the  tissue  has  the  appearance  in  section  of  a  network  of  six-rayed 
stars  (Fig.  211),  linked  together  by  the  tips  of  the  rays,  and  separated 
by  symmetrical,  air-filled  intercellular  spaces,  which  give  its  snow- 
like whiteness  to  the  pith.  In  thick  sections,  the  solid  twelve-rayed 
"  star-dodecahedra "  may  be  very  beautifully  seen  under  the 
binocular  microscope.  They  are  not  difficult  to  understand. 
Imagine,  as  before,  a  system  of  equal  spheres  in  close  contact,  each 
one  touching  its  twelve  neighbours,  six  of  them  in  the  equatorial 


Fig.  211.     Stellate  cells  in  pith  of  Juncus. 

plane ;  and  let  the  cells  be  not  only  in  contact,  but  become  attached 
at  the  points  of  contact.  Then,  instead  of  each  cell  expanding  so 
as  to  encroach  on  and  fill  up  the  intercellular  spaces,  let  each  tend 
to  shrink  or  shrivel  up  by  the  withdrawal  of  fluid  from  its  interior. 
The  result  will  be  to  enlarge  the  intercellular  spaces ;  the  attachments 
of  each  cell  to  its  neighbours  will  remain  fixed,  but  the  walls  between 
these  points  of  attachment  will  be  withdrawn  in  a  symmetrical 
fashion  towards  the  centre.  As  the  final  result  w^e  have  the  star- 
dodecahedron,  which  appears  in  plane  section  as  a  six-rayed  figure. 
It  is  necessary  not  only  that  the  pith-cells  should  be  attached  to 
one  another,  but  also  that  the  outermost  should  be  attached  to  a 
boundary  wall,  to  preserve  the  symmetry  of  the  system.     What 


548  THE  FORMS  OF  TISSUES  [ch. 

actually  occurs  in  the  rush  is  tantamount  to  this,  but  not  absolutely 
identical.  It  is  not  so  much  the  pith-cells  which  tend  to  shrink 
within  a  boundary  of  constant  size,  but  rather  the  boundary  wall 
which  continues  to  expand  after  the  pith-cells  which  it  encloses  have 
ceased  to  grow  or  to  multiply.  The  points  of  attachment  on  the 
surface  of  each  little  pith-cell  are  drawn  asunder,  but  the  content 
of  the  cell  does  not  correspondingly  increase;  and  the  remaining 
portions  of  the  surface  shrink  inwards,  accordingly,  and  gradually 
constitute  the  complicated .  figure  which  Kepler  called  a  star- 
dodecahedron,  which  is  still  a  symmetrical  figure,  and  is  still  a 
surface  of  minimal  area  under  the  new  and  altered  conditions. 

The  tetrakaidekahedron 

A  few  years  after  the  pubhcation  of  Plateau's  book,  Lord  Kelvin 
shewed,  in  a  short  but  very  beautiful  paper*,  that  we  must  not 
hastily  assume  from  such  arguments  as  the  foregoing  that  a  close- 
packed  assemblage  of  rhombic  dodecahedra  will  be  the  true  and 
general  solution  of  the  problem  of  dividing  space  with  a  minimum 
partitional  area,  or  will  be  present  in  a  liquid  ''foam,"  in  which  the 
general  problem  is  completely  and  automatically  solved.  The 
general  mathematical  solution  of  the  problem  (as  we  have  already 
indicated)  is,  that  every  interface  or  partition-wall  must  have  con- 
stant mean  curvature  throughout ;  that  where  these  partitions  meet 
in  an  edge,  they  must  intersect  at  angles  such  that  equal  forces,  in 
planes  perpendicular  to  the  line  of  intersection,  shall  balance ;  that 
no  more  than  three  such  interfaces  may  meet  in  a  line  or  edge, 
whence  it  follows  (for  symmetry)  that  the  angle  of  intersection  of 
all  surfaces  or  facets  must  be  120°;  and  that  neither  more  nor  less 
than  four  edges  meet  in  a  point  or  corner.  An  assemblage  of  rhombic 
dodecahedra  goes  far  to  meet  the  case.  It  fills  space;  its  surfaces 
or  interfaces  are  planes,  and  therefore  surfaces  of  constant  curvature 
throughout;  and  they  meet  together  at  angles  of  120°.  Never^ 
theless,  the  proof  that  the  rhombic  dodecahedron  (which  we  find 
exemplified  in  the  bee's  cell)  is  a  figure  of  minimal  area  is  not 
a  comprehensive  proof;    it  is  Hmited  to  certain  conditions,  and 

*  Sir  W.  Thomson,  On  the  division  of  space  with  minimum  partitional  acea, 
Phil.  Mag.  (5),  xxiv,  pp.  503-514,  Dec.  1887;  cf.  Baltimore  Lectures,  1904,  p.  615; 
Molecular  tactics  of  a  crystal  (Robert  Boyle  Lecture),  1894,  pp.  21-25. 


VII 


OF  THE  PARTITIONING  OF  SPACE 


549 


practically  amounts  to  no  more  than  this,  that  of  the  ordinary 
space-filling  soHds  with  all  sides  plane  and  similar,  this  one  has  the 
least  surface  for  its  solid  content. 

The  rhombic  dodecahedron  has  six  tetrahedral  angles  and  eight 
trihedral  angles.  At  each  of  the  latter  three,  and  at  each  of  the 
former  six,  dodecahedra  meet  in  a  point  in  close  packing;  and  foiir 
edges  meet  in  a  point  in  the  one  case  and  eight  in  the  other.  This 
is  enough  to  shew  that'  the  conditions  for  minimal  area  are  not 
rigorously  metr  In  one  of  Plateau's  most  beautiful  experiments*, 
a  wire  cube  is  dipped  in  soap-solution.  When  hfted  out,  a  film  is 
seen  to  pass  inwards  from  each  of  the  twelve  edges  of  the  cube,  and 
these  twelve  films  meet,  three  by  three,  in  eight  edges,  running 
inwards  from  the  eight  corners  of  the 
cube;  but  the  twelve  films  and  their 
eight  edges  do  not  meet  in  a  point,  but 
are  grouped  around  a  small  central 
quadrilateral  fihn  (Fig.  212).  Two  of 
the  eight  edges  run  to  each  corner  of  the 
little  square,  and,  with  the  two  sides  of 
the  square  itself,  make  up  the  four  edges 
meeting  in  a  point  which  the  theory  of 
area  minima  requires.      We  may  sub-  ^^' 

stitute  (by  a  second  dip)  a  little  cube  for  the  little  square;  now  an 
edge  from  each  corner  of  the  outer  cube  runs  to  the  corresponding 
corner  of  the  inner  one,  and  with  the  three  adjacent  edges  of  the 
little  cube  itself  the  number  four  is  still  maintained.  Twelve  films, 
and  eight  edges  meeting  in  a  point,  w^ere  essentially  unstable;  but 
the  introduction  of  the  little  square  or  cube  meets  most  of  the 
conditions  of  stabihty  which  Plateau  was  the  first  to  lay  down. 
One  more  condition  has  to  be  met,  namely  the  equality  of  angles  at 
which  the  four  edges  meet  in  each  conjunction.  These  co-equal 
"Maraldi  angles"  at  each  corner  of  the  square  can  only  be  con- 
structed by  help  of  a  slight  curvature  of  the  sides,  and  the  httle 
square  is  seen  to  have  its  sides  curved  into  circular  arcs  accordingly; 
moreover  its  size  and  shape,  as  that  of  all  the  other  films  in  the 
system,   are  perfectly  definite.     It  is  all  one,   according  to  the 

*  Also  discovered  independently  by  Sir  David  Brewster,  Trans.  R.S.E.  xxrv, 
p.  505,  1867;   xxv,  p.  115,  1869. 


550  THE  FORMS  OF  TISSUES  [ch. 

symmetry  of  the  figure,  to  which  side  of  the  skeleton  cube  the 
square  lies  parallel;  wherever  it  may  be,  if  we  blow  gently  on  it, 
then  (as  M.  Van  Rees  discovered)  it  alters  its  place  and  sets  itself 
parallel  now  to  one  and  now  to  another  of  the  paired  faces  of  the 
cube. 

The  skeleton  cube,  Hke  the  tetrahedron  which  we  have  already 
studied,  is  only  one  of  many  interesting  cases;  for  we  may  vary 
the  shape  of  our  wire  cages  and  obtain  other  and  not  less  beautiful 
configurations.  An  hexagonal  prism,  if  its  sides  be  square  or  nearly 
so,  gives  us  six  vertical  triangular  films,  whose  apices  meet  the 
corners  of  a  horizontal  hexagon*;  also  six  pairs  of  truncated 
triangles,  which  link  the  top  and  bottom  edges  of  the  cage  to  the 
sides  of  the  median  hexagon.  But  if  the  height  of  the  hexagonal 
prism  be  increased,  the  six  vertical  films  become  curvihnear  triangles, 
with  sides  concave  towards  the  apex;  and  the  twelve  remaining 
films,  which  spring  from  the  top  and  bottom  of  the  hexagon,  are 
curved  surfaces,  looking  like  a  sort  of  hexagonal  hourglass  f. 

There  is  a  deal  of  elegant  geometry  in  these  various  configurations. 
Lamarle  shewed  that  if,  in  a  figure  represented  by  our  wire  cage, 
we  suppress  (in  imagination)  one  face  and  all  the  other  faces 
adjoining  it,  then  the  faces  which  remain  are  those  which  appear 
in  the  centre  of  the  figure  after  the  cage  has  been  withdrawn  from 
the  soap-solution.  Thus,  in  a  cube,  we  suppress  one  face  and  the 
four  adjacent  to  it;  only  one  remains,  and  it  reappears  as  the  central 
square  in  the  middle  of  the  new  configuration;  in  the  tetrahedron, 
when  we  have  suppressed  one  face  and  the  three  adjacent  to  it, 
there  is  nothing  left — save,  a  median  point,  corresponding  to  the 
opposite  corner.  In  a  regular  dodecahedron,  if  we  suppress  one 
pentagonal  face  and  its  five  neighbours,  the  other  half  of  the  whole 
figure  remains;  and  the  dodecahedral  cage,  after  immersion  in  the 
soap,  shews  a  central  and  symmetrical  group  of  six  pentagons  J. 

Moreover,  while  the  cage  is  carrying  its  configuration  of  films, 
we  may  blow  a  bubble  within  it,  and  so  insert  a  new  polyhedron 

*  The  angles  of  a  hexagon  are  too  big,  as  those  of  a  square  were  too  small,  to 
form  the  Maraldi  angles  of  symmetry;  hence  the  sides  of  the  hexagon  are  found 
to  be  concave,  as  those  of  the  square  bulged  out  convexly. 

t  Cf.  Dewar,  op.  cit.  1918. 

X  That  is  to  say,  if  nF„^  be  a  polyhedron  (of  n  m-faced  sides),  the  corresponding 
wire  cage  will  exhibit  (w  -  m  +  1)  i^^  as  central  fenestrae. 


VII]  OF  THE  TETRAKAIDEKAHEDRON  551 

within  the  old,  and  set  it  in  place  of  the  former  fenestra.  The  inner 
polyhedral  bubble  so  produced  may  be  of  any  dimensions,  but  it 
resembles  the  outer  polyhedral  cage  precisely,  except  in  the 
curvature  of  its  sides ;  it  has  all  its  faces  spherical,  and  all  of  equal 
radius  of  curvature;  its  edges  are  either  arcs  of  circles  or  straight 
lines.  Later  on,  we  shall  see  that  there  is  no  small  biological 
interest  attaching  to  these  configurations. 

Lord  Kelvin  made  the  remarkable  discovery  that  the  square 
fenestra  with  the  four  quadrilateral  films  impinging  on  its  sides,  in 
Plateau's  experiment,  represented  the  one-sixth  part  of  a  symmetrical 
figure;  that  this  figure  when  complete  was  bounded  by  six  squares 
and  eight  hexagons;    that  by  means  of  an  assemblage  of  these 


Fig.  213.     A  set  of  14-he(lra,  to  shew  close-packing.     From  F.  T.  Lewis 


fourteen-sided  figures,  or  "tetrakaidekahedra,"  space  is  filled  and 
homogeneously  partitioned — into  equal,  similar  and  similarly  situated 
cells — with  an  economy  of  surface  in  relation  to  volume  even  greater 
than  in  an  assemblage  of  rhombic  dodecahedra*. 

The  tetrakaidekahedron,  in  its  most  generalised  case,  is  bounded 
by  three  pairs  of  equal  and  opposite  quadrilateral  faces,  and  four 
pairs  of  equal  and  opposite  hexagonal  faces,  neither  the  quadri- 
laterals nor  the  hexagons  being  necessarily  plane.  In  its  simplest 
case,  with  all  its  facets  plane  and  equilateral,  it  is  Kelvin's  "ortho- 
tetrakaidekahedron " ;  and  also  (though  Kelvin  was  unaware  of  the 
fact)  one  of  the  thirteen  semi-regular  and  isogonal  polyhedra,  or 
"Archimedean  bodies."  In  a  particular  case,  the  quadrilaterals  are 
plane  surfaces  with  curved  edges,   but  the  hexagons  are  slightly 

*  Kelvin,  Boyle  Lecture  and  Baltitnore  Lectures.  In  the  first  of  these  Kelvin 
described  the  plane-faced  tetrakaidekahedron;  in  the  second  he  shewed  how  that 
figure  must  have  its  faces  warped  and  edges  curved  to  fulfil  all  the  conditions  of 
minimal  area. 


552  THE  FORMS  OF  TISSUES  [ch. 

curved  " anticlastic "  surfaces;-  and  these  latter  have  at  every  point 
equal  and  opposite  curvatures,  and  are  surfaces  of  minimal  curvature 
for  a  boundary  of  six  curved  edges.  This  figure  has  the  remarkable 
property  that,  hke  the  plane  rhombic  dodecahedron,  it  so  partitions 
space  that  three  faces  meeting  in  an  edge  do  so  everywhere  at 
co-equal  angles  of  120°;  and,  unlike  the  rhombic  dodecahedron, 
four  edges  meet  in  each  point  or  corner  at  co-equal  angles  of 
109°  28'*. 

We  may  take  it  as  certain  tha^),  in  a  homogeneous  system  of  fluid 
films  Hke  the  interior  of  a  froth  of  soap-bubbles,  where  the  films 
are  perfectly  free  to  glide  or  turn  over  one  another  and  are  of 
approximately  co-equal  size,  the  mass  is  actually  divided  into  cells 
of  this  remarkable  conformation:  and  the  possibility  of  such  a 
configuration  being  present  even  in  the  cells  of  an  ordinary  vegetable 
pareilchyma  was  suggested  in  the  first  edition  of  this  book.  It  is 
all  a  question  of  restraint,  of  degrees  of  mobihty  or  fluidity.  If  we 
squeeze  a  mass  of  clay  pellets  together,  hke  Buffon's  peas,  they  come 
out,  or  all  the  inner  ones  do,  in  neat  garnet-shape,  or  rhombic 
dodecahedra.  But  a  young  student  once  shewed  me  (in  Yale)  that 
if  you  wet  these  clay  pellets  thoroughly,  so  that  they  slide  easily 
on  one  another  and  so  acquire  a  sort  of  pseudo-fluidity  in  the  mass, 
they  no  longer  come  out  as  regular  dodecahedra,  but  with  square 
and ,  hexagonal  facets  recognisable  as  those  of  ill-formed  or  half- 
formed  tetrakaidekahedra. 

Dr  F.  T.  Lewis  has  made  a  long  and  careful  study  of  various 
vegetable  parenchymas,   by  simple  maceration,   wax-plate  recon- 

*  Von  Fedorov/  had  already  described  (in  Russian),  unaware  that  Archimedes 
had  done  so,  the  same  figure  under  the  name  of  cubo-octahedron,  or  hepta- 
parallelohedron^  hmited  however  to  the  case  where  all  the  faces  are  plane  and 
regular.  This  cubo-octahedron,  together  with  the  cube,  the  hexagonal  prism, 
the  rhombic  dodecahedron  and  the  "elongated  dodecahedron,"  constitute  the 
five  plane-faced,  parallel-sided  figures  by  which  space  is  capable  of  being  completely 
filled  and  uniformly  partitioned;  the  series  so  forming  the  foundation  of  Von 
Fedorow's  theory  of  crystalline  structure — though  the  space-fillers  are  not  all,  and 
cannot  all  be,  crystalline  forms.  All  of  these  figures,  save  the  hexagonal  prism, 
are  related  to  and  derivable  from  the  cube,  so  we  end  by  recognising  two  principal 
types,  cubic  and  hexagonal.  We  have  learned  to  recognise  the  dodecahedron, 
and  we  may  find  in  still  closer  packing  the  cubo-octahedron,  in  a  parenchyma; 
the  elongated  dodecahedron  is,  essentially,  the  figure  of  the  bee's  cell;  the  cube 
we  have,  in  essence,  in  cambium -tissue;  the  hexagonal  prism,  dwarf  or  tall,  simple 
or  recognisably  deformed,  we  see  in  every  epithelium. 


vn]  OF  THE  TETRAKAIDEKAHEDRON  553 

struction  and  otherwise,  and  has  succeeded  in  shewing  that  the 
tetrakaidekahedral  form  is  closely  approached,  or  even  attained,  in 
certain  simple  and  homogeneous  tissues.  After  reconstructing  a 
large  model  of  the  cells  of  elder-pith,  he  finds  that  the  fourteen-sided 
figure  clearly  manifests  itself  as  the  characteristic  or  typical  form 
to  which  the  cells  approximate,  in  spite  of  repeated  cell-divisions 
and  consequent  inequalities  of  size.  Counting  in  a  hundred  cells 
the  number  of  contacts  which  each  made  with  its  neighbours,  that 
is  to  say  the  total  number  both  of  actual  and  potential  facets,  Lewis 
found  that  74  per  cent,  of  the  cells  were  either  12,  13,  14,  15  or 
16-sided,  56  per  cent,  either  13,  14  or  15-sided,  and  that  the  averagje 


-•-^ 


Fig.  214.     Reconstructed  models  of  cells  of  elder-pith,  shewing  a  certain 
approximation  to  14-hedral  form.     From  F.  T.  Lewis. 

number  of  facets  or  contacts  was,  in  this  instance,  just  13-96.  These 
figures  indicate  the  general  symmetry  of  the  cells,  their  departure 
from  the  dodecahedral,  and  their  tendency  towards  the  tetra- 
kaidekahedral, form*. 

But  after  all,  the  geometry  of  the  14-hedron,  displayed  to  per- 
fection by  our  soap-films  in  the  twinkhng  of  an  eye,  is  only  roughly 
developed  in  an  organic  structure,  even  one  so  delicate  as  elder- 
pith;   the  conditions  are  no  longer  simple,  for  friction,  viscosity  and 

*  F.  T.  Lewis,  The  typical  shape  of  polyhedral  cells  in  vegetable  parenchyma, 
and  the  restoration  of  that  shape  following  cell-division,  Proc.  Amer.  Acad,  of  Arts 
and  Sci.  lviii,  pp.  537-552,  1923,  and  other  papers.  See  also  {int.  al.)  J.  W.  Marvin, 
The  aggregation  of  orthis-tetrakaidekahedra.  Science,  lxxxiii,  p.  188,  1936; 
E.  B.  Metzger,  An  analysis  of  the  orthotetrakaidekahedron,  Bull.  Torrey  Bot.  Club, 
Liv,  pp.  341-348,  1927.  Professor  van  Iterson  of  Delft  tells  me  that  Asparagus 
Sprengeri  (a  common  greenhouse  plant)  is  a  good  subject  for  shewing  the  14-hedral 
cells. 


554  THE  FORMS  OF  TISSUES  [ch. 

solidification  have  vastly  complicated  the  case.  We  get  a  curious 
and  an  unexpected  variant  of  the  same  phenomenon  in  the  micro- 
scopic foam-like  structure  assumed  as  molten  metal  cools.  If  these 
foam-cells  were  again  14-hedra,  their  facets  would  all  be  either 
squares  or  hexagons;  but  pentagonal  facets  are  commoner  than 
either,  and  the  cells  often  approach  closely  to  the  form  of  a  regular 
pentagonal  dodecahedron !  The  edges  of  this  figure  meet  at  angles 
of  108°,  not  far  from  the  characteristic  Maraldi  angle  of  109°  28'; 
and  the  faces  meet  at  an  angle  not  far  removed  from  120°.  A  slight 
curvature  of  the  sides  is  enough  to  turn  our  pentagonal  dodecahedron 
into  a  possible  figure  of  equihbrium  for  a  foam-cell.  We  cannot 
close-pack  pentagonal  dodecahedra,  whether  equal  or  unequal,  so 
as  to  fill  space;  but  still  the  figure  may  be,  and  seems  to  be,  common, 
interspersed  among  the  polyhedra  of  various  shapes  and  sizes  which 
are  packed  together  in  a  metallic  foam*. 

A  somewhat  similar  result,  and  a  curious  one,  was  found  by 
Mr  J.  W.  Marvin,  who  compressed  leaden  small-shot  in  a  steel 
cylinder,  as  Buffon  compressed  his  peas;  but  this  time  the  pressure 
on  the  plunger  ran  from  1000  to  35,000  lb.  or  nearly  twenty  tons  to 
the  square  inch.  When  the  shot  was  introduced  carefully,  so  as 
to  lie  in  ordinary  close  packing,  the  result  was  an  assemblage  of 
regular  rhombic  dodecahedra,  as  might  be  expected  and  as  Buffon 
had  found.  But  the  result  was  very  different  when  the  shot  was 
poured  at  random  into  the  cyhnder,  for  the  average  number  of 
facets  on  each  grain  now  varied  with  the  pressure,  from  about  8-5 
at  1000  lb.  to  12-9  at  10,000  lb.,  and  to  no  less  than  14-16  facets  or 
contacts  after  all  interstices  were  eliminated,  which  took  the  full 
pressure  of  35,000  lb.  to  do.  An  average  of  just  over  fourteen 
facets  might  seem  to  indicate  a  tendency  to  the  production  of 
tetrakaidekahedra,  just  as  in  the  froth  of  soap-bubbles;  but  this 
is  not  so.  The  squeezed  grains  are  irregular  in  shape,  and  pentagonal 
facets  are  much  the  commonest,  just  as  we  found  them  to  be  in 
the  microscopic  structure  of  a  once-molten  metal.  At  first  sight 
it  might  seem  that,  though  the  experiment  has  something  to  teach 
us  about  random  packing  in  a  limited  space,  it  has  no  biological 
significance;     but    it    is    curious    to    find    that    the    pith-cells    of 

*  Cf.  Cecil  H.  Desch,  The  solidification  of  metals  from  the  liquid  state,  Joarn. 
Inst,  of  Metals,  xxii,  p.  247,  1919. 


VII]  OF  THE  PARTITIONING  OF  SPACE  555 

Eupatorium  have  a  similar  average  configuration,  with  the  same 
predominance  of  pentagonal  facets*. 

We  learn,  in  short,  from  Lewis  and  from  Marvin  that  the 
mechanical  result  of  mutual  pressure,  even  in  an  assemblage  of 
co-equal  spheres,  is  more  varied  and  more  complex  than  we  had 
supposed.  The  two  simple  and  homogeneous  configurations — the 
rhombo-dodecahedral  and  tetrakaidekahedral  assemblages — are 
easily  and  commonly  produced,  the  one  by  the  compression  of  solid 
spheres  in  ordinary  close-packing,  the  other  when  a  liquid  system 
of  spheres  or  bubbles  is  free  to  slide  and  glide  into  a  packing  which 
is  closer  still.  Between  these  two  configurations  there  is  no  othei 
symmetrical  or  homogeneous  arrangement  possible;  but  random 
packing  and  degrees  of  compression  leave  their  random  effects, 
among  which  are  traces  here  and  there  of  regular  shape  and 
symmetry. 

As  a  froth  has  its  histological  lessons  for  us,  throwing  light  on 
the  structure  of  a  parenchyma,  so  may  we  draw  an  illustration 
or  two  from  the  analogous  characteristics  of  an  emulsion.  Both 
alike  are  "states  of  aggregation";  both  are  "two-phase  systems," 
one  phase  being  dispersed  and  the  other  the  medium  of  dispersion. 
Both  phases  are  liquid  in  the  emulsion,  in  the  froth  the  dispersed 
phase  is  a  gas;  our  living  tissue  is,  so  far,  more  likely  to  be  an 
emulsion  than  a  froth.  The  concept  widens.  A  colony  of  bacteria, 
the  blood  corpuscles  in  their  plasma,  the  filaments  of  an  alga,  the 
heterogeneous  texture  of  any  ordinary  tissue,  may  all  be  brought 
under  the  general  concept  of  "phase  systems,"  and  share  the 
common  character  that  one  phase  exposes  a  large  "interface"  to 
the  other.  If  we  take  milk  as  a  simple  emulsion,  we  see  its  hquid 
oil-globules  dispersed  in  a  watery  medium  and  rounded  by  surface 
tension  into  spheres.  The  watery  medium,  as  is  usual  in  such 
emulsions,  contains  dissolved  substances  which  tend  to  lower  the 
interfacial  tension;  for  were  that  tension  high  the  globules  would 
tend  to  be  larger  and  their  aggregate  surface  less.  Suppose  the 
"phase-ratio"  to  alter,  the  globules  becoming  more  numerous  and 
the  disperse  medium  less  and  less,  the  globules  will  be  close-packed 

*  J.  W.  Marvin,  The  shape  of  compressed  lead-shot,  etc.;  Amer.  Journ.  of 
Botany  xxvi,  pp.  280-288,  1939;  Cell-shape  studies  in  the  pith  of  Eupatorium, 
ibid.  pp.  487-504. 


556  THE  FORMS  OF  TISSUES  [ch. 

at  last.  Then  each  (provided  they  be  of  equal  size)  will  be  in  touch 
with  twelve  neighbours;  and  if  the  spheres  were  solid— were  the 
system  not  an  emulsion  but  a  "  suspension  "—the  matter  would  end 
here.  But  our  hquid  globules  are  capable  of  deformation,  and  the 
points  of  contact  are  flattened  in  still  closer  packing  into  planes. 
Ttey  become  polyhedral,  and  tend  to  take  the  form  of  rhombic 
dodecahedra,  or  it  may  be  even  of  14-hedra,  and  the  dispersion- 
medium  is  reduced  to  mere  films  or  pellicles  between.  At  the  stage 
of  mere  twelve-point  contact,  the  spherules  constitute  about  74  per 
cent.,  and  the  disperse  .medium  26  per  cent.,  of  the  whole.  But  in 
the  final  stage  the  phase-ratio  has  so  altered  that  the  disperse- 
medium  is  but  a  small  fraction  of  the  whole,  the  thin  film  to  which 
it  has  been  reduced  has  the  appearance  of  a  cell-membrane  separating 
the  cells,  and  the  microscopic  structure  of  the  whole  corresponds  to 
the  cellular  configuration  of  a  parenchymatous  tissue*. 

Of  certain  groupings  of  cells 
It  follows  from  all  that  we  have  said  that  the  problems  connected 
with  the  conformation  of  cells,  and  with  the  manner  in  which  a 
given  space  is  partitioned  by  them,  soon  become  complex;  and 
while  this  is  so  even  when  all  our  cells  are  equal  and  symmetrically 
placed,  it  becomes  vastly  more  so  when  cells  varying  even  slightly 
in  size,  in  hardness,  rigidity  or  other  quahties,  are  packed  together. 
The  mathematics  of  the  case  very  soon  become  too  hard  for  us, 
but  in  its  essence  the  phenomenon  remains  the  same.  We  have 
little  reason  to  doubt,  and  no  just  cause  to  disbelieve,  that  the 
whole  configuration,  for  instance  of  an  egg  in  the  advanced  stages 
of  segmentation,  is  accurately  determined  by  simple  physical  laws: 
just  as  much  as  in  the  early  stages  of  two  or  four  cells,  during  which 
early  stages  we  are  able  to  recognise  and  demonstrate  the  forces 
and  their  effects.  But  when  mathematical  investigation  has  become 
too  difficult,  physical  experiment  can  often  reproduce  the  pheno- 
mena which  Nature  exhibits,  and  which  we  are  striving  to  com- 
prehend. In  an  admirable  research,  M.  Robert  not  only  shewed 
some  years  ago  that  the  early  segmentation  of  the  egg  of  Trochus 
(a  marine  univalve  mollusc)  proceeded  in  accordance  with  the  laws 

*  Cf.  E.  Hatschek,  Homogeneous  partitionings,  etc.,  Phil.  Mag.  xxxiii,  p.  83, 
1917. 


VII]  OF  POLAR  FURROWS  557 

of  Surface-tension,  but  he  also  succeeded  in  imitating  by  means  of 
soap-bubbles  one  stage  after  another  of  the  developing  egg. 

M.  Robert  carried  his  experiments  as  far  as  the  stage  of  sixteen 
cells,  or  bubbles.  It  is  not  easy  to  carry  the  artificial  system  quite 
so  far,  but  in  the  earHer  stages  the  experiment  is  easy;  we  have 
merely  to  blow  our  bubbles  in  a  httle  dish,  adding  one  to  another, 
and  adjusting  their  sizes  to  produce  a  symmetrical  system.  One 
of  the  simplest  and  prettiest  parts  of  his  investigation  concerned 
the  "polar  furrow  "  of  which  we  have  spoken  on  p.  489.  On  blowing 
four  httle  contiguous  bubbles  he  found  (as  we  may  all  find  with  the 
greatest  ease)  that  they  form  a  symmetrical  system,  two  in  contact 
with  one  another  by  a  laminar  film,  and  two  which  are  elevated 
a  httle  above  the  others  and  are  separated  by  the  length  of 
the  aforesaid  lamina.  The  bubbles  are  thus  in  contact  three  by 
three,  their  partition-walls  making  with  one  another  equal  angles 
of  120°.  The  upper  and  lower  edges  of  the  intermediate  lamina 
(the  lower  one  visible  through  the  transparent  system)  constitute 
the  two  polar  furrows  of  the  embryologist  (Fig.  215,  1-3).  The 
lamina  itself  is  plane  when  the  system  is  symmetrical,  but  it  responds 
by  a  corresponding  curvature  to  the  least  inequahty  of  the  bubbles 
on  either  side.  In  the  experiment,  the  upper  polar  furrow  is  usually 
a  httle  shorter  than  the  lower,  but  parallel  to  it;  that  is  to  say, 
the  lamina  is  of  trapezoidal  form :  this  lack  of  perfect  symmetry 
being  due  (in  the  experimental  case)  to  the  lower  portion  of  the 
bubbles  being  somewhat  drawn  asunder  by  the  tension  of  their 
attachments  to  the  sides  of  the  dish  (Fig.  215,  4).  A  similar 
phenomenon  is  usually  found  in  Trochus,  according  to  -Robert, 
and  many  otlier  observers  have  hkewise  found  the  upper  furrow 
to  be  shorter  than  the  one  below.  In  the  various  species  of  the 
genus  Crepidula,  Conklin  asserts  that  the  two  furrows  are  equal 
in  C.  convexa,  ^'^hat  the  upper  one  is  the  shorter  in  C.  fornicata, 
and  that  the  upper  one  all  but  disappears  in  C.  plana;  but  we 
may  well  be  permitted  to  doubt,  without  the  evidence  of  very 
special  investigations,  whether  these  slight  physical  differences  are 
actually  characteristic  of,  and  constant  in,  particular  species. 
Returning  to  the  experimental  case,  Robert  found  that  by  with- 
drawing a  little  air  from,  and  so  diminishing  the  bulk  of  the  two 
terminal  bubbles  (i.e.  those  at  the  ends  of  the  intermediate  lamina), 


558 


THE  FORMS  OF  TISSUES 


[CH. 


the  upper  polar  furrow  was  caused  to  elongate,  till  it  became  equal 
in  length  to  the  lower;  and  by  continuing  the  process  it  became 
the  longer  in  its  turn.  These  two  conditions  have  again  been 
described  by  investigators  as  characteristic  of  this  embryo  or  that; 
for  instance  in  Unio,  Lillie  has  described  the  two  furrows  as 
gradually  altering  their  respective  lengths*;    and  Wilson  (as  LilHe 


Fig.  215.     Aggregations  of  four  soap-bubbles,  to  shew  various  arrangements  of 
the  intermediate  partition  and  polar  furrows.     After  Robert. 

remarks)  had  already  pointed  out  that  "the  reduction  of  the  apical 
cross-furrow,  as  compared  with  that  at  the  vegetative  pole  in 
molluscs  and  annelids,  'stands  in  obvious  relation  to  the  different 
size  of  the  cells  produced  at  the  two  polesf." 

When  the  two  lateral  bubbles  are  gradually  reduced  in  size,  or 
the  two  terminal  ones  enlarged,  the  upper  furrow  becomes  shorter 

*  F.  R.  Lillie,  Embryology  of  the  Unionidae,  Journ.  Morph.  x,  p.  12,  1895. 
t  E.  B.  Wilson,  The  cell-lineage  of  Nereis,  Journ.  Morph.  vi,  p.  452,  1892. 


VII]  OF  POLAR  FURROWS  559 

and  shorter;  and  at  the  moment  when  it  is  about  to  vanish,  a  new 
furrow  makes  its  instantaneous  appearance  in  a  direction  perpen- 
dicular to  the  old  one;  but  the  inferior  furrow,  constrained  by  its 
attachment  to  the  base,  remains  unchanged,  and  it  looks  as  though 
our  two  polar  furrows,  which  were  formerly  parallel,  were  now  at 
right  angles  to  one  another.  But  in  fact,  the  geometry  of  the 
whole  system  is  entirely  altered.  Before,  two  furrows  left  each 
end  of  one  polar  furrow  for  the  same  end  of  the  other  polar  furrow, 
and  the  two  cells  at  either  end  were  shaped  like  "liths"  of  an  orange. 
Under  the  new  arrangement,  two  furrows  leave  each  end  of  one  for 
the  two  ends  of  another.  The  figure  is  now  divided  by  six  similar 
furrows  into  four  similar  curviHnear  triangles*;  it  has  become 
(approximately)  a  spherical  tetrahedron,  and  the  four  cells  into 
which  it  is  divided  are  four  similar  and  symmetrical  figures,  also 
tetrahedral,  all  meeting  in  a  point  at  the  centroid  of  the  figure. 
Such  a  four-celled  embryo,  described  as  having  two  polar  furrows 
arranged  in  a  cross,  has  often  been  seen  and  figured  by  the 
embryologists.  Robert  himself  found  this  condition  in  Trochus,  as 
an  occasional  or  exceptional  occurrence :  it  has  been  described  as 
normal  in  Asterina  by  Ludwig,  in  Branchipus  by  Spangenberg,  and 
in  Podocoryne  and  Hydractinia  by  Bunting. 

So,  by  slight  and  delicate  modifications,  we  pass  through  many, 
and  perhaps  through  all,  of  the  possible  arrangements  of  external 
furrows  and  internal  partitions  which  divide  the  four  cells  from  one 
another  in  a  four-celled  egg  or  embryo;  and  many,  or  most,  or 
possibly  all  of  these  arrangements  have  been  more  or  less  frequently 
observed  in  the  four-celled  stages  of  various  embryos.  iVnd  all 
these  configurations,  which  the  embryologists  have  witnessed  and 
described,  belong  to  that  large  class  of  phenomena  whose  distribution 
among  embryos,  or  among  organisms  in  general,  bears  no  relation 
to  the  boundaries  of  zoological  classification;  through  molluscs, 
worms,  coelenterates,  vertebrates  and  what  not,  we  meet  with  now 
one  and  now  another,  in  a  medley  which  defies  classification.  They 
are  not  ''vital  phenomena,"  or  "functions"  of  the  organism,  or 
special  characteristics  of  this  organism  or  that,  but  purely  physical 

*  That  the  sphere  can  be  symmetrically  divided  into  four  equilateral  triangles, 
after  the  manner  of  these  embryos  (or  of  many  pollen-grains),  is  an  elementary 
fact  of  great  importance  in  geometry  and  trigonometry. 


560  THE  FORMS  OF  TISSUES  [ch. 

phenomena.  The  kindred  but  more  compUcated  phenomena  analogous 
to  the  polar  furrow,  which  arise  when  a  larger  number  of  cells 
than  four  are  associated  together,  we  shall  deal  with  in  the  next 
chapter. 

Having  shewn  that  the  capillary  phenomena  are  patent  and 
unmistakable  during  the  earlier  stages  of  embryonic  development, 
but  soon  become  more  obscure  and  less  capable  of  experimental 
reproduction  in  the  later  stages  when  the  cells  have  increased  in 
number,  various  writers  including  Robert  himself  have  been  inclined 
to  argue  that  the  physical  phenomena  die  away,  and  are  over- 
powered and  cancelled  by  agencies  of  a  different  order.  Here  we 
pass  into  a  region  where  observation  and  experiment  are  not  at 
hand  to  guide  us,  and  where  a  man's  trend  of  thought,  and  way  of 
judging  the  whole  evidence  in  the  case,  must  shape  his  philosophy. 
We  must  always  remember  that  even  in  a  froth  of  soap-bubbles 
we  can  apply  an  exact  analysis  only  to  the  simplest  cases  and 
conditions;  we  cannot  describe,  but  can  only  imagine,  the  forces 
which  in  such  a  froth  control  the  respective  sizes,  positions  and 
curvatures  of  the  innumerable  bubbles  and  films  of  which  it  con- 
sists; but  our  knowledge  is  enough  to  leave  us  assured  that  what 
we  have  learned  by  investigation  of  the  simplest  cases  includes  the 
principles  which  determine  the  most  complex.  In  the  case  of  the 
growing  embryo  we  know  from  the  beginning  that  surface-tension 
is  only  one  of  the  physical  forces  at  work;  and  that  other  forces, 
including  those  displayed  within  the  interior  of  each  living  cell,  play 
their  part  in  the  determination  of  the  system.  But  we  have  no 
evidence  whatsoever  that  at  this  point,  or  that  point,  or  at  any,  the 
dominion  of  the  physical  forces  over  the  material  system  gives  place 
to  a  new  condition  where  agencies  at  present  unknown  to  the 
physicist  impose  themselves  on  the  living  matter,  and  become 
responsible  for  the  conformation  of  its  material  fabric. 

Before  we  leave  for  the,  present  the  subject  of  the  segmenting 
egg,  we  may  take  brief  note  of  two  associated  problems:  viz. 
(1)  the  formation  and  enlargement  of  the  segmentation  cavity,  or 
central  interspace  around  which  the  cells  tend  to 'group  themselves 
in  a  single  layer,  and  (2)  the  formation  of  the  gastrula,  that  is  to 
say  (in  a  typical  case)  the  conversion  by  "invagination,"  of  the 


VII]  OF  THE  GASTRULA  561 

one-layered  ball  into  a  two-layered  cup.  Neither  problem  is  free 
from  difficulty,  and  all  we  can  do  meanwhile  is  to  state  them  in 
general  terms,  introducing  some  more  or  less  plausible  assumptions. 

The  former  problem  is  comparatively  easy,  as  regards  the  tendency 
of  a  segmentation  cavity  to  enlarge,  when  once  it  has  been  estab- 
lished. We  may  then  assume  that  subdivision  of  the  cells  is  due 
to  the  appearance  of  a  new-formed  septum  within  each  cell,  that 
this  septum  has  a  tendency  to  shrink  under  surface-tension,  and 
that  these  changes  will  be  accompanied  on  the  whole  by  a  diminu- 
tion of  surface-energy  in  the  system.  This  being  so,  it  may  be 
shewn  that  the  volume  of  the  divided  cells  must  be  less  than  it  was 
prior  to  division,  or  in  other  words  that  part  of  their  contents  must 
exude  during  the  process  of  segmentation*.  Accordingly,  the  case 
where  the  segmentation  cavity  enlarges  and  the  embryo  developes 
into  a  hollow  blastosphere  may,  under  the  circumstances,  be  simply 
described  as  the  case  where  that  outflow  or  exudation  from  the  cells 
of  the  blastoderm  is  directed  on  the  whole  inwards. 

The  physical  forces  involved  in  the  invagination  of  the  cell-layer 
to  form  the  gastrula  have  been  repeatedly  discussed  f,  but  the 
several  explanations  are  conflicting,  and  are  far  from  clear.  There 
is,  however,  a  certain  homely  phenomenon  which  goes  some  way, 
perhaps  a  long  way,  to  explain  this  remarkable  configuration.  An 
ordinary  gelatine  lozenge,  or  jujube,  has  (Uke  the  developing 
gastrula)  a  more  or  less  spherical  form,  depressed  or  dimpled  at  one 
side;  this  is  a  very  noteworthy  conformation,  and  it  arises,  auto- 
matically, by  the  shrinkage  of  a  sphere.  Were  the  initial  sphere  of 
gelatine  perfectly  homogeneous,  and  so  situated  as  to  shrink  with 
absolute  uniformity,  it  would  merely  shrink  into  a  smaller  sphere; 
it  does  nothing  of  the  kind.  There  is  always  some  part  or  other 
which  shrinks  a  little  'hore  than  the  rest  J ;  and  the  dimple  so  formed 
goes  on  increasing,  until  at  last  a  very  perfect  cup-shaped  figure  is 
formed.     I   imagine   that   the   gastrula   is   formed   in   much    the 

*  Professor  Peddie  has  given  me  this  interesting  result,  but  the  mathematical 
reasoning  is  too  lengthy  to  be  set  forth  here. 

t  Cf.  Butschli,  Arch.f.  Entw.  Mech.  v,  p.  592,  1897;  Rhumbler,  ibid,  xiv,  p.  401, 
1902;  Assheton,  ibid,  xxxi,  p.  46,  1910. 

X  Just  as  there  may  be  some  small  part  which  shrinks  a  little  less.  But  this 
we  should  not  distinguish  from  the  common  case  where  one  small  part  grows 
a  little  more,  and  so  "produces  a  6ud,"  as  in  the  yeast-cell  on  p.  363. 


562  THE  FORMS  OF  TISSUES  [ch. 

same  way,  save  only  that  the  initial  dimple,  instead  of  being 
fortuitous,  has  its  constant  place,  determined  by  the  physico- 
chemical  heterogeneity  of  the  embryo.  We  may  even  go  one  step 
further,  and  see  (or  imagine  we  see)  in  the  formation  of  the  gastrula 
a  physico-chemical  or  physiological  turning-point,  the  segmentation 
cavity  being  due  (as  we  have  seen)  to  an  inward  flow,  and  a  reversal 
of  the  current  leading  to  that  shrinkage  which  produces  the  gastrula. 


Fig.  216.    Effect  of  shrinkage  on  a  globule  of  gelatine. 
After  E.  Hatschek. 


A  note  on  shrinkage 

We  have  dealt  much  with  growth,  but  the  fact  is  that  negative 
growth,  or  shrinkage,  is  also  an  important  matter;  and  just  as  we 
find  a  whole  series  of  phenomena  to  be  based  on  the  extension  or 
expansion  of  bubbles,  vesicles,  etc.,  so  there  is  another  series, 
physically  ahke  and  mathematically  identical,  which  depend  on  the 
shrinkage  of  a  soHd  or  semi-fluid  mass.  After  all,  growth  and  its 
converse  go  hand  in  hand,  and  a  special  case  of  shrinkage  is  that 
surface-tension  to  which  all  the  Plateau  configurations  are  due. 
One  clear  case,  the  gastrula,  we  have  touched  on,  and  we  have 
discussed  another  which  led  to  the  stellate  dodecahedra  of  the  Rush. 

As  a  cube  of  gelatine,  or  of  paraffin,  dries,  and  shrinks,  it  alters 
its  shape  in  a  remarkable  way*.  Its  corners  become  more  salient, 
its  sides  become  concave ;   its  cross-section  has  the  form  of  a  four- 

*  Emil  Hatschek.  Kolloid  Ztschr.  xxxv,  pp.  67-76,  1924;   Nature,  1st  Nov.  1924. 


VII 


OF  THE  EFFECTS  OF  SHRINKAGE 


563 


rayed  star  with  rounded  angles.  The  block  has  dried  unequally; 
its  corners  and  its  edges  were  naturally  the  first  to  dry*,  and  the 
twelve  dried  and  hardened  edges  began  to  play  the  part  of  the  wire 
frame  in  Plateau's  soap-bubble  experiment.  The  shrinking  cube  is 
tending  towards  the  identical  configuration  shewn  in  Fig.  212;  it  is 
a  minimal  configuration,  partially  reahsed  in  a  coarse  material,  but 
reahsable  to  perfection  in  a  film. 

A  shrinking  cylinder  (as  Plateau  knew)  she)vs  various 
phenomena,  depending  on  its  proportions.  A  low,  squat  cyhnder 
begins  to  show  a  pulley-hke  groove — a  catenoid— around  its  periphery, 
precisely  like  the  soap-film  between  its  two  wire  rings  in  Fig.  108 ; 
and  as  the  groove  deepens,  the  plane  surfaces  of  the  cyhnder  also  begin 


Fig.  217.    Shrinkage  of  cube  and  cylinder. 


to  dimple  in.  They  become  spherical  as  they  grow  more  concave,  and 
the  deepening  groove  of  the  pulley  passes  from  a  catenoid  to  a  nodoid 
curve — so  at  least  theory  tells  usf,  for,  beautiful  as  the  experimental 
configurations  are,  they  hardly  lend  themselves  to  precise  measure- 
ments of  curvature.  But  this  shrunken  cyhnder  is  now  wonderfully 
like  the  "  amphicoelous  "  vertebra  of  a  cartilaginous  fish,  the  simplest 
and  most  ''primitive"  vertebra  of  all.  A  series  of  cracks,  or  sphts, 
around  the  circular  groove  in  the  vertebra  seem  to  be  a  final  result 
of  irregular  shrinkage,  not  shewn  in  the  more  homogeneous  gelatine. 
A  long  cylinder,  or  thread,  of  gelatine  tends  to  become  fluted, 
with  three  or  more  ribs  or  folds,  and  it  is  in  this  way  that  threads 

*  Just  as,  conversely,  the  prominent  parts  of  a  crystal  tend  to  grow  more 
rapidly  than  the  rest  in  a  super-saturated  solution,  and  to  dissolve  more  rapidly 
in  one  below  saturation;  cf.  0.  Lehmann,  Ueber  das  Wachstum  der  Krystalle, 
Ztschr.  f.  Krystallogr.  i,  p.  453. 

t  See  p  369. 


564  THE  FORMS  OF  TISSUES  [ch. 

of  viscose,  or  artificial  silk,  tend  likewise  to  have  a  ridged  or  fluted 
structure,  and  gain  in  lustre  thereby.  The  subject  is  new,  and 
hardly  ripe  for  full  discussion ;  but  it  holds  out  promise  (as  it  seems 
to  me)  of  many  biological  lessons  and  illustrations. 

We  glanced  in  passing  at  such  "shrinkage-patterns"  as  are  found, 
for  instance,  on  the  little  shells  of  Lagena,  or  on  those  other  hanging 
drops  which  constitute  Emil  Hatschek's  artificial  medusae;  it 
is  no  small  subject.  A  stretched  elastic  membrane,  circular  or 
spherical,  remains  spherical  or  circular  when  we  let  its  tension 
relax;  but  if,  to  begin  with,  we  coat  the  rubber  with  a  pliant  but 
non-elastic  material  such  as  wax,  the  waxen  layer,  failing  to  con- 


Fig.  218.     Amphicoelous  vertebrae  of  a  shark. 

tract,  is  thrown  into  more  or  less  characteristic  folds.  In  a  dried 
pea  the  seed  has  shrunken  through  loss  of  moisture,  and  the  loose 
outer  coat  wrinkles  up*.  The  pretty  pattern  of  a  poppy-seed  arises 
in  the  same  way;  but  so  do  the  wrinkles  on  an  old  man's  withered 
skin.  When  our  experimental  elastic  with  its  non-contractile  coat 
is  suffered  to  contract,  the  first  sign  of  the  coat's  inability  to  keep 
pace  is  the  appearance  of  little  domes,  or  hummocks,  or  bhsters; 
and  soon  from  each  of  these  there  run  out  folds,  which  tend  to  fork, 
and  the  angles  between  the  three  branches  tend  to  equalise.  They 
tend,  in  simple  and  symmetrical  cases,  to  form  a  pattern  of  hexagons, 
with  occasional  pentagons  or  quadrilaterals  between ;  but  where  the 
surface  is  larger  and  the  coat  more  flexible  the  folds  form  an  irregular 
network,  still  with  the  various  anticlines  mostly  meeting  in  three- 

*  The  difference  between  a  smooth  and  a  wrinkled  pea,  familiar  to  Mendelians, 
merely  depends,  somehow,  on  amount  and  rate  of  shrinkage. 


VII]  OF  SHRINKAGE-PATTERNS  565 

way  nodes*.  Nature  will  ring  the  changes  on  the  resultant  patterns, 
according  as  the  surface  be  plane  or  curved,  spherical  or  cylindrical, 
coarse  or  fine,  fragile  or  tough.  But  on  these  general  fines  very 
many  structures,  both  regular  and  irregular,  spines,  bristles,  ridges, 
tubercles  and  wrinkled  patterns,  bid  fair  to  find  their  physical  or 
mechanical  interpretation;  and  it  is  in  the  more  or  less  hardened 
parts  of  plant  or  animal  that  we  find  them  one  and  all  displayed. 
On  the  egg  of  a  butterfly,  on  the  grooved  and  dotted  elytron  of  a 
beetle,  on  the  notched  forehead  of  a  scarab,  in  thesaw-fike  teeth  on 
a  grasshopper's  leg,  in  the  little  fines  of  dotted  tubercles  on  the 
shell  of  a  Rissoa,  more  crudely  in  the  lozenged  bark  of  elm  or  pine, 
we  see  a  very  few  of  this  innumerable  class  of  "  shrinkage-patterns." 

*  The  fact  that  such  triplets  of  divergent  ridges  or  crests  are  not  a  feature  in 
the  topography  of  mountain-ranges  is  a  strong  argument  against  the  view  that 
general  shrinkage  accounts  for  the  pattern  of  the  earth's  crust.  Cf.  A.  J.  Bull, 
The  pattern  of  a  contracting  earth,  Geolog.  Mag.  lxix,  pp.  73-75,  1932;  A.  E.  B. 
de  Chancourtois,  <^.R.  lix,  p.  348,  1903. 


CHAPTER  VIII 

THE  FORMS  OF  TISSUES  OR  CELL-AGGREGATES  {continued) 

The  problems  which  we  have  been  considering,  and  especially  that 
of  the  bee's  cell,  belong  to  a  class  of  "  isoperimetrical "  problems, 
which,  deal  with  figures  whose  surface  is  a  minimum  for  a  definite 
content  or  volume.  Such  problems  soon  become  difficult*,  but  we 
may  find  many  easy  examples  which  lead  us  towards  the  explanation 
of  biological  phenomena;  and  the  particular  subject  which  we  shall 
find  most  easy  of  approach  is  that  of  the  division,  in  definite  pro- 
portions, of  some  definite  portion  of  space,  by  a  partition- wall  of 
minimal  area.  The  theoretical  principles  so  arrived  at  we  shall  then 
attempt  to  apply,  after  the  manner  of  Berthold  and  Errera,  to  the 
biological  phenomena  of  cell-division. 

This  investigation  may  be  approached  in  two  ways:  by  con- 
sidering the  partitioning  off  from  some  given  space  or  area  of  one-half 
(or  some  other  fraction)  of  its  content;  or  again,  by  dealing  with  the 
partitions  necessary  for  the  breaking  up  of  a  given  space  into  a 
definite  numbet  of  compartments. 

If  We  begin  with  the  simple  case  of  a  cubical  cell,  it  is  obvious  that, 
to  divide  it  into  two  halves,  the  smallest  partition-wall  is  one  which 
runs  parallel  to,  and  midway  between,  two  of  its  opposite  sides. 
If  we  call  a  the  length  of  one  of  the  edges  of  the  cube,  then  a^  is  the 
area,  ahke  of  one  of  its  sides  and  of  the  partition  which  'we  have 
interposed  parallel  thereto.  But  if  we  now  consider  the  bisected 
cube,  and  wish  to  divide  the  one-half  of  it  again,  it  is  obvious  that 
another  partition  parallel  to  the  first,  so  far  from  being  the  smallest 
possible,  is  twice  the  size  of  a  cross-partition  perpendicular  to  it; 
for  the  area  of  this  new  partition  is  a  x  a/2.  And  again,  for  a 
third  bisection,  our  next  partition  must  be  perpendicular  to  the  other 
two,  and  is  obviously  a  Httle  scj^uare,  with  an  area  of  (|a)2  =  ^a^. 

*  Minkowski,  and  others  have  shewn  how  hard  it  is,  for  instance,  to  prove  the 
seemingly  obvious  proposition  that  the  sphere,  of  all  figures,  has  the  greatest  volume 
for  a  given  surface;  cf.  (e.g.)  T.  Bonneson,  Les  problemes  des  isoperimetres  et  des 
iseplphanes,  Paris,  1929.  For  a  historical  account  of  this  class  of  problems,  see 
G.  Enestrom,  in  Bibl.  Math.  1888. 


CH.viii]  OF  SACHS'S  RULES  567 

From  this  we  may  draw  the  simple  rule  that,  for  a  rectangular 
body  or  parallelepiped  to  be  bisected  by  means  of  a  partition  of 
minimal  area,  (1)  the  partition  must  cut  across  the  longest  axis  of 
the  figure;  and  (2)  in  successive  bisections,  each  partition  must  run. 
at  right  angles  to  its  immediate  predecessor. 


X 


/\ 

t; 

/ 

/         X 

V 

/ 

/ 

/ 

^' 

/ 

/ 

7 

/ 

Fig.  219.     After  Berthold. 

We  have  already  spoken  of  "Sachs's  Rules,"  which  are  an 
empirical  statement  of  the  method  of  cell-division  in  plant-tissues ; 
and  we  may  now  set  them  forth  as  follows : 

(1)  The  cell  tends  to  divide  into  two  co-equal  parts. 

(2)  Each  new  plane  of  division  tends  to  intersect  thCj  preceding 
plane  of  division  at  right  angles. 

The  first  of  these  rules  is  a  statement  of  physiological  fact,  not 
without  its  exceptions,  but  so  generally  true  that  it  will  justify  us 
in  limiting  our  enquiry  for  the  most  part  to  cases  of  equal  sub- 
division That  it  is  by  no  means  universally  true  for  cells  generally 
is  shewn,  for  instance,  by  such  well-known  cases  as  the  unequal 
segmentation  of  the  frog's  egg.  It  is  true,  when  the  dividing  cell 
is  homogeneous  and  under  the  influence  of  symmetrical  forces ;  but 
it  ceases  to  be  true  when  the  field  is  no  longer  dynamically  sym- 
metrical, as  when  the  parts  differ  in  surface  tension  or  internal 
pressure,  or,  speaking  generally,  in  their  chemico-physical  properties 


568  THE  FORMS  OF  TISSUES  [ch. 

and  conditions.  This  latter  condition,  of  asymmetry  of  field,  is 
frequent  in  segmenting  eggs*,  and  it  then  covers  or  includes  the 
principle  upon  which  Balfour  laid  stress  as  leading  to  "unequal"  or 
to  "partial"  segmentation  of  the  egg — viz,  the  unequal  or  asym- 
metrical distribution  of  protoplasm  and  of  food-yolk. 

The  second  rule,  which  also  has  its  exceptions,  is  true  in  a  large 
number  of  cases,  and  owes  its  vahdity,  as  we  may  judge  from  the 
illustration  of  the  repeatedly  bisected  cube,  to  the  guiding  principle 
of  minimal  areas.  It  is  in  short  subordinate  to  a  much  more 
important  and  fundamental  rule,  due  not  to  Sachs  but  to  Errera; 
that  (3)  the  incipient  partition- wall  of  a  dividing  cell  tends  to  be 
such  that  its  area  is  the  least  possible  by  which  the  given  space-content 
can  be  enclosed. 

Let  us  return  to  the  case  of  our  cube,  and  suppose  that,  instead 
of  bisecting  it,  we  desire  to  shut  off  some  small  portion  only  of  its 
volume.  It  is  found  in  the  course  of  experiments  upon  soap-films, 
that  if  we  try  to  bring  a  partition-film  too  near  to  one  side  of  a 
cubical  (or  rectangular)  space  it  becomes  unstable,  and  is  then  easily 
shifted  to  a  new  position  in  which  it  constitutes  a  curved  cylindrical 
wall  cutting  oif  one  corner  of  the  cube.  It  still  meets  the  sides  of 
the  cube  at  right  angles  (for  reasons  which  we  have  already  con- 
sidered); and,  as  we  may  see  from  the  symmetry  of  the  case,  it 
constitutes  one-quarter  of  a  cylinder.  Our  plane  transverse  parti- 
tion had  always  the  same  area,  wherever  it  was  placed,  viz.  a^\ 
and  it  is  obvious  that  a  cylindrical  wall,  if  it  cut  oif  a  small  corner, 
may  be  much  less  than  this.  We  want,  accordingly,  to  determine 
what  volume  might  be  partitioned  oif  with  equal  economy  of  wall- 
space  in  one  way  as  the  other,  that  is  to  say,  what  area  of  cyhndrical 

*  M.  Robert  {loc.  cit.  p.  305)  has  compiled  a  long  list  of  cages  among  the  molluscs 
and  the  worms,  where  the  initial  segmentation  of  the  egg  proceeds  by  equal  or 
unequal  division.  The  two  cases  are  about  equally  numerous.  But  like  most 
other  writers  of  his  time,  he  would  ascribe  this  equality  or  inequality  rather  to 
a  provision  for  the  future  than  to  a  direct  effect  of  immediate  physical  causation : 
"II  semble  assez  probable,  comme  on  I'a  dit  sou  vent,  que  la  plus  grande  taille 
d'un  blastomere  est  liee  a  I'importance  et  au  developpement  precoce  des  parties 
du  corps  qui  doivent  en  naitre :  il  y  aurait  la  une  sorte  de  reflet  des  stades  posterieures 
du  developpement  sur  les  premieres  phenomenes,  ce  que  M.  Ray  Lankester  appelle 
precocious  segregation.  II  faut  avouer  pourtant  qu'on  est  parfois  assez  embarrasse 
pour  assigner  une  cause  a  pareilles  differences." 


VIII]  OF  AREAE  MINIMAE  ,  569 

wall  would  be  neither  more  nor  less  than  the  area  a^.   The  calculation 
is  easy. 

The  surface-area  of  a  cylinder  of  length  a  is  27rr .  a,  and  that 
of  our  quarter-cylinder  is,  therefore,  a.7Trl2;  and  this  being,  by 
hypothesis,  =  a^,  we  have  a  =  7rr/2,  or  r  =  2a/7r. 

The  volmne  of  a  cyhnder  of  length  a  is  anr^,  and  that  of  our 
quarter-cyhnder  is  a .  Trr^/i,  which  (by  substituting  the  value  of  r) 
is  equal  to  a^JTi^ 

Now  precisely  this  same  volume  is,  obviously,  shut  off  by  a 
transverse  partition  of  area  a^  if  the  third  side  of  the  rectangular 
space  be  equal  to  a/n;  and  this  fraction, 
if  we  take  a  =  1,  is  equal  to  0-318... ,  or 
rather  less  than  one- third.  And,  as  we 
have  just  seen,  the  radius,  or  side,  of 
the  corresponding  quarter-cylinder  will 
be  twice  that  fraction,  or  equal  to  0-636 
times  the  side  of  the  cubical  cell. 

If  then,  in  the  process  of  division  of 
a  cubical  cell,  it  so  divide  that  the  two 
portions  be  not  equal  in  volume  but 
that  one  portion  be  anything  less  than 
about  three-tenths  of  the  whole  or  three- 
sevenths  of  the  other  portion,  there  will  be  a  tendency  for  the  cell 
to  divide,  not  by  means  of  a  plane  transverse  partition,  but  by  means 
of  a  curved,  cylindrical  wall  cutting  off  one  corner  of  the  original 
cell;   and  the  part  so  cut  off  will  be  one- quarter  of  a  cylinder. 

By  a  similar  calculation  we  can  shew  that  a  spherical  wall,  cutting 
oiF  one  soHd  angle  of  the  cube  and  constituting  an  octant  of  a  sphere, 
would  likewise  be  of  less  area  than  a  plane  partition  as  soon  as  the 
volume  to  be  enclosed  was  not  greater  than  about  one- quarter  of 
the  original  cell*.     But  while  both  the  cyhndrical  wall  and  the 


Fig.  220. 


*  The  principle  is  well  illustrated  in  an  experiment  of  Sir  David  Brewster's 
{Trans.  R.S.E.  xxv,  p.  Ill,  1869).  A  soap-film  is  drawn  over  the  rim  of  a  wine- 
glass, and  then  covered  by  a  watch-glass.  The  film  is  inclined  or  shaken  till  it 
becomes  attached  to  the  glass  covering,  and  it  then  immediately  changes  place, 
leaving  its  transverse  position  to  take  up  that  of  a  spherical  segment  extending 
from  one  side  of  the  wine-glass  to  its  cover,  and  so  enclosing  the  same  volume  of 
air  as  formerly  but  with  a  great  economy  of  surface,  precisely  as  in  the  case  of  our 
spherical  partition  cutting  oflF  one  corner  of  a  cube. 


570 


THE  FORMS  OF  TISSUES 


[CH. 


spherical  wall  would  be  of  less  area  than  the  plane  transverse 
partition  after  that  Hmit  (of  one-quarter  volume)  was  passed,  the 
cylindrical  would  still  be  the  better  of  the  two  up  to  a  further  hmit. 
It  is  only  when  the  volume  to  be  partitioned  off  is  no  greater  than 


1 1 1 1 

Fraction    of  cube  partitioned  off. 

15 

\^ 

- 

t'4 

H' 

■ 

1-3 
1-2 
M 

Plane          \^A    \ 

R     transverse  partition 

9 

\ 

X  \                                         '^ 

- 

\ 

.0            "^ 

\  " 

u' 

•2 

1 

_J 

\ 

A'^     B'       ^ 
Fig:  221. 


C 


about  0-15,  or  somewhere  about  one-seventh  of  the  whole,  that  the 
spherical  cell-wall  in  a  corner  of  the  cubical  cell,  that  is  to  say  the 
octant  of  a  sphere,  is  definitely  of  less  area  than  the  quarter-cylinder. 
In  the  accompanying  diagram  (Fig.  221)  the  relative  areas  of  the 
three  partitions  are  shewn  for  all  fractions,  less  than  one-half,  of 
the  divided  cell. 

In  this  figure,  we  see  that  the  plane  transverse  partition,  whatever  fraction 
of  the  cube  it  cut  ofF,  is  always  of  the  same  dimensions,  that  is  to  say  is 


VIII]  OF  AREAE  MINIMAE  571 

always  equal  to  a^,  or  =1.  If  one-half  of  the  cube  have  to  be  cut  off,  this 
plane  transverse  partition  is  much  the  best,  for  we  see  by  the  diagram  that  a 
cylindrical  partition  cutting  off  an  equal  volume  would  have  an  area  about 
25  per  cent,  and  a  spherical  partition  would  have  an  area  about  50  per  cent, 
greater.  The  point  A  in  the  diagram  corresponds  to  the  point  where  the 
cylindrical  partition  would  begin  to  have  an  advantage  over  the  plane,  thati  i 
to  say  (as  we  have  seen)  when  the  fraction  to  be  cut  off  is  about  one-third, 
or  0-318  of  the  whole.  In  like  manner,  at  B  the  spherical  octant  begins  to 
have  an  advantage  over  the  plane;  and  it  is  not  till  we  reach  the  point  C 
that  the  spherical  octant  becomes  of  less  area  than  the  quarter-cylinder. 

The  case  we  have  dealt  with  is  of  little  practical  importance  to 
the  biologist,  because  the  cases  in  which  a  cubical,  or  rectangular, 
cell  divides  unequally  and  unsymmetrically  are  apparently  few ;  but 
we  can  find,  as  Berthold  pointed  out,  a  few  examples,  as  in  the  hairs 
within  the  reproductive  "  conceptacles "  of 
certain  Fuci  (Sphacelaria,  etc..  Fig.  222),  or 
in  the  "paraphyses"  of  mosses  (Fig.  226). 
But  it  is  of  great  theoretical  importance :  as 
serving  to  introduce  us  to  a  large  class  of 
cases  in  which,  under  the  guiding  principle 
of  minimal  areas,  the  shape  and  relative 
dimensions  of  the  original  cavity  lead  to 
cell-division  in  very  definite  and  sometimes 
unexpected  ways.  It  is  not  easy,  nor  indeed  possible,  to  give  a  general 
account  of  these  cases,  for  the  limiting  conditions  are  somewhat, 
complex  and  the  mathematical  treatment  soon  becomes  hard.  But 
it  is  easy  to  comprehend  a  few  simple  cases,  which  carry  us  a  good 
long  way;  and  which  will  go  far  to  persuade  the  student  that,  in 
other  cases  which  we  cannot  fully  master-,  the  same  guiding  principle 
is  at  the  root  of  the  matter. 

The  bisection  of  a  solid  (or  its  subdivision  in  other  definite  propor- 
tions) soon  leads  us  into  a  geometry  which,  if  not  necessarily  difficult, 
is  apt  to  be  unfamiliar ;  ^  but  in  such  problems  we  can  go  some  way, 
and  often  far  enough  for  our  purpose,  if  we  merely  consider  the 
plane  geometry  of  a  side  or  section  of  our  figure.  For  instance,  in 
the  case  of  the  cube  which  we  have  just  been  considering,  and  in 
the  case  of  the  plane  and  cyhndrical  partitions  by  which  it  has  been 
divided,  it  is  obvious,  since  these  two  partitions  extend  symmetrically 
from  top  to  bottom  of  our  cube,  that  we  need  only  have  considered 


572  THE  FORMS  OF  TISSUES  [ch. 

the  manner  in  which  they  subdivide  the  base  of  the  cube;  in  short 
the  problem  of  the  sohd,  up  to  a  certain  point,  is  contained  in  our 
plane  diagram  of  Fig.  221.  And  when  our  particular  soUd  is  a 
solid  of  revolution^  then  it  is  equally  obvious  that  a  study  of  its 
plane  of  symmetry  (that  is  to  say  any  plane  passing  through  its 
axis  of  rotation)  gives  us  the  solution  of  the  whole  problem.  The 
right  cone  is  a  case  in  point,  for  here  the  investigation  of  its  modes 
of  symmetrical  subdivision  is  completely  met  by  an  examination 
of  the  isosceles  triangle  which  constitutes  its  plane  of  symmetry. 

The  bisection  of  an  isosceles  triangle  by  a  Hne  which  shall  be  the 
shortest  possible  is  an  easy  problem;  for  it  is  obvious  that,  if  the 
triangle  be  low,  a  vertical  partition  will  be  shortest;  if  it  be  high, 
a  horizontal  one ;  if  it  be  equilateral,  the  partition  may  run  parallel 
to  any  side;  and  if  it  be  right-angled,  the  partition  may  bisect  the 
right  angle  or  run  parallel  to  either  side  equally  well. 

Let  ABC  be  an  isosceles  triangle  of  which  A  is  the  apex ;  it  may 
be  shewn  that,  for  its  shortest  Hne  of  bisection,  we  are  limited  to 
three  cases:  viz.  to  a  vertical  line  AD,  bisecting  the  angle  at ^  and 
the  side  BC\  to  a  transverse  line  parallel  to  the  base  BC\  or  to  an 
obUque  Hne  paraUel  to  AB  or  to  AC.  The  lengths  of  these  partition 
Hues  follow  at  once  from  the  magnitudes  of  the  angles  of  our  triangle. 
We  know,  to  begin  with,  since  the  areas  of  similar  figures  vary  as 
the  squares  of  their  linear  dimensions,  that,  in  order  to  bisect  the 
area,  a  line  parallel  to  one  side  of  our  triangle  must  always  have 
a  length  equal  to  1/a/2  of  that  side.  If  then,  we  take  our  base, 
BC,  in  all  cases  of  a  length  =  2,  the  transverse  partition,  EF,  drawn 
parallel  to  it  wiU  always  have  a  length  equal  to  2/V2,  or  =  a/2. 
The  vertical  partition,  AD,  since  BD  =  1,  will  always  equal  tan^; 
and  the  oblique  partition,  GH,  being  equal  to  ABjV^,  =  l/'v/2  cos  j8. 
If  then  we  call  our  vertical,  transverse  and  obUque  partitions  F, 
T,  and  0,  we  have  F  =  tan  j8;    T  ==  a/2;    and  0  -  1/V2  cos  j8,  or 

V:T:0  =  tan^/V2  : 1 : 1/2  cos^S. 

And,  working  out  these  equations  for  various  values  of  j8,  we  soon 
see  that  the  vertical  partition  (F)  is  the  least  of  the  three  until 
P  =  45°,  at  which  limit  F  and  0  are  each  equal  to  1/ V2  =  0-707 ; 
that  0  then  becomes  the  least  of  the  three,  and  remains  so  until 


VIII]  OF  AREAE  MINIMAE  573 

^  =  60"",  when  cos  ^  =  0-5,  and  0=  T;  after  which  T  (whose  value 
always  =  1)  is  the  shortest  of  the  three  partitions.    And,  as  we  have 


80°    90° 


10°      20°      30°      40"      50°      60"      70' 

Basal  angle  of  triangle 

Fig.  224.     Comparative  length  of  the  partitions,  transverse,  oblique 
or  vertical,  bisecting  an  isosceles  triangle. 

seen,  these  results  are  at  once  apphcable,  not  only  to  the  case  of 
the  plane  triangle,  but  also  to  that  of  the  conical  or  pyramidal 
cell. 


574  THE  FORMS  OF  TISSUES  [ch. 

In  like  manner,  if  we  have  a  spheroidal  body  less  than  a  hemi- 
sphere, such  for  instance  as  a  low,  watchglass-shaped  cell  (Fig. 
225,  A),  it  is  obvious  that  the  smallest  partition  by  which  we  can 
divide  it  into  two  halves  is  (as  in  our  flattened  disc)  a  median 
vertical  one;  jand  likewise,  the  hemisphere  itself  can  be  bisected 
by  no  smaller  partition  meeting  the  walls  at  right  angles  than  that 
median  one  which  divides  it  into  two  similar  quadrants  of  a  sphere. 
But  if  we  produce  our  hemisphere  into  a  more  elevated  conical 
body,  or  into  a  cylinder  with  spherical  cap,  there  comes  a  point 
where  a  transverse  horizontal  partition  will  bisect  the  figure  with 
less  area  of  partition-wall  than  a  median  vertical  one  (C).  And 
furthermore,  there  will  be  an  intermediate  region,  a  region  where 
height  and  base  have  their  relative  dimensions  nearly  equal  (as 


Fig.  225. 

in  B),  where  an  oblique  partition  will  be  better  than  either 'the 
vertical  or  the  transverse ;  though  here  the  analogy  of  our  triangle 
does  not  suffice  to  give  us  the  precise  hmiting  values. 

We  need  not  examine  these  hmitations  in  detail,  but  we  must 
look  at  the  curvatures  which  accompany  the  several  conditions.  We 
have  seen  that  a  film  tends  to  set  itself  at  equal  angles  to  the  surface 
which  it  meets,  and  therefore,  when  that  surface  is  a  sohd,  to  meet 
it  (or  its  tangent)  at  right  angles.  Our  vertical  partition  is,  there- 
fore, a  plane  surface,  everywhere  normal  to  the  original  cell-walls. 
But  in  the  taller,  conical  cell  .with  transverse  partition,  the  latter 
still  meets  the  opposite  sides  of  the  cell  at  right  angles,  and  it 
follows  that  it  must  itself  be  curved;  moreover,  since  the  tension, 
and  therefore  the  curvature,  of  the  partition  is  everywhere  uniform, 
it  follows  that  its  curved  surface  must  be  a  portion  of  a  sphere, 
concave  towards  the  apex  of  the  original  cell.  In  the  intermediate 
case,  where  we  have  an  obhque  partition  meeting  both  the  base 
and  the  curved  sides  of  the  mother-cell,  the  contact  must  still  be 


viTi]  OF  SIGMOID  PARTITIONS  575 

everywhere  at  right  angles:  provided  we  continue  to  suppose  that 
the  walls  of  the  mother-cell  (hke  those  of  our  diagrammatic  /cube) 
have  become  practically  rigid  before  the  partition  appears,  and  are 
therefore  not  affected  and  deformed  by  the  tension  of  the  latter. 
In  such  a  case,  and  especially  when  the  cell  is  elliptical  in  cross- 
section  or  still  more  comphcated  in  form,  the  partition  may  have 
to  assume  a  complex  curvature  in  order  to  remain  a  surface  of 
minimal  area. 


Fig.  226.     S-shaped  partitions:    A,  Taonia  atomaria  (after  Rcinke);    B,  paraphya 
of  Fucus;   C,  rhizoids  of  moss;   D,  paraphyses  oi  Polytrichmn. 


While  in  very  many  cases  the  partitions  (like  the  walls  of  the 
original  cell)  will  be  either  plane  or  spherical,  a  more  complex 
curvature  will  sometimes  be  alssumed.  It  will  be  apt  to  occur  when 
the  mother-cell  is  irregular  in  shape,  and  one  particular  case  of 
such  asymmetry  will  be  that  in  which  (as  in  Fig.  227)  the  cell  has 
begun  to  branch  before  division  takes  place.  And  again,  whenever 
we  have  a  marked  internal  asymmetry  of  the  cell,  leading  to  irregular 
and  anomalous  modes  of  division,  in  which  the  cell  is  not  necessarily 
divided  into  two  equal  halves  and  in  which  the  partition-wall  may 


576 


THE  FORMS  OF  TISSUES 


[CH. 


assume  an  oblique  position,  then  equally  anomalous  curvatures  will 
tend  to  make  their  appearance*. 

Suppose  an  oblong  cell  to  divide  by  means  of  an  obhque  partition 
(as  may  happen  through  various  causes  or  conditions  of  asymmetry), 
such  a  partition  will  still  have  a  tendency  to  set  itself  at  right  angles 
to  the  rigid  walls  of  the  mother-cell :  and  it  follows  that  our  oblique 
partition,  throughout  its  whole  extent,  will  assume  the  form  of  a 
complex,  saddle-shaped  or  anticlastic  surface. 

Many  such  partitions  of  complex  or  double  curvature  exist,  but 
they  are  not  always  easy  of  recognition,  nor  do  they  often  appear 
in  a  terminal  cell.     We  may  see  them  in  the  roots  (or  rhizoids)  of 


t 

" 


A  B  C  D 

Fig.  227.     Diagrammatic  explanation  of  S-shaped  partition.  , 

mosses,  especially  at  the  point  of  development  of  a  new  rootlet 
(Fig.  226,  C);  and  again  among  mosses,  in  the  "paraphyses"  of  the 
male  plants  (e.g.  in  Polytrichum),  we  find  more  or  less  similar 
partitions  (D).  They  are  frequent  also  among  Fuci,  as  in  the  hairs 
or  paraphyses  of  Fucus  itself  (B).  In  Taonia  atofnaria,  as  figured 
in  Reinke's  memoir  on  the  Dictyotaceae  of  the  Gulf  of  Naples  |, 
we  see,  in  like  manner,  oblique  partitions,  which  on  more  careful 
examination  are  seen  to  be  curves  of  double  curvature  (Fig.  226,  A). 
The  physical  cause  and  origin  of  these  S-shaped  partitions  is 
somewhat  obscure,  but  we  may  attempt  a  tentative  explanation. 
When  we  assert  a  tendency  for  the  cell  to  divide  transversely  to 
its  long  axis,  we  are  not  only  stating  empirically  that  the  partition 


*  Cf  Wildeman,  Attache  des  cloisons,  etc.,  pis.  1,  2. 
f  Nova  Acta  K.  Leop.  Akad.  xi,  1,  pi.  iv. 


VIII]  OF  SIGMOID  PARTITIONS  577 

tends  to  appear  in  a  small,  rather  than  a  large  cross-section  of  the 
cell:  but  we  are  also  ascribing  to  the  cell  a  longitudinal  polarity 
(Fig.  227,  A),  and  imphcitly  asserting  that  it  tends  to  divide  (just 
as  the  segmenting  egg  does),  by  a  partition  transverse  to  its  polar 
axis.  Such  a  polarity  may  conceivably  be  due  to  a  chemical 
asymmetry,  or  anisotropy,  such  as  we  have  learned  of  (from 
Macallum's  experiments)  in  our  chapter  on  Adsorption.  Now  if  the 
chemical  concentration,  on  which  this  anisotropy  or  polarity  (by 
hypothesis)  depends,  be  unsymmetrical,  one  of  its  poles  being  as  it 
were  deflected  to  one  side  where  a  little  branch  or  bud  is  being 
(or  about  to  be)  given  off — all  in  precise  accordance  with  the 
adsorption  phenomena  described  on  p.  4607-then  oiir  "polar  axis" 
would  necessaiily  be  a  curved  axis,  and  the  partition,  being  con- 
strained (again  ex  hypothesi)  to  arise  transversely  to  the  polar  axis, 
would  lie  obHquely  to  the  apparent  axis  of  the  cell  (as  in  B  or  C). 
And  if  the  oblique  partition  be  so  situated  that  it  has  to  meet  the 
opposite  walls  (as  in  C),  then,  in  order  to  do  so  symmetrically  (i.e. 
either  perpendicularly,  as  when  the  cell-wall  is  already  solidified,  or 
at  least  at  equal  angles  on  either  side),  it  is  evident  that  the  partition, 
in  its  course  from  one  side  of  the  cell  to  the  other,  must  necessarily 
assume  a  more  or  less  S-shaped  curvature  (D). 

The  complex  curvature  of  the  partition-walls  in  such  cases  as 
these  m^y  be  illustrated  by  the  following  experiment.     Set  two 

plates  of  glass  (as  in  Fig.  228)  in  a  wire  . 

frame,  so  that  they  may  he  parallel  or        /  / 

at  any  angle  to  one  another;  and  dip      ^ — ■ — 
the  whole  thing  in  soap-solution,  so  that 

a  sheet  of  film  is  formed  between  the        

two  plates  and  is  framed  by  the  two     / 

wires  which  carry  them.     The  film  is,  ^ 

of  course,  a  surface  of  minimal  area ;  its  ^^' 

mean  curva,ture  is  constant  everywhere,  and  (since  the  film  is  an 
open  surface  with  identical  pressure  on  both  sides)  the  mean  curvature 
is  everywhere  nil.  A  related  condition  is  that  the  film  must  meet 
its  solid  framework,  glass  or  wire,  everywhere  at  right  angles  or 
*' orthogonally";  and  this  last  constraint  leads  to  curvatures  of 
extreme  complexity,  which  continually  vary  as  we  rotate  one  plate 
on  the  plane  of  the  other. 


578  THE  FORMS  OF  TISSUES  [ch. 

As  a  matter  of  fact,  while  we  have  abundant  simple  illustrations 
of  the  principles  which  we  have  now  begun  to  study,  apparent 
exceptions  to  this  simplicity,  due  to  an  asymmetry  of  the  cell  itself 
or  of  the  system  of  which  the  single  cell  is  a  part,  are  by  no  means 
rare.  We  know  that  in  cambium-cells  division  often  takes  place 
parallel  to  the  long  axis  of  the  cell,  though  a  partition  of  much  less 
area  would  suffice  if  it  were  set  cross- ways:  and  it  is  only  when  a 
considerable  disproportion  has  been  set  up  between  the  length  and 
breadth  of  the  cell  that  the  balance  is  in  part  redressed  by  the- 
appearance  of  a  transverse  partition.  It  was  owing  to  such  excep- 
tions that  Berthold  was  led  to  quahfy  and  even  to  depreciate  the 
importance  of  the  law  of  minimal  areas  as  a  factor  in  cell-division, 
after  he  himself  had  done  so  much  to  demonstrate  and  elucidate 
it*.  He  was  deeply  and  rightly  impressed  by  the  fact  that  other 
forces  besides  surface  tension,  both  external  and  internal  to  the  cell, 
play  their  part  in  determining  its  partitions,  and  that  the  answer 
to  our  problem  is  not  to  be  given  in  a  word.  Hpw  fundamentally 
important  it  is,  however,  in  spite  of  all  conflicting  tendencies  and 
apparent  exceptions,  we  shall  see  better  and  better  as  we  proceed. 

But  let  us  leave  the  exceptions  and  consider  the  simpler  and 
more  general  phenomena.  And  let  us  leave  the  case  of  the  cubical, 
quadrangular  or  cylindrical  cell,  and  examine  that  of  a  spherical 
cell  and  of  its  successive  divisions,  or  the  still  simpler  case  of  a 
circular,  discoidal  cell. 

When  we  attempt  to  investigate  mathematically  the  place  and 
form  of  a  partition  of  minimal  area,  it  is  plain  that  we  shall  be 
dealing  with  comparatively  simple  cases  wherever  even  one  dimen- 
sion of  the  cell  is  much  less  than  the  other  two.  Where  two 
dimensions  are.small  compared  with  the  third,  as  in  a  thin  cylindrical 
filament  like  that  of  Spirogyra,  we  have  the  problem  at  its  simplest ; 
for  it  is  obvious,  then,  that  the  partition  must  lie  transversely  to 
the  long  axis  of  the  thread.  But  even  where  one  dimension  only 
is  relatively  small,  as  for  instance  in  a  flattened  plate,  our  problem 

*  Cf.  Protoplasmamechanik,  p.  229:  "Insofern  liegen  also  die  Verhaltnisse  hier 
wesentlich  anders  als  bei  der  Zertheilung  hohler  Korperformen  durclj  fliissige 
Lamellen.  Wenn  die  Membran  bei  der  Zelltheilung  die  von  dem  Prinzip  der 
kleinsten  Flachen  geforderte  Lage  und  Kriimmung  annimmt,  so  werden  wit  den 
Grund  dafiir  in  andrer  Weise  abzuleiten  haben." 


VIII]  OF  DISCOID  CELLS  579 

is  so  far  simplified  that  we  see  at  once  that  the  partition  cannot  be 
parallel  to  the  extended  plane,  but  most  cut  the  cell,  somehow,  at 
right  angles  to  that  plane.  In  short,  the  problem  of  dividing  a 
much  flattened  sohd  becomes  identical  with  that  of  dividing  a 
simple  surface  of  the  same  form. 

There  are  a  number  of  small  algae  growing  in  the  form  of  small 
flattened  discs,  and  consisting  (for  a  time  at  any  rate)  of  but  a 
single  layer  of  cells,  which,  as  Berthold  shewed,  exempUfy  this 
comparatively  simple  problem;*  and  we  shall  find  presently  that 
it  is  admirably  illustrated  in  the  cell-divisions  which  occur  in  the 
egg  of  a  frog  or  a  sea-urchin,  when  it  is  flattened  out  under  artificial 
pressure.  These  same  httle  algae  which  serve  to  exempHfy  the 
partitioning  of  a  disc  also  illustrate,  now 
and  then,  a  curious  feature  of  its  contour. 
Such  a  small^  green  alga  as  Castagna 
(Fig.  229)  shews,  and  many  Desmids 
shew  just  as  well,  a  sinuous  border 
running  out  into  rounded  crenations  or 
lobes.  This  is  a  surface-tension  phe- 
nomenon.    A  little  milk  poured  over  an 

apple-pie  gives  a  homely  illustration  of  Fig.  .229.       Castagna  pdycarpa. 

the  same  sinuous  outlines;  a  drop  on  a  ^^^^^^^^ZlT^  ^''''"^  ^^^"*'' 
greasy  plate  spreads  in  the  same  uneven 

way,  and  does  so  indeed  unless  the  utmost  care  be  taken  to  ensure 
absolute  cleanhness  and  surface  equihbrium*. 

Fig.  230 1  represents  younger  and  older  discs  of  the  little  alga 
Erythrotrichia  discigera;  and  it  will  be  seen  that  in  all  stages  save 
the  first  we  have  an  arrangement  of  cell-partitions  which  looks 
somewhat  complex,  but  into  which  we  must  attempt  to  throw  some 
light  and  order.  Starting  with  the  original  single,  and  flattened, 
cell,  we  have  no  difficulty  with  the  first  two  cell-divisions;  for  we 
know  that  no  bisecting  partitions  can  possibly  be  shorter  than  the 
two  diameters,  which  divide  the  cell  into  halves  and  into  quarters. 
We  have  only  to  remember  that,  for  the  sum  total  of  partitions  to 

*  Cf.  Quincke's  "  Ausbreitungserscheinungen,"  in  Poggendorff^s  Annalen,  cxxxix, 
p.  37,  1870;  also  Tomlinson's  papers  in  Phil.  Mag.  vm-xxxix;  and  Van  der 
Mensbrugghe,  Mem.  Cour.  de  VAcad.  R.  Belgique,  xxxiv,  1870;   xxxvii,  1873. 

t  From  Berthold's  Monograph  of  the  Naples  Bangiaceae,  1882. 


580 


THE  FORMS  OF  TISSUES 


[CH. 


be  a  minimum,  three  only  must  meet  in  a  point;  and  therefore, 
the  four  quadrantal  walls  must  shift  a  Uttle,  producing  the  usual 
little  median  partition,  or  cross-furrow,  instead  of  one  common 
central  point   of  junction.     This  intermediate  partition,  however, 


Fig.  230.     Development  of  Erythrotrichia.     After  Berthold. 

will  be  small,  and  to  all  intents  and  purposes  we  may  deal  with  the 
case  as  though  we  had  now  to  do  with  four  equal  cells,  each  one 
of  them  a  perfect  quadrant;  so  our  piroblem  is,  to  find  the  shortest 
line  which  shall  divide  the  quadrant  of  a  circle  into  two  halves  of 


V 

.--^^ 

\^ 

/ 

'"•. 

\ 

/ 

\ 

•A 

v 

j 

X 

\/ 

V 

/__ 

2 

Fig.  231. 

equal  area.  A  radial  partition  (Fig.  231,  A),  starting  from  the 
apex  of  the  quadrant,  is  at  once  excluded,  for  the  reason  just  referred 
to ;  our  choice  must  lie  between  two  modes  of  division  such  as  are 
illustrated  in  Fig.  231,  where  the  partition  is  either  (as  in  B)  con- 
centric with  the  outer  border  of  the  cell,  or  else  (as  in  C)  cuts  that 


VIIl] 


OF  THE  SEGMENTATION  OF  A  DISC 


581 


outer  border;  in  oth^r  words,  our  partition  may  (B)  cut  both  radial 
walls,  or  (C)  may  cut  one  radial  wall  and  the  periphery.  These  are 
the  two  methods  of  division  which  Sachs  called,  respectively, 
(B)  periclinal,  and  (C)  anticlinal*.  We  may  either  treat  the  wa&s 
of  the  dividing  quadrant  as  already  sohdified,  or  at  least  as  having 
a  tension  compared  with  which  that  of  the  incipient  partition  film 
is  inconsiderable;  in  either  case  the  new  partition  must  meet  the 
old  wall,  on  either  side,  at  right  angles,  and  (its  own  tension  and 
curvature  being  everywhere  uniform)  must  take  the  form  of  a 
circular  arc. 


jgX 


M 


Fig.  232. 


We  find  that  a  flattened  cell  which  is  approximately  a  quadrant 
of  a  circle  invariably  divides  ^fter  the  manner  of  Fig.  231,  C,  that 
is  to  say,  by  an  approximately  circular,  anticlinal  wall,  and  this 
we  now  recognise  in  the  eight-celled  stage  of  Eryihrotrichia  (Fig. 
230) ;  let  us  then  consider  that  Nature  has  solved  our  problem,  and 
let  us  work  out  the  actual  geometrio  conditions. 

Let  the  quadrant  OAB  (in  Fig.  232)  be  divided  into  two  parts 
of  equal  area,  by  the  circular  arc  MP.     It  is  required  to  determine 

*  There  is,  I  think,  some  ambiguity  or  disagreement  among  botanists  as  to  the 
use  of  this  latter  term :  the  sense  in  which  I  am  using  it,  viz.  for  any  partition 
which  meets  the  outer  or  peripheral  wall  at  right  angles  (the  strictly  radial  partition 
being  for  the  present  excluded),  is,  however,  clear. 


582  THE  FORMS  OF  TISSUES  [ch. 

(i)  the  position  of  P  upon  the  arc  of  the  quadrant,  that  is  to  say 
the  angle  BOP\  (2)  the  position  of  the  point  M  on  the  side  OA; 
and  (3)  the  length  of  the  arc  MP  in  terms  of  a  radius  of  the  quadrant. 
(1)  Draw  0P\  also  PC  a  tangent,  meeting  OA  in  C;  and  PN , 
perpendicular  to  OA.  Let  us  call  a  a  radius;  and  6  the  angle  at  C, 
which  is  equal  to  OPN ,  or  POB.     Then 

CP  =  a  cot  ^;  PN  =  a  cos  ^;  NO  =  CP  cos  6  =  a  .  cos^  ^/sin  6. 

The  area  of  the  portion  PMN 

=  iCP^  d  -  iPN  .  NO 
=  Ja2  0  cot2  d-iacosd  .a  cos^  ^/sin  6 
=  ia^  (d  cot2  d  -  cos3  e-lsin  6). 

And  the  area  of  the  portion  PNA 

=  ia^  (77/2  -6)-  \0N  .  NP 

=  U^  (77/2  -  (9)  -  Ja  sin  ^  .  a  cos  ^ 

=  \a^  (7r/2  -  ^  -  sin  ^  .  cos  d). 

Therefore  the  area  of  the  whole  portion  PMA 

=  a^l2  (77/2  -  e  +  d  cot2  d  -  cos3  ^/sin  ^  -  sin  ^  .  cos  9) 
=  a2/2  (77/2  -  ^  +  ^  cot2  d  -  cot  <9), 

and  also,  by  hypothesis,  =  J  .  area  of  the  quadrant,  =  Tra'^/S. 
Hence  6  is  defined  by  the  equation 

072  (77/2  -  e+  d  cot2  e  -  cot  d)  =  77a2/8, 

or  77/4:-  6+  d  cot2  ^  -  cot  ^  =  0. 

We  may  solve  this  equation  by  constructing  a  table  (of  which 
the  following  is  a  small  portion)  for  various  values  of  d. 


■d 

7r/4 

-d 

-cot^ 

+ d  cot2  e 

=  x 

34°  34' 

0-7854  - 

-  0-6033  - 

-  1-4514 

+   1-2709  = 

0-0016 

35' 

0-7854 

0-6036 

1-4505 

1-2700 

0-0013 

36' 

0-7854 

0-6039 

1-4496 

1-2690 

0-0009 

37' 

0-7854 

0-6042 

1-4487 

1-2680 

0-0005 

38' 

0-7854 

0-6045 

1-4478 

1-2671 

00002 

39' 

0-7854 

0-6048 

1-4469 

1-2661 

-00002 

40' 

0-7854 

0-6051 

1-4460 

1-2652 

-0-0005 

We  see  accordingly  that  the  equation  is  solved  (as  accurately 
as  need  be)  when  6  is  an  angle  somewhat  over  34°  38',  or  say 


VIII]  THE  BISECTION  OF  A  QUADRANT  583 

34°  3^1'.  That  is  to  say,  a  quadrant  of  a  circle  is  bisected  by  a 
circular  arc  cutting  the  side  and  the  periphery  of  the  quadrant 
at  right  angles,  when  the  arc  is  such  as  to  include  (90°  —  34°  38'), 
i.e.  55°  22'  of  the  quadrantal  arc.  This  determination  of  ours  is 
practically  identical  with  that  which  Berthold  arrived  at  by  a 
rough  and  ready  method,  without  the  use  of  mathematics.  He 
simply  tried  various  w^ays  of  dividing  a  quadrant  of  paper  by  means 
of  a  circular  arc,  and  went  on  doing  so  till  he  got  the  weights  of 
his  two  pieces  of  paper  approximately  equal.  The  angle,  as  he 
thus  determined  it,  was  34-6°,  or  say  34°  36'. 

(2)  The  position  of  M  on  the  side  of  the  quadrant  OA  is  given 
by  the  equation  OM  =  a  cosec  6  —  a  cot  6 ;  the  value  oi  which 
expression,  for  the  angle  which  we  have  just  discovered,  is  0-3028. 
That  is  to  say,  the  radius  (or  side)  of  the  quadrant  will  be  divided 
by  the  new  partition  into  two  parts,  in  the  proportions,  nearly,  of 
three  to  seven. 

(3)  The  length  of  the  arc  MP  is  equal  to  ad  cot  d;  and  the 
value  of  this  for  the  given  angle  is  0-8751.  This  is  as  much  as  to 
say  that  the  curved  partition- wall  which  we '  are  considering  is 
shorter  than  a  radial  partition  in  the  proportion  of  8|  to  10,  or 
seven-eighths,  almost  exactly. 

But  we  must  also  compare  the  length  of  this  curved  antichnal 
partition-wall  (MP)  with  that  of  the  concentric,  or  perichnal,  one 
{RS,  Fig.  233)  by  which  the  quadrant  might  also 
be  bisected.  The  length  of  this  partition  is 
obviously  equal  to  the  arc  of  the  quadrant  (i.e. 
the  peripheral  wall  of  the  cell)  divided  by  V2; 
or,  in  terms  of  the  radius,  =  77/2  a/2  =  1-111. 
So  that,  not  only  is  the  anticlinal  partition  (such 
as  we  actually  find  in  nature)  notably  the  best, 
but  the  periclinal  one,  when  it  comes  to  dividing 
an  entire  quadrant,  is  very  considerably  larger  even  than  a  radial 
partition. 

The  two  cells  into  which  our  original  quadrant  is  now  divided 
are  equal  in  volume,  but  of  very  different  shapes;  the  one  is  a 
triangle  (MAP)  with  circular  arcs  for  two  of  its  sides,  and  the  other 
is  a  four-sided  figure  (MOBP),  which  we  may  call  approximately 
oblong.     How  will  they  continue  to  divide?     We  cannot  say  as 


584 


THE  FORMS  OF  TISSUES 


[CH, 


yet  how  the  triangular  portion  ought  to  divide;  but  it  is  obvious 
that  the  least  possible  partition-wall  which  shall  bisect  the  other 
must  run  across  the  long  axis  of  the  oblong,  that  is  to  say  periclihally. 
This  is  precisely  what  tends  actually  to  take  place.  In  the  following 
diagrams  (Fig.  234)  of  a  frog's  egg  dividing  under  pressure,  that 
is  to  say  when  reduced  to  the  form  of  a  flattened  plate,  we  see, 
firstly  (A),  the  division  into  four  quadrants  (by  the  partitions  L  2); 
secondly  (B),  the  division  of  each  quadrant  by  means  of  an  anti- 
clinal circular  arc  (3,  3),  cutting  the  peripheral  wall  of  the  quaarant 
approximately  in  the  proportions  of  three  to  seven;    and  thirdly 


3 

ABO 
Fig.  234.     Segmentation  of  frog's  egg,  under  artificial  compression. 
After  Roux. 

(C),  we  see  that  of  the  eight  cells  (four  triangular  and  four  oblong) 
into  which  the  whole  egg  is  now  divided,  the  four  which  we  have 
called  oblong  now  proceed  to  divide  by  partitions  transverse  to 
their  long  axes,  or  roughly  parallel  to'  the  periphery  of  the  egg. 


The  question  how  the  other,  or  triangular,  portion  of  the  divided 
quadrant  will  next  divide  leads  us  to  a  well-defined  problem  which 
is  only  a  shght  extension,  making  allowance  for  the  circular  arcs, 
of  that  elementary  problem  of  the  triangle  we  have  already  con- 
sidered. We  know  now  that  an  entire  quadrant  (in  order  that  its 
bisecting  wall  shall  have  the  least  possible  area)  must  divide  by 
means  of  an  antichnal  partition,  but  how  about  any  smaller  sectors 
of  circles?  It  is  obvious  in  the  case  of  a  small  prismatic  sector, 
such  as  that  shewn  in  Fig.  235,  that  a  periclinal  partition  is  the 
least  by  which  we  can  bisect  the  cell;  we  want,  accordingly,  to 
know  the  limits  below  which  the  periclinal  partition  is  always  the 


VIII]  THE  BISECTION  OF  A  QUADRANT  585 

best,  and  above  which  the  anticHnal  arc  has  the  advantage,  as  in 
the  case  of  the  whole  quadrant. 

This  may  be  easily  determined;  for  the  preceding  investigation 
is  a  perfectly  general  one,  and  the  results  hold  good  for  sectors  of 
any  other  arc,  as  well  as  for  the  quadrant,  or  arc  of  90°.  That  is 
to*  say,  the  length  of  the  partition- wall  MP  is  always  determined 
by  the  angle  9,  according  to  our  equation  MP  -^  ad  cot  6;  and  the 
angle  6  has  a  definite  relation  to  a,  the  angle  of  arc. 


Fig.  235. 

Moreover,  in  the  case  of  the  periclinal  boundary,  RS  (Fig.  233) 
(or  ab,  Fig.  235),  we  know  that,  if  it  bisects  the  cell, 

RS  =  a.  a/V2. 

Accordingly,  the  arc  RS  will  be  just  equal  to  the  arc  MP  when 

^  cot  ^  =  a/v/2. 

When  ^  cot  ^  >  a/ V2,  or  MP  <  RS, 

then  division  will  take  place  as  in  RS,  or  perichnally. 

When  6*  cot  ^  <  a/ V2,  or  MP  >  RS, 

then  division  will  take  place  as  in  MP,  or  anticlinally. 

In  the  accompanying  diagram  (Fig.  236),  I  have  plotted  the 
various  magnitudes  with  which  we  are  concerned,  in  order  to 
exhibit  the  several  limiting  values.  Here  we  see,  in  the  first  place, 
the  curve  marked  a,  which  shews  on  the  (left-hand)  vertical  scale 
the  various  possible  magnitudes  of  that  angle  (viz.  the  angle  of  arc 
of  the  whole  sector  which  we  wish  to  divide),  and  on  the  horizontal 
scale  the  corresponding  values  of  6,  or  the  angle  which  determines 
the  point  on  the  periphery  where  it  is  cut  by  the  partition-wall. 


586 


THE  FORMS  OF  TISSUES 


[CH. 


MP.  Two  limiting  cases  are  to  be  noticed  here:  (1)  at  90°  (point 
A  in  diagram),  because  we  are  at  present  only  dealing  with  arcs 
no  greater  than  a  quadrant;  and  (2),  the  point  (B)  where  the  angle  6 


Fig.  23G. 


comes  to  equal  the  angle  a,  for  after  that  point  the  construction 
becomes  impossible,  since  an  anticlinal  bisecting  partition-wall 
would  be  partly  outside  the  cell.  The  only  partition  which,  after 
that  point,  can  possibly  exist  is  a  periclinal  one;    and  this  point, 


VIII]  THE  BISECTION  OF  A  QUADRANT  587 

as  our  diagram  shews  us,  occurs  when  the  two  angles  (a  and  9)  are 
both  rather  under  52°. 

Next  I  have  plotted  on  the  same  diagram,  and  in  relation  to 
the  same  scale  of  angles,  the  corresponding  lengths  of  the  two 
partitions,  viz.  RS  and  MP,  their  lengths  being  expressed  (on  the 
right-hand  side  of  the  diagram)  in  terms  of  the  radius  of  the  circle 
(a),  that  is  to  say  the  side  wall,  OA,  of  our  cell. 

The  hmiting  values  here  are  (1),  C,  C[,  where  the  angle  of  arc 
is  90°,  and  where,  as  we  have  already  seen,  the  two  partition-walls 
have  the  relative  magnitudes  of  MP  :  J?*S  =  0-875  : Mil:  (2)  the 
point  D,  where  RS  equals  unity,  that  is  to  say  where  the  periclinal 
partition  has  the  same  length  as  a  radial  one;  this  occurs  when 
a  is  rather  under  82°  (cf.  the  points  D,  D'):  (3)  the  point  E,  where 
RS  and  MP  intersect,  that  is  to  say  the  point  at  which  the  two 
partitions,  periclinal  and  anticlinal,  are  of  the  same  magnitude; 
this  is  the  case,  according  to  our  diagram,  when  the  angle  of  arc 
is  just  over  62 J°.  We  see  from  this  that  what  we  have  called  an 
antichnal  partition,  as  MP,  is  only  Ukely  to  occur  in  a  triangular 
or  prismatic  cell  whose  angle  of  arc  hes  between  90°  and  62|°;  in 
all  narrower  or  more  tapering  cells  the  periclinal  partition  will  be 
of  less  area,  and  will  therefore  be  more  and  more  likely  to  occur. 

The  case  (F)  where  the  angle  a  is  just  60°  is  of  some  interest. 
Here,  owing  to  the  curvature  of  the  peripheral  border,  and  the 
consequent  fact  that  the  peripheral  angles  are  somewhat  greater 
than  the  apical  angle  a,  the  periclinal  partition  has  a  very  slight 
and  almost  imperceptible  advantage  over  the  anticlinal,  the  relative 
proportions  being  about  as  MP\  RSwO'lZ.O-n.  But  if  the 
triangle  be  a  plane  equiangular  triangle,  bounded  by  circular  arcs, 
then  we  see  that  there  is  no  longer  any  distinction  at  all  between 
our  two  partitions;   MP  and  RS  are  now  identical. 

On  the  same  diagram,  I  have  inserted  the  curve  for  values  of 
cosec  ^  —  cot  ^  =  OM,  that  is  to  say  the  distances  from  the  centre, 
along  the  side  of  the  cell,  of  the  starting-point  (M)  of  the  anticlinal 
partition.  The  point  C"  represents  its  position  in  the  case  of  a 
quadrant,  and  shews  it  to  be  (as  we  have  already  said)  about  3/10 
of  the  length  of  the  radius  from  the  centre.  If  on  the  other  hand 
our  cell  be  an  equilateral  triangle,  then  we  have  to  read  oif  the 
point  on  this  curve  corresponding  to  a  --  60°;    and  we  find  it  at 


588 


THE  FORMS  OF  TISSUES 


[CH. 


the  point  F'"  (vertically  under  F),  wbich  tells  us  that  the  partition 
now  starts  45/100,  or  nearly  halfway,  along  the  radial  wall. 

The  foregoing  considerations  carry  us  a  long  way  in  our  investi- 
gation of  the  simpler  forms  of  cell-division.  Strictly  speaking  they 
are  hmited  to  the  case  of  flattened  cells,  in  which  we  can  treat  the 
problem  as  though  we  were  partitioning  a  plane  surface.  But  it  is 
obvious  that,  though  they  do  not  teach  us  the  whole  conformation 
of  the  partition  which  divides  a  more  comphcated  solid  into  two 
halves,  yet,  even  in  such  a  case  they  so  far  enlighten  us  as  to  tell 
us  the  appearance  presented  in  one  plane  of  the  actual  solid.  And, 
as  this  is  all  that  we  see  in  a  microscopic  section,  it  follows  that 
the  results  we  have  arrived  at  will  help  us  greatly  in  the  interpreta- 
tion of  microscopic  appearances,  even  in  comparatively  complex 
cases  of  cell-division. 

Let  us  now  return  to  our  quadrant  cell  (OAPB),  which  we  have 
found  to  be  divided  into  a  triangular  and  a  quadrilateral  portion, 
as  in  Figs.  233  or  237 ;  and  let  us  _, 
now  suppose  the  whole  system  to 
grow,  in  a  uniform  fashion,  as  a 
prelude  to  further  subdivision.  The 
whole  quadrant,  growing  uniformly 
(or  with  equal  radial  increments), 
will  still  remain  a  quadrant,  and  it 
is  obvious,  therefore,  that  for  every 
new  increment  of  size,  more  will  be 
added  to  the  margin  of  its  triangular 
portion  than  to  the  narrower  margin 

of  the   quadrilateral;  and   the  in- 

.         Ml    1       •  ^'         ^  Fig-  237. 

crements  will  be  m  proportion  to 

the  angles  of  arc,  viz.  55°  22' :  34°  38',  or  as  0-96  :  0-60,  i.e.  as  8  :  5. 

Accordingly,   if  we  may  assume   (and  the  assumption  is  a  very 

plausible  one),  that,  just  as  the  quadrant  itself  divided  into  two 

halves  after  it  got  to  a  certain  size,  so  each  of  its  two  halves  will 

reach  the  same  size  before  again  dividing,  it  is  obvious  that  the 

triangular  portion  will  be  doubled  in  size,  and  therefore  ready  to 

divide,  a  considerable  time  before  the  quadrilateral  part.     To  work 

out  the  problem  in  detail  would  lead  us  into  troublesome  mathe- 


VIII]  THE  BISECTION  OF  A  QUADRANT  589 

matics;  but  if  we  simply  assume  that  the  increments  are  propor- 
tional to  the  increasing  radii  of  the  circle,  we  have  the  following 
equations : 

Call  the  triangular  cell  T,  and  the  quadrilateral  Q  (Fig.  237); 
let  the  radius,  OA,  of  the  original  quadrantal  cell  =  a  =  1;  and  let 
the  increment  which  is  required  to  add  on  a  portion  equal  to  T 
(such  as  PP'A'A)  be  called  x,  and  let  that  required,  similarly,  for 
the  doubhng  of  Q  be  called  x' . 

Then  we  see  that  the  area  of  the  original  quadrant 

=  T  +  Q  =  Itto"  =  0-7854a2, 

while  the  area  of  T  =Q  =  0-3927a2. 

The  area  of  the  enlarged  sector,  P'OA', 

=  {a  +  x)^  X  (55°  22')  ^  2  =  0-4831  (a  +  x)^, 

and  the  area  OP  A 

=  a^x  (55°  22')  -^  2  =  0-4831a2. 

Therefore  the  area  of  the  added  portion,  T\ 

=  0-4831  {{a  +  x)^  -  a^}. 

And  this,  by  hypothesis, 

=  T  =  0-3927a2. 

We  get,  accordingly,  since  a  =  I, 

x^  +  2x=  0-3927/0-4831  =  0-810, 
and,  solving, 

x-{-l  =  Vrsi  =  1-345,  or  x^  0-345. 

Working  out  x'  in  the  same  way,  we  arrive  at  the  approximate 
value,  x'  +  1  =  1-517. 

This  is  as  much  as  to  say  that,  supposing  each  cell  tends  to 
divide  into  two  halves  when  (and  not  before)  its  original  size  is 
doubled,  then,  in  our  flattened  disc,  the  triangular  cell  T  will  tend 
to  divide  ,when  the  radius  of  the  disc  has  increased  by  about  a 
third  (from  1  to  1-345),  but  the  quadrilateral  cell,  Q,  will  not  tend 
to  divide  until  the  Hnear  dimensions  of  the  disc  have  increased  by 
fully  a  half  (from  1  to  1-517). 


590 


THE  FORMS  OF  TISSUES 


[CH. 


The  case  here  illustrated  is  of  no  small  importance.  For  it  shews 
us  that  a  uniform  and  symmetrical  growth  of  the  organism  (sym- 
metrical, that  is  to  say,  under  the  limitations  of  a  plane  surface, 
or  plane  section)  by  no  means  involves  a  uniform  or  symmetrical 
growth  of  the  individual  cells,  but  may  under  certain  conditions 
actually  lead  to  inequahty  among  these;  and  this  phenomenon 
(or  to  be  quite  candid,  this  hypothesis,  which  is  due  to  Berthold) 
is  independent  of  any  change  or  variation  in  surface  tensions, 
and  is  essentially  different  from  that  unequal  segmentation  (studied 
by  Balfour)  to  which  we. have  referred  on  p.  568. 

After  this  fashion  we  might  go  on  to  consider  the  manner,  and 
the  order  of  succession,  in  which  subsequent  cell-divisions  should 
tend  to  take  place,  as  governed  by  the  principle  of  minimal  areas. 


I 


Fig.  238. 

The  calculations  would  grow  more  difficult,  and  the  results  •got 
by  simple  methods  would  grow  less  and  less  exact;  at  the  same 
time  some  of  the  results  would  be  of  great  interest,  and  well  worth 
our  while  to  obtain.  For  instance,  the  precise  manner  in  which  our 
triangular  cell,  T,  would  next  divide  would  be  interesting  to  know, 
and  a  general  solution  of  this  problem  is  certainly  troublesome  to 
calculate.  But  in  this  particular  case  we  see  that  the  width  of  the 
triangular  cell  near  P  (Fig.  238)  is  so  obviously  less  than  that  near 
either  of  the  other  two  angles,  that  a  circular  arc  cutting  off  that 
angle  is  bound  to  be  the  shortest  possible  bisecting  line;  and  that, 
in  short,  our  triangular  cell  will  tend  to  subdivide,  just  hke  the 
original  quadrant,  into  a  triangular  and  a  quadrilateral  portion. 

But  the  case  will  be  different  next  time,  because  in  this  new 
triangle,'  PRQ,  the  least  width  is  near  the  innermost  angle,  that 
at  Q\  and  the  bisecting  circular  arc  will  therefore  be  opposite  to  Q, 
or  (approximately)  parallel  to  PR.  The  importance  of  this  fact  is 
at  once  evident;    for  it  means  to   say  that  there  comes  a  time 


VIII]  THE  SEGMENTATION  OF  A  DISC  591 

when,  whether  by  the  division  of  triangles  or  of  quadrilaterals,  we 
find  only  quadrilateral  cells  adjoining  the  periphery  of  our  circular 
disc.  In  the  subsequent  division  of  these  quadrilaterals,  the  parti- 
tions will  arise  transversely  to  their  long  axes,  that  is  to  say,  radially 
(as  U,  V) ;  and  we  shall  consequently  have  a  superficial  or  peripheral 
layer  of  quadrilateral  cells,  with  sides  approximately  parallel,  that 
is  to  say  what  we  are  accustomed  to  call  an  epidermis.  And  this 
epidermis  or  superficial  layer  will  be  in  clear  contrast  with  the  more 
irregularly  shaped  cells,  the  products  of  triangles  and  quadrilaterals, 
which  make  up  the  deeper,  underlying  layers  of  tissue. 

In  following  out  these  theoretic  principles,  and  others  Hke  to 
them,  in  the  actual  division  of  living  cells,  we  must  bear  in  mind 
certain  conditions  and  qualifications.  In  the  first  place,  the  law 
of  minimal  area  and  the  other  rules  which  we  have  arrived  at  are 
not  absolute  but  relative :  they  are  Unks,.  and  very  important  links, 
in  a  chain  of  physical  causation ;  they  are  always  £tt  work,  but  their 
effects  may  be  overridden  and  concealed  by  the  operation  of  other 
forces.  Secondly,  we  must  remember  that,  in  most  cases,  the  cell- 
system  which  we  have  in  view  is  constantly  increasing  in  magnitude 
by  active  growth;  and  by  this  means  the  form  and  also  the  propor- 
tions of  the  cells  are  continually  altering,  of  which  phenomenon  we 
have  already  had  an  example.  Thirdly,  we  must  carefully  remember 
that,  until  our  cell-walls  become  absolutely  solid  and  rigid,  they  are 
always  apt  to  be  modified  in  form  owing  to  the  tension  of  the 
adjacent  walls;  and  again,  that  so  long  as  our  partition  films  are 
fluid  or  semifluid,  their  points  and  fines  of  contact  with  one  another 
may  shift,  like  the  shifting  outlines  of  a  system  of  soap-bubbles. 
This  is  the  physical  cause  of  the  movements  frequently  seen  among 
segmenting  cells,  like  those  to  which  Rauber  called  attention  in 
the  segmenting  ovum  of  the  frog,  and  hke  those  more  striking 
movements  or  accommodations  which  give  rise  to  a  so-called 
"spiral"  type  of  segmentation. 

Bearing  in  mind  these  considerations,  let  us  see  what  our  fla|itened 
disc  is  hkely  to  look  like,  after  a  few  successive  Cell-divisions.  In 
Fig.  239  a,  we  have  a  diagrammatic  representation  of  our  disc,  after 
it  has  divided  into  four  quadrants,  and  each  quadrant  into  a 
triangular  and  a  quadrilateral  portion;    but  as  yet,  this,  figure  has 


592 


THE  FORMS  OF  TISSUES 


[CH. 


scarcely  anything  like  the  normal  look  of  an  aggregate  of  hving 
cells.  But  let  us  go  a  little  further,  still  hmiting  ourselves  to  the 
consideration  of  the  eight-celled  stage.  Wherever  one  of  our 
radiating  partitions  meets  the  peripheral  wall,  there  will  (as  we 
know)  be  a  mutual  tension  between  the  three  convergent  films, 
which  will  tend  to  set  their  edges  at  equal  angles  to  one  another, 
angles  that  is  to  say  of  120°.  In  consequence  of  this,  the  outer  wall 
of  each  individual  cell  will  (in  this  surface  view  of  our  disc)  be  an 
arc  of  a  circle  of  which  we  can  determine  the  centre  by  the  method 
used  on  p.  485;  and,  furthermore,  the  narrower  cells,  that  is  to  say 
the   quadrilaterals,   will   have   this   outer   border  somewhat   more 


Fig.  239.     Diagram  of  flattened  or  discoid  cell  dividing  into  octants :   to  shew 
gradual  tendency  towards  a  position  of  equilibrium. 

curved  than  their  broader  neighbours.  We  arrive,  then,  at  the 
condition  shewn  in  Fig.  239  b.  Within  the  cell,  also,  wherever 
wall  meets  wall,  the  angle  of  contact  must  tend,  in  every  case,  to 
be  an  angle  of  120°;  in  no  case  may  more  than  three  films  (as  seen 
in  section)  meet  in  a  point  (c) ;  and  this  condition,  of  the  partitions 
meeting  three  by  three  and  at  co-equal  angles,  will  involve  the 
curvature  of  some,  if  not  all,  of  the  partitions  (d)  which  to  begin 
with  we  treated  as  plane.  To  solve  this  problem  in  a  general  way 
is  no  easy  matter;  but  it  is  a  problem  which  Nature  solves  in 
every  case  where,  as  in  the  case  we  are  considering,  eight  bubbles 
or  eight  cells  meet  together  in  a  plane  or  curved  surface.  An 
approximate  solution  has  been  given  in  d;  and  it  will  at  once  be 
recognised  that  this  figure   has   vastly  more   resemblance   to   an 


VIII 


THE  SEGMENTATION  OF  A  DISC 


593 


aggregate  of  living  cells  than  had  the  diagram  of  a,  with  whi«h  we 
began. 

Just  as  we  have  constructed  in  this  case  a  series  of  purely 
diagrammatic  or  schematic  figures,  so  will  it  be  possible  as  a  rule 
to  diagrammatise,  with  but  little  alteration,  the  complicated  ap- 
pearances presented  by  any  ordinary  aggregate  of  cells.  The 
accompanying  httle  figure  (Fig.  240),  of  a  germinating 
spore  of  a  Liverwort  (Riccia),  after  a  drawing  of  D.  H. 
Campbell's,  scarcely  needs  further  explanation:  for  it  is 
well-nigh  a  typical  diagram  of  the  method  of  space- 
partitioning  which  we  are  now  considering.  The  same  is 
equally  true  of  any  one  of  Hanstein's  figures  of  the 
hairs  on  a  leaf-bud*,  or  Berthold's  of  the  small  discoid  algae.  Let 
us  look  again  at  our  figures  of  Erythrotrichia  or  Chaetopeltis  from 
Berthold's    Monograph,   and   redraw   some   of   the   earher   stages. 


Ficr.  240. 


Fig.  241.     Embryo-stages  of  Chaetopeltis  orbicularis.     After  Berthold. 

In  the  following  diagrams  (Fig.  242)  the  new  partitions,  or  those  just 
about  to  form,  are  in  each  case  outhned ;  and  in  the  next  succeeding 
stage  they  are  shewn  after  settling  down  into  position,  and  after 
exercising  their  respective  tractions  on  the  walls  previously  kid 
down.  It  is  clear,  I  think,  that  these  four  diagrammatic  figures 
represent  all  that  is  shewn  in  the  first  five  stages  drawn  by  Berthold 
from  the  plant  itself;  ,but  the  correspondence  cannot  in  this  case 
be  precisely  accurate,  for  the  reason  that  Berthold's  figures  are 
taken  from  different  individuals,  and  so  are  not  strictly  and  con- 
secutively continuous.  The  last  of  the  six  drawings  in  Fig.  230  is 
already  too  comphcated  for  diagrammatisation,  that  is  to  say  it  is 
too  comphcated  for  us  to  decipher  with  certainty  the  order  of 
appearance  of  the  numerous  partitions  which  it  contains.  But  in 
Fig.  243  I  shew  one  more  diagrammatic  figure,  of  a  disc  which  has 
*  Bot.  Zeitung,  xxvi,  p.  11,  xi,  xii,  1868. 


594 


THE  FORMS  OF  TISSUES 


[CH. 


divided,  according  to  the"  theoretical  plan,  into  about  sixty-four 
cells;  and  making  due  allowance  for  the  changes  which  mutual 
tensions  and  tractions  bring  about,  increasing  in  complexity  with 
each  succeeding  stage,   we  can  see,   even  at  this  advanced  and 


"DC  d 

Fig.  242.     Theoretical  arrangement  of  successive  partitions  in  a  discoid  cell; 
for  comparison  with  Figs.  230  and  241. 

complicated  stage,  a  very  considerable  resemblance  between  the 
actual  picture  and  the  diagram  which  we  have  here  constructed 
in  obedience  to  a  few  simple  rules. 


Fig.  243.     Theoretical  division  of  a  discoid  cell  into  sixty-four  chambers:   no 
allowance  being  made  for  the  mutual  tractions  of  the  cell- walls. 

In  like  manner,  in  the  annexed  figures  representing  sections 
through  a  young  embryo  of  a  moss,  we  have  little  difficulty  in 
discerning  the  successive  stages  which  must  have  intervened  between 
the  two  stages  shewn:  so  as  to  lead  from  the  just  divided  or  dividing 
quadrants  (a),  to  the  stage  (6)  in  which  a  well-marked  epidermal 


VIII]  THE  SEGMENTATION  OF  A  DISC  595 

layer  surrounds  an  at  first  sight  irregular  agglomeration  of  "funda- 
mental tissue". 

In  the  last  paragraph  but  one,  I  have  spoken  of  the  difficulty  of 
so  arranging  the  meetmg-places  of  a  number  of  cells  that  at  each 
junction  only  three  cell- walls  shall  meet  in  a  point,  and  all  three 
shall  meet  at  equal  angles  of  120°.  As  a  matter  of  fact,  the  problem 
is  soluble  in  a  number  of  ways;  that  is  to  say,  when  we  have  a 
number  of  cells  enclosed  in  a  common  boundary,  say  eight  as  in 
the  case  considered,  there  are  various  ways  in  which  their  walls 
may  meet  internally,  three  by  three,  at  equal  angles;  and  these 
differences  will  entail  differences  also  in  the  curvature  of  the  walls, 
and  consequently  in  the  shape  of  the  cells.     The  question  is  some- 


a  — ^-------^  b 

Fig.  244.     Sections  of  embryo  of  a  moss.     After  Kienitz-Gerlofif. 

what  complex ;    it  has  been  dealt  with  by  Plateau, "  and  treated 
mathematically  by  M.  Van  Rees*. 

If  within  our  boundary  we  have  only  three  cells  all  meeting 
internally,  they  must  meet  in  a  point;  furthermore,  they  tend  to 
do  so  at  equal  angles  of  120°,  and  there  is  an  end  of  the  matter. 
If  we  have  four  cells,  then,  as  we  have  already  seen,  the  conditions 
are  satisfied  by  interposing  a  little  intermediate  wall,  the  two 
extremities  of  which  constitute  the  meeting-points  of  three  cells 
each,  and  the  upper  edge  of  which  marks  the  "polar  furrow." 
In  the  case  of  five  cells,  we  require  two  httle  intermediate  walls, 
and  two  polar  furrows;  and  we  soon  arrive  at  the  rule  that,  for 
n  cells,  we  require  ti  —  3  Httle  longitudinal  partitions  (and  corre- 
sponding polar  furrows),  connecting  the  triple  junctions  of  the  cells f ; 
and  these  Httle  walls,  Hke  all  the  rest  within  the  system,  tend  to 

*  Cit.  Plateau,  Statique  des  Liquides,  i,  p.  358. 

t  There  is  an  obvious  analogy  between  this  rule  for  the  number  of  internal 
partitions  within  a  polygonal  system,  and  Lamarle's  rule  for  the  number  of  "free 
films  "  within  a  polyhedron.     Vide  supra,  ^.  b5Q. 


596 


THE  FORMS  OF  TISSUES 


[CH. 


incline  to  one  another  at  angles  of  120°.  Where  we  have  only  one 
such  wall  (in  the  case  of  four  cells),  or  only  two  (in  the  case 
of  five  cells),  there  is  no  room  for  ambiguity.  But  where  we  have 
three  little  connecting- walls,  in  the  case  of  six  cells,  we  can 
arrange  them  in  three  different  ways,  as  Plateau*  found  his  six 

3  4  5     "*        6  7  8      Cells' 


etc. 
Fig.  245.    Various  possible  arrangements  of  intermediate  positions, 
in  groups  of  3,  4,  5,  6,  7  or  8  cells. 

soap-films  to  do  (Fig.  246).  In  the  system  of  seven  cells,  the  four 
partitions  can  be  arranged  in  four  ways;  and  the  five  partitions 
required  in  the  case  of  eight  cells  can  be  arranged  in  no  less  than 


Fig.  246.. 

twelve  different  waysf.  It  does  not  follow  that  these  various 
arrangements  are  all  equally  good;  some  are  known  to  be  more 
stable  than  others,  and  some  are  hard  to  realise  in  actual  experiment. 
Examples  of  these  various  arrangements  meet  us  at  every  turn, 
in  all  sorts  of  partitioning,  whether  there  be  actual  walls  or  mere 

*  Plateau  experimented  with  a  wire  frame  or  "cage",  in  the  form  of  a  low 
hexagonal  prism.  When  this  was  plunged  in  soap-solution  and  withdrawn  upright, 
a  vertical  film  occupied  its  six  quadrangular. sides  and  nothing  more.  But  when 
it  was  drawn  out  sideways,  six  films  starting  from  the  six  vertical  edges  met  some- 
how in  the  middle,  and  divided  the  hexagon  into  six  cells.  Moreover  the  partition- 
films  automatically  solved  the  problem  of  meeting  one  another  three-by-three,  at 
co-equal  angles  of  120°;  and  did  so  in  more  ways  than  one,  which  could  be  controlled 
more  or  less,  according  to  the  manner  and  direction  of  lifting  the  cage. 

t  Plateau,  on  Van  Rees's  authority,  says  thirteen;  but  this  is  wrong — unless 
he  meant  to  include  the  case  where  one  cell  is  wholly  surrounded  by  the  seven 
others. 


VIII]  THE  PARTITIONING  OF  SPACE  597 

rifts  and  cracks  in  a  broken  surface.  The  phenomeiion  is  in  the 
first  instance  mathematical,  in  the  second  physical ;  and.  the  limited 
number  of  possible  arrangements  appear  and  reappear  in  the  most 
diverse  fields,  and  are  capable  of  representation  by  the  same 
diagrams.  We  have  seen  in  Fig.  196  how  the  cracks  in  drying  mud 
exhibit  to  perfection  the  polar  furrow  joining  two  three-way  nodes, 
which  is  the  characteristic  feature  of  the  four-celled  stage  of  a 
segmenting  egg. 

The  possible  arrangements  of  the  intermediate  partitions  becomes 
a  question  of  permutations.  Let  us  call  the  flexure  between  two 
consecutive  furrows  a  or  6,  according  to  its  direction,  right  or  left; 
and  let  a  triple  conjunction  be  called  c.  Then  the  three  possible 
arrangements  in  a  system  of  six  cells  are  aa,  ah,  c;  the  four  in  a 
system  of  seven  cells  are  aaa,  aab,  aba,  ac;  and  the  twelve  possible 
arrangements  in  a  system  of  eight  cells  are  as  follows* : 


a 

aac 

f 

aahb 

b 

ahc 

9 

aaba 

c 

ad) 

h 

aaab 

d 

aca 

i 

aaaa 

e 

bcb 

J 

abab 

k 

abba 

I         cc 

i 

We  may  classify,  and  may  denote  or  symbolise,  these  several 
arrangements  in  various  ways.  In  the  following  table  we  see: 
A,  the  twelve  arrangements  of  the  five  intermediate  partitions 
which  are  necessary  to  enable  all  the  boundary  walls  of  a  plane 
assemblage  of  eight  cells  (none  being  "insular")  to  meet  in  three- 
way  junctions;  B,  the  literal  permutations  which  symbolise  the 
same;  C,  the  number  of  sides  (other  than  the  external  boundary) 
which  in  each  case  each  cell  possesses,  i.e.  the  number  of  contacts 
each  makes  with  its  neighbours.^  The  total  number  of  contacts 
(as  we  shall  see  presently)  is  26,  and  the  mean  number  3-25;  if  we 
take  the  departures  from  the  mean,  and  sum  them  irrespective  of 
sign,  the  sum  is  shewn  under  D. 

*  I  believe  that  Kirkman,  in  a  paper  of  more  than  80  years  ago,  said  that  the 
number  of  S-sided  convex,  Eulerian  polyhedra,  with  trihedral  comers,  was  thirteen. 


598  THE  FORMS  OF  TISSUES  [ch. 

A  B  C  D 

aac  222  33  44  6  8-5 

aca  222  33  44  6  8-5 


acb 

222  33  4  55 

8-5 

bcb 

22233  455 

8-5 

abc 

222  3  4445 

8-0 

abab 

22  33  4444 

60 

aabb 

22  3333  55 

7-0 

aaba 

22  333  445 

6-5 

abba 

22  333  44  5 

6-5 

aaab 

22  3333  46 

7-0 

aaaa 

2233333  7 

7-5 

cc 

2222  44  55 

10-0 

I 

9 
k 


Nine  cells  may  be  arranged  in  twenty-seven  ways.  In  higher 
series  the  numbers  increase  very  rapidly,  but  the  cells  will  tend  to 
overlap,  and  so  introduce  a  new  complication* 

We  may  draw  help  from  the  theory  of  polyhedra  (in  an  elementary, 
way)  if  we  treat  our  group  of  eight  cells  (none  of  them  "insular") 
as  part  of  a  polyhedron,  to  be  completed  by  one  eight-sided  cell, 

*  Max  Bruckner  states  the  number  of  possible  arrangements  of  thirteen  cells, 
with  trihedral  junctions,  as  nearly  50,000;  of  sixteen,  nearly  30  miUions;  and  of 
eighteen,  "bereits  iiber  einige  Billionen"  (Proc.  Math.  Congress,  Bologna,  1930,  vol. 
IV,  p.  11),  It  is  plain  that  the  study  of  "  cell-lineage,"  or  the  mapping  out  in  detail 
of  the  cell-arrangements  after  repeated  cell-divisions,  is  only  possible  under  severe 
limitations. 


VIIl] 


THE  PARTITIONING  OF  SPACE 


599 


serving  (so  to  speak)  as  a  base  under  all  the  rest.  Then,  calling 
^3,  ^4,  etc.,  the  number  of  three-sided  and  four-sided  facets,  we  re- 
classify our  twelve  configurations  as  follows : 

Polyhedral  arrangements  of  eight  cells  (none  of  them  ''insular'*)^ 
considered  as  part  of  a  nine-faced  polyhedron,  whose  ninth  face 
is  octagonal. 


^3  ^4 


F,  F, 


I 

ae 

cd 

b 

i 

h 

f 
gk 

J 


2  — 

—  1 

2  — 

1  — 


It  is  of  interest,  and  of  more  than  mere  mathematical  interest, 
to  know,  not  only  that  these  possible  arrangements  are  few,  but 
that  they  are  strictly  defined  as  to  the  number  and  form  of  the 
respective  faces.  For  we  know  that  we  are  limited  to  three-way 
corners  or  nodes;  and,  that  being  so,  the  fpllowing  simple  rule  holds 
for  the  facets — a  rule  which  we  shall  use  later  on  in  still  more 
curious  circumstances,  and  which  may  be  easily  verified  in  any  line 
of  the  foregoing  table : 

ZF^  +  2F,  +  F,±  O.F,  -F,-  2F,  -  etc.  =  12. 

We  may  produce  and  illustrate  all  these  configurations  by  blowing 
bubbles  in  a  dish  and  here  (Fig.  247)  is  the  complete  series,  up  to 
seven  cells.  They  correspond  precisely  to  the  diagrams  shewn  on 
p.  596,  and  their  resemblance  to  embryological  diagrams  is  only 
cloaked  a  little  by  the  circular  outhne,  the  artificial  boundary  of 
the  system.  Of  the  twelve  eight-celled  arrangements,  four  seem 
unstable;  these  include  the  one  case  (i)  where  one  cell  of  the  eight 
is  in  contact  with  all  seven  others,  and  the  three  cases  (a,  e,  h) 
where  one  is  in  contact  with  six  others.  The  reason  of  this  insta- 
bility is,  I  imagine,  that  the  internal  angles  cannot  be  angles  of 
120°,  as  equilibrium  demands,  unless  the  sides  be  curved,  and 
convex  inwards;  but  this  implies  a  combined  pressure  from  without 
on  the  large  cell  in  the  middle.  While  it  adjusts  its  walls,  then, 
to  the  required  angles,  the  large  cell  tends  to  close  up,  to  lose  hold 


600 


THE  FORMS  OF  TISSUES 


[CH. 


of  the  boundary  of  the  system,  and  to  become  an  island-cell  entirely 
surrounded  by  the  rest. 


Fig.  247.  Group  of  soap-bubbles,  blown  in  Petri  dishes,  a,  b,  c,  the  normal 
partitioning^  of  groups  of  three,  four  or  five  cells  or  bubbles,  d,  the  three  ways 
of  partitioning  a  group  of  six  cells  or  bubbles,  e,  three  of  the  four  ways  of 
partitioning  a  group  of  seven  cells. 

Among  the  published  figures  of  embryonic  stages  and  other  cell 
aggregates,  we  only  discern  the  httle  intermediate  partitions  in 
cases  where  the  investigator  has  drawn  carefully  just  what  lay 
before  him,  without  any  preconceived  notions  as  to  radial  or  other 
symmetry;  but  even  in  other  cases  we  can  often  recognise, 
without  much  difficulty,  what  the  actual  arrangement  was  whereby 


VIIl] 


THE  SEGMENTATION  OF  THE  EGG 


601 


the  cell- walls  met  together  in  equihbrium.  I  suspect  that  a  leaning 
towards  Sachs's  Rule,  that  one  cell-wall  tends  to  set  itself  at  right 
angles  to  another  cell-wall  (a  rule  whose  strict  hmitations  and 
narrow  range  of  apphcation  we  have  already  considered)  is  responsible 
for  many  inaccurate  or  incomplete  representations  of  the  mutual 
arrangement  of  associated  cells. 

In  the  accompanying  series  of  figures  (Figs.  248-255)  I  have  set 
forth  a  few  aggregates   of  eight  cells,   mostly  from  drawings  of 


Figj  248.     Segmenting  egg 
oiTrochus.  After  Robert. 


Fig.  249.     Two  views  of  segmenting  egg  of 
Cynthia  partita.     After  Conklin. 


Fig.  250.     [a)  Section  of  apical  cone  of  Salvinia.     After  Pringsheim  *. 
[h)  Diagram  of  probable  actual  arrangement. 

segmenting  eggs.  In  some  cases  they  shew  clearly  the  manner  in 
which  the  cell-walls  meet  one  another,  always  by  three-way  junctions, 
at  angles  of  about  120°,  more  or  less,  and  always  with  the  help  of 
five  intermediate  boundary  walls  within  the  eight-celled  system; 
in  other  cases  I  have  added  a  slightly  altered  drawing,  so  as  to  shew, 
with  as  Httle  change  as  possible,  the  arrangement  of  boundaries 

*  This,  like  many  similar  figures,  is  manifestly  drawn  under  the  influence  of 
Sachs's  theoretical  views,  or  assumptions,  regarding  orthogonal  trajectories,  coaxial 
circles,  confocal  ellipses,  etc. 


602  THE  FORMS  OF  TISSUES  [ch. 

which  may  have  existed,  and  given  rise  to  the  appearance  which  the 
observer  drew.  These  drawings  may  be  compared  with  the  diagrams 
on  p.  598,  in  which  the  twelve  possible  arrangements  of  five  inter- 
mediate partitions  for  a  system  of  eight  cells  have  been  set  forth. 

It  will  be  seen  that  Robert-Tornow's  figure  of  the  segmenting  egg 
of  Trochus  (Fig.  248)  clearly  shews  the  cells  grouped  after  the  fashion 
of  l\  while  Conkhn's  figure  of  the  ascidian  egg  (Cynthia)  shews 
equally  clearly  the  arrangement  e.    A  sea-urchin  egg  segmenting 


Fig.  251.     Egg  of  Pyrosoma. 
After  KorotneflF. 


Fig.  252.     Egg  of  Echinus,  segmenting 
under  pressure.     After  Driesch. 


Fig.  253.     (a)  Part  of  segmenting  egg  of  Cephalopod  (after  Watase) ; 
(6)  probable  actual  arrangement. 

under  pressure,  as  figured  by  Driesch,  scarcely  wants  any  modifica- 
tion of  the  drawing  to  appear  in  one  case  as  type  /,  in  another  as  g. 
Turning  to  a  botanical  illustration,  we  have  a  figure  of  Pringsheim's 
shewing  an  eight-celled  stage  in  the  apex  of  the  young  cone  of 
Salvinia:  it  is  ill  drawn,  but  may  be  referable,  as  in  my  diagram, 
to  type  /;  after  it  is  figured  a  very  different  object,  a  segmenting 
egg  of  the  ascidian  Pyrosoma,  after  Korotneif,  also,  but  still  more 
doubtfully,  referred  to  /.  In  the  cuttlefish  egg  there  is  again  some 
uncertainty,  but  it  is  probably  referable  to  g.  Lastly,  I  have 
copied  from  Roux  a  curious  figure  of  the  frog's  egg,  viewed  from 
the  animal  pole;  it  is  obviously  inaccurate,  but  may  perhaps 
belong  to  type  e.     Of  type  i,  in  which  the  five  partitions  form 


VIII]  THE  SEGMENTATION  OF  THE  EGG  603 

four  re-entrant  angles,  that  is  to  say  a  figure  representing  the  five 
sides  of  a  hexagon,  and  one  cell  is  in  touch  with  seven  others, 
I  have  found  no  examples  among  pubhshed  figures  of  segmenting 
It  is  obvious  enough,  without  more  ado,  that  these  phenomena 
2  2 


Fig.  254.     (a)  Egg  oi  Echinus;   (6)  do.  oi  Nereis,  under  pressure. 
After  Driesch. 


a  b 

Fig.  255.     (a)  Eg^of  frog,  under  pressure  (after  Roux); 
(6)  probable  actual  arrangement. 

are  in  the  strictest  and  completest  way  common  to  both  plants  and 
animals,  in  which  respect  they  tally  with,  and  further  extend,  the 
fundamental  conclusions  laid  down  by  Schwann  wellnigh  a  hundred 
years  ago,  in  his  Mikroskopische  Untersuthungen  iiher  die  Ueberein- 
stimmung  in  der  Struktur  und  dem  Wachsthwn  der  Thiere  und 
Pflanzen*. 

But  now  that  we  have  seen  how  a  certain  limited  number  of 

types  of  eight-celled  segmentation   (or  of  arrangements  of  eight 

cell-partitions)  appear  and  reappear  here  and  there  throughout  the 

whole  world  of  organisms,  there  still  remains  the  very  important 

*  Berlin,  1839;  Sydenham  Society,  1847. 


604  THE  FORMS  OF  TISSUES  [ch. 

question,  whether  in  each  particular  organisin  the  conditions  are 
such  as  to  lead  to  one  particular  arrangement  being  predominant, 
characteristic,  or  even  invariable.  In  short,  is  a  particular  arrange- 
ment of  cell-partitions  to  be  looked  upon  (as  the  published  figures 
of  the  embryologist  are  apt  to  suggest)  as  a  specific  character,  or  at 
least  a  constant  or  normal  character,  of  the  particular  organism? 
The  answer  to  this  question  is  a  direct  negative,  but  it  is  only  in 
the  work  of  the  most  careful  and  accurate  observers  that  we  find 
it  revealed.  Rauber  (whom,  we  have  more  than  once  had  occasion 
to  quote)  was  one  of  those  embryologists  who  recorded  just  what 
he  saw,  without  prejudice  or  preconception;  as  Boerhaave  said 
of  Swammerdam,  quod  vidit  id  asseruit.  Now  Rauber  has  put  on 
record  a  considerable  number  of  variations  in  the  arrangement  of 
the  first  eight  cells,  which  form  a  discoid  surface  about  the  dorsal 
(or  "animal")  pole  of  the  frog's  egg.  In  a  certain  number  of 
cases  these  figures  are  identical  with  one  another  in  type,  identical 
(that  is  to  say)  save  for  slight  differences  in  magnitude,  relative 
proportions,  or  orientation.  But  I  have  selected  (Fig.  256)  six 
diagrammatic  figures,  which  are  all  essentially  different,  and  these 
diagrams  seem  to  me  to  bear  intrinsic  evidence  of  their  accuracy: 
the  curvatures  of  the  partition-walls  and  the  angles  at  which 
they  meet  agree  closely  with  the  requirements  of  theory,  and  when 
they  depart  from  theoretical  symmetry  they  do  so  only  to  the 
slight  extent  which  we  might  expect  in  a  material  system*. 
Of  these  six  illustrations,  two  are  exceptional.  In  Fig.  256,  5, 
we  observe  that  one  of  the  eight  cells  is  insular,  and  surrounded 
by  the  other  seven.  This  is  a  perfectly  natural  condition,  and 
represents,  like  the  rest,  a  phase  of  partial  or  conditional  equili- 
brium; but  it  is  not  included  in  the  series  we  are  now  considering, 
which  is  restricted  to  the  case  of  eight  cells  extending  outwards 
to  a  common  boundary.     The  condition  shewn  in  Fig.  256,  6,  is 

*  Such  preconceptions  as  Rauber  entertained  were  all  in  a  direction  likely  to 
lead  him  away  from  such  phenomena  as  he  has  faithfully  depicted.  Rauber  had 
no  idea  whatsoever  of  the  principles  by  which  we  are  guided  in  this  discussion, 
nor  does  he  introduce  at  all  the  analogy  of  surface-tension,  or  any  other  purely 
physical  concept;  but  he  was  deeply  under  the  influence  of  Sachs's  rule  of 
rectangular  intersection,  and  he  was  accordingly  disposed  to  look  upon  the 
configuration  represented  above  in  Fig.  256,  6,  as  the  most  typical  or  primitive. 
His  articles  on  Thier  und  Pflanze,  in  Biol.  Cbt.  iv,  1881,  tell  us  much  about  this  and 
other  biological  theories  of  his  time. 


VIIl] 


THE  SEGMENTATION  OF  THE  EGG 


605 


again  peculiar,  and  is  probably  rare,  but  it  is  included  under  the 
cases  considered  on  p.  491,  in  which  the  cells  are  not  in  complete 
fluid  contact  but  are  separated  by  Httle  droplets  of  extraneous 
matter;  it  needs  no  further  comment.  But  the  other  four  cases 
are  beautiful  diagrams  of  space-partitioning,  similar  to  those  we 
have  just  been  considering,  but  so  exquisitely  clear  that  they  need 
no  modification,  no  "touching-up,"  to  exhibit  their  mathematical 
regularity.     It  will  easily  be  recognised  that  in  Fig.  256,  1  and  2, 


4  5  6 

Fig.  256.     Various  modes  of  grouping  of  eight  cells,  at  the  dorsal  or 
epiblastic  pole  of  the  frog's  egg.     After  Rauber. 

we  have  the  arrangements  corresponding  to  I  and  g,  and  in  3  and  4  to  c 
in  our  table  on  p.  598.  One  thing  stands  out  as  very  certain  indeed: 
that  the  elementary  diagram  of  the  frog's  segmenting  egg  given  in 
textbooks  of  embryology — in  which  the  cells  are  depicted  as 
uniformly  symmetrical  and  more  or  less  quadrangular  bodies — is 
entirely  inaccurate  and  grossly  misleading*. 

*  Cf.  Rauber,  Neue  Grundlegungen  z.  K.Mer  Zelle,  Morphol.  Jahrb.  viii,  p.  273, 
1883 :  "  leh  betone  noch,  dass  unter  meinen  Figuren  diejenige  gar  nicht  enthalten  ist, 

welche  zum  Typus  der  Batrachierfurchung  gehorig  am  meisten  bekannt  ist Es 

haben  so  ausgezeichnete  Beobachter  sie  als  vorhanden  besehrieben,  dass  es  mir 
nicht  einfallen  kann,  sie  liberhaupt  nicht  anzuerkennen."  See  also  0.  Hertwig, 
Ueber  den  Werth  d.  erste  Furchungszelle  fiir  die  Organbildung  des  Embryo, 
Arch.  f.  A7iat.  XLin,  1893;  here  0.  Hertwig  maintains  that  there  is  no  such  thing 
as  "cellular  homology." 


606 


THE  FORMS  OF  TISSUES 


[CH. 


We  begin  to  realise  the  remarkable  fact,  which  may  even  appear 
a  startHng  one  to  the  biologist,  that  all  possible  groupings  or 
arrangements  whatsoever  of  eight  cells  in  a  single  layer  or  surface 
(none  being  submerged  or  wholly  enveloped  by  the  rest)  are  referable 
to  one  or  another  of  twelve  types  or  patterns;  and  that  all  the 
thousands  and  thousands  of  drawings  which  dihgent  observers  have 
made  of  such  eight-celled  embryos  or  blastoderms,  or  other  eight- 
celled  structures,  animal  or  vegetable,  anatomical,  histological  or 


Fig.  257.     Photographs  of  lVf)<fs'  eggs,  slunving  various  arrangements,  or 
partitionings,  of  the  lirst  eight  cells. 

embryological,  are  one  and  all  of  them  representations  of  some  one 
or  other  of  these  twelve  types— or  rather  for  the  most  part  of  less 
than  the  whole  twelve;  for  a  certain  small  number  are  essentially 
unstable,  and  have  at  best  but  a  transitory  and  evanescent  existence. 
But  that  even  the  unstable  cases  should  now  and  then  be  seen  is 
not  to  be  wondered  at :  when  viscidity  and  friction,  and  in  general 
the  imperfect  fluidity  of  the  system,  retard  the  adjustment  of  the  cells 
and  delay  the  advent  of  equilibrium. 

As  soon  as  we  reahse  that  the  number  of  cell-patterns,  for  instance 
in  a  segmenting  egg,  is  strictly  limited,  we  want  to  know  how  many 


VIII]  THE  SEGMENTATION  OF  THE  EGG  607 

patterns  actually  occur  and  in  what  proportions  they  do  so  in  a 
random  sample  of  identical  eggs.  Some  years  ago  Mr  Martin 
Adamson  photographed  more  than  a  thousand  frogs'  eggs  in  my 
laboratory,  all  at  the  stage  shewing  an  eight-celled  group  of  epiblastic 
cells:  with  the  remarkable  result  that  every  one  of  the  twelve 
possible  arrangements  was  found  to  occur^  but  some  were  common 
and  some  rare,  and  the  following  were  their  comparative  frequencies : 


ype 

Frequency 

Type 

Frequency 

c 

19-0  % 

d 

6.70/0 

j 

170 

h 

6-6 

b 

12-8 

f 

51 

9 

10-3 

k 

3-0 

a 

7-8 

I 

2-4 

e 

6-9 

i 

1-8 

In  six  separate  batches  of  eggs  (combined  in  the  above  list)  one 
or  other  of  the  first  two  types  (c  or^')  was  always  the  commonest; 
and  the  first  four  taken  together  made  up  from  50  to  80  per  cent,  of 
each  separate  sample.  On  the  other  hand,  when  Roux,  many  years 
ago,  shewed  how  various  cell-configurations  might  be  simulated  by 
oil-drops* — as  we  have  done  by  means  of  soap-bubbles — he  found 
that  the  type  i  was  essentially  unstable,  the  large  drop  with  its 
seven  contacts  easily  shpping  into  the  centre  of  the  system,  and 
there  taking  up  a  stable  position  of  equihbrium.  That  the  latter 
is  the  more  stable,  and  therefore  the  more  probable,  configuration, 
seems  obvious  enough;  and  indeed  type  i  seems  so  obviously 
unstable  that  we  are  not  surprised  to  find  it  at  the  bottom  of 
Martin  Adamson"s  fist  of  frequencies.  The  order  in  which  the  rest 
occur  is  by  no  means  so  easy  of  explanation. 

There  is  a  point  worth  considering  in  regard  to  the  number  of 
contacts  between  cell  and  cell.  In  a  system  of  eight  cells,  all 
reaching  the  boundary  and  all  with  three-way  junctions,  there  are, 
besides  the  eight  peripheral  boundary-walls,  thirteen  internal  parti- 
tions, or  2  (n  —  2)  -|-  1 ;  the  number  of  interfacial  contacts  is  double 
that  number,  or  twenty-six;  and  Ihe  mean  number  of  contacts  for 
each  cell  is  26/8,  or  3-25.  But,  looking  at  the  diagrams  in  Fig.  259 
(which  represent  three  out  of  our  twelve  possible  arrangements  of 

*  Roux's  experiments  were  performed  with  drops  of  paraffin  suspended  in 
dilute  alcohol,  to  which  a  little  calcium  acetate  was  added  to  form  a  soapy  pellicle 
over  the  drops  and  prevent  them  from  reuniting  with  one  another. 


608 


THE  FORMS  OF  TISSUES 


[CH. 


eight  cells),  we  see  that,  in  type  j,  two  cells  are  each  in  contact  with 
two  others,  two  with  three  others,  and  four  each  with  four  other 
cells;    in  type  /,  four  cells  are  each  in  contact  with  two,  two  with 


4 

Fig.  258.     Aggregations  of  oil-drops.     After  Roux. 
Nos.  5,  6  represent  successive  changes  in  a  single  system. 

four  and  two  with  five;  and  in  type  i,  two  are  in  contact  with  two, 
four  with  three  and  one  with  no  less  than  seven.  And  if  we  sum 
up,  irrespective  of  sign,  the  differences  from  the  mean  in  these  three 
cases,  the  sum  amounts  in  /  to  6,  in  i  to  7-5,  and  in  /  to  no  less  than 


10.  We  might  expect  to  find  in  such  arrangements,  that  the  com- 
monest and  most  stable  types  were  those  in  which  the  cell-contacts 
were  most  evenly  distributed,  and  the  fact  that  j  is  (according  to 
Martin  Adamson's  results)  one  of  the  commonest,  and  I  one  of  the 


VIII]  OF  LISTING'S  TOPOLOGY  609 

rarest  of  all  looks  Hke  supporting  the  conjecture.  Moreover,  in  all 
the  commonest  types  we  have  a  more  or  less  equable  division;  but 
on  the  other  hand,  the  number  of  contacts  in  "type /is  just  the  same 
as  in  b,  but  the  latter  occurred  thrice  as  often,  and  k,  which  is  as 
equable  as  any,  was  one  of  the  least  frequent  of  all.  Coincidences 
are  weighed  down  by  discrepancies,  and  we  are  left  pretty  much  in 
the  dark  as  to  why  some  types  are  much  commoner  than  others. 

The  rules  and  principles  which  we  have  arrived  at  from  the  point 
of  view  of  surface  tension  have  a  much  wider  bearing  than  is  at 
once  suggested  by  the  problems  to  which  we  have  apphed  them; 
for  in  this  study  of  a  segmenting  egg  we  are  on  the  verge  of  a  subject 
adumbrated  by  Leibniz,  studied  more  deeply  by  Euler,  and  greatly 
developed  of  recent  years.  It  is  the  Geotnetria  Situs  of  Gauss,  the 
Analysis  Situs  of  Riemann,  the  Theory  of  Partitions  of  Cayley,  of 
Spatial  Complexes  or  Topology  of  Johann  Benedict  Listing*.  It 
begins  with  regions,  boundaries  and  neighbourhoods,  but  leads 
to  abstruse  developments  in  modern  mathematics.  Leibniz 
had  pointed  outf  that  there  was  room  for  an  analysis  of  mere 
position,  apart  from  magnitude:  "je  croy  qu'il  nous  faut  encor  une 
autre  analyse,  qui  nous  exprime  directement  situm,  comme  I'Algebre 
exprime  magnitudinem.''  There  were  many  things  to  which  the 
new  Geometria  Situs  could  be  applied.  Leibniz  used  it  to  explain 
the  game  of  sohtaire,  Euler  to  explain  the  knight's  move  on  the 
chess-board,  or  the  routes  over  the  bridges  of  a  town.  Vandermonde 
created  a  geometric  de  tissage%,  which  Leibniz  himself  had  foreseen, 
to  describe  the  intricate  complexity  of  interwoven  threads  in  a  satin 
or  a  brocade§.  Listing,  in  a  famous  paper ||,  admired  by  Maxwell, 
Cayley  and  Tait,  gave  a  new  name  to  this  new  "algorithm,"  and 
shewed  its  apphcation  to  the  curvature  of  a  twining  stem  or  tendril, 

*  Cf.  Clerk  Maxwell,  On  reciprocal  figures.  Trans.  R.S.E.  xxvi,  p.  9,  1870. 

t  In  a  letter  to  Huygens,  Sept.  8,  1679;  see  Hugenii  Exercitationes  math,  et 
philos.,  etc.,  ed.  Uylenbroeck,  p.  9,  1833. 

X  Remarques  sur  les  problemes  de  situation,  Mem.  Acad.  Sci.  Paris  (1771), 
1774,  p.  566. 

§  A  problem  developed  by  many  eminent  mathematicians,  and  which  Edouard 
Lucas  shewed  to  be  intimately  related  to  the  construction  of  Magic  Squares: 
Recreations  mathem.  i,  p.  xxii,  1891. 

II  Vorstudien  iiber  Topologie,  Gottinger  Studien,  i,  pp.  811-75,  1847;  Der  Census 
raumlicher  Complexe,  ibid,  x,  pp.  97  seq.,  1861. 


610 


THE  FORMS  OF  TISSUES 


[CH. 


the  aestivation  of  a  flower,  the  spiral  of  a  snail-shell,  the  scales  on 
a  fir-cone,  and  many  other  common  things.  The  theory  of  "spatial 
complexes,"  as  illustrated  especially  by  knots,  is  a  large  part  of 
the  subject. 

Topological  analysis  seems  somewhat  superfluous  here;  but  it 
may  come  into  use  some  day  to  describe  and  classify  such  com- 
pUcated,  and  diagnostic,  patterns  as  are  seen  in  the  wings  of  a 
butterfly  or  a  fly.  Let  us  look  for  a  moment  at  how  the  topologist 
might  begin  to  study  one  of  our  groups  of  cells ;   he  would  probably 


Fig.  260. 


call  it  an  island  divided  into  n  counties,  all  maritime  (i.e.  none 
encircled  by  the  rest),  and  having  inland  none  but  three-way 
junctions*.  Here  (in  Fig.  260  a)  is  an  island  with  nine  counties; 
and  here  (b)  is  a  9-gon,  whose  corners  represent  the  same  counties,  and 
the  lines  connecting  these  (whether  sides  or  chords)  represent  the 
contacts  between.  The  polygon  is  now  divided  by  six  chords  into 
seven  triangles.  Three  of  these  are  peripheral,  BCD,  FGH,  HJA ; 
mark  their  vertices,  C,  G,  J,  each  with  the  symbol  4,  and  obliterate 
these  three  triangles  (as  in  c).  The  remaining  polygon  has  two 
peripheral  triangles  BDE,   EFH;    obHterate  these,  after  marking 

♦  This,  like  many  another  thing,  comes  from  my  good  friend  Dr  G.  T.  Bennett. 


VIII]  OF  TOPOLOGICAL  ANALYSIS  611 

their  vertices  with  the  symbol  3.  There  remains  the  quadrilateral 
ABEH,  containing  two  peripheral  triangles  ABE,  EH  A;  mark  B 
and  H  each  with  the  symbol  2.  The  residual  points  A,  E  are  to  be 
marked  1  and  1.  The  polygon  ABCDEFGHJ  may  now  be  read 
off:  124313424;  and  this  formula,  resulting  from  the  triangula- 
tion,  defines  completely  the  system  of  chords  and  the  topology  of  the 
"island."  This  is  one  of  the  twenty-seven  cases  of  a  nine-celled 
arrangement;  and  here  are  our  twelve  arrangements  of  eight  cells, 
recatalogued  under  the  new  method: 

a        11231323    |   g        12341243 


b  1 1321323 

c  12312313 

d  12313132 

e  123  132  13 

/  1234  1234 


h  12341342 

i  12  3  4  14  3  2 

j  12  4  3  12  4  3 

k  12431342 

I  1323  1323 


The  crucial  point  for  the  biologist  to  comprehend  is,  that  in  a 
closed  surface  divided  into  a  number  of  faces,  the  arrangement  of 
all  the  faces,  lines  and  points  in  the  system  is  capable  of  analysis, 
and  that,  when  the  number  of  faces  or  areas  is  small,  the  number 
of  possible  arrangements  is  small  also.  This  is  the  simple  reason 
why  we  meet  in  such  a  case  as  we  have  been  discussing  (viz.  the 
arrangement  of  a  group  or  system  of  eight  cells)  with  the  same  few 
types  recurring  again  and  again  in  all  sorts  of  organisms,  plants  as 
well  as  animals,  and  with  no  relation  to  the  fines  of  biological 
classification:  and  why,  further,  we  find  similar  configurations 
occurring  to  mark  the  symmetry,  not  of  cells  merely,  but  of  the 
parts  and  organs  of  entire  animals.  The  phenomena  are  not 
"functions,"  or  specific  characters,  of  this  or  that  tissue  or  organism, 
but  involve  general  principles,  even  "properties  of  space,"  which 
lie  within  the  province  of  the  mathematician. 

The  theory  of  space-partitioning,  to  which  the  segmentation  of 
the  egg  gives  us  an  easy  practical  introduction,  is  illustrated  in 
innumerable  ways,  some  simple,  some  extremely  compHcated,  in 
other  fields  of  natural  history ;  and  some  serve  the  better  to  illustrate 
the  mathematical,  and  others  the  physical  groundwork  of  the 
phenomenon. 


612  THE  FORMS  OF  TISSUES  [ch. 

Very  beautiful  instances  are  to  be  found  in  insects'  wings.  In  the 
dragonfly's  wing  (which  we  have  already  spoken  of  on  p.  476)  we 
see  at  first  sight  a  vague  assemblage  of  reticulate  cells;  but  their 
arrangement  is  both  orderly  and  simple.  The  long  narrow  wing  is 
stiffened  by  longitudinal  "  veins,"  which  in  front  lie  near  and  parallel, 
for  reasons  well  known  to  the  student  of  aerodynamics*,  but  become 
remote  and  divergent  over  the  rest  of  the  wing;  finer  veinlets, 
running  between  the  veins,  break  up  the  surface  into  cells  or 
areolae.  Where  two  large  veins  run  parallel,  and  so  near  together 
that  there  is  only  room  for  one  row  of  cells  between,  the  walls  of 
these  meet  the  large  veins  at  right  angles,  for  the  reason  that  the 


Fig.  261.     Wing  of  "demoiselle"  dragonfly  (Agrion). 

tension  in  these  latter  is  much  greater  than  their  own ;  and  this 
happens  nearly  all  over  the  delicate  wings  of  the  little  dragonflies 
called  "demoiselles."  But  in  the  big  dragonflies  (Aeschna),  and 
in  general  wherever  there  is  space  enough  between  two  strong  veins 
to  hold  a  double  row  of  cells,  the  walls  of  these  intercalate  with  one 
another  at  co-equal  angles  of  120°,  while  still  impinging  at  right 
angles  on  the  strong  longitudinal  partitions.  Wherever,  as  in  the 
hinder  parts  of  the  wing,  the  great  veins  are  few,  the  cells  numerous, 
and  their  walls  equally  delicate,  then  the  reticulum  of  cells  becomes 
an  hexagonal  network  of  all  but  perfect  regularity.  In  a  cicada  and 
in  many  others  there  is  less  contrast  between  great  veins  and  small ; 
the  cells  are  few,  the  veins  meet  neither  orthogonally  nor  at  co-equal 
angles,  and  the  shape  of  the  cells  suggests  a  common  deformation 
under  strain.  In  this  last  case,  and  generally  in  flies,  bees  and 
butterflies,  the  few  cells  form  a  complex  space-arrangement,  simplified 

*  Sir  George  Cayley  was  the  first  to  shew  that  in  a  sail — or  wing — set  at  an 
acute  angle  to  the  wind,  the  centre  of  pressure  lay  near  the  front  edge,  which 
had,  therefore,  to  be  supported  or  stiffened  {Nicholson's  Journal,  xxv,  1810). 


VIII]  OF  THE  VENATION  OF  WINGS  613 

only  by  the  condition  that  the  walls  impinge  on  one  another  three 
by  three;  and  this  being  so,  the  assemblage  includes  a  number  of 
small  intermediate  partitions  analogous  to  the  "polar  furrows"  of 
embryology.  We  have  seen  how  complex  such  configurations  become 
as  the  cells  increase  in  number;  and  another  source  of  complexity 
comes  in  when  the  veins  are  of  varying  thickness  and  unequal  tension, 
and  hence  meet  one  another  at  varying  angles. 

The  entomologist  is  much  concerned  with  the  number  and 
arrangement  of  these  veins.  In  Fig.  263  we  shew  three  forewings 
of  a  certain  stonefiy,  which  serve  first  to  shew  how  constantly  the 
veins  meet  in  three-way  junctions;  and  then  we  notice  how  the 


Fig.  262.     Forewing  of  cicada. 

three  wings  are  not  exactly  alike,  for  all  their  close  resemblance, 
because  in  two  of  them  there  is  one  cell  less  than  in  the  third,  a  being 
confluent  with  h  in  one  of  these  cases  and  with  c  in  the  gther*.  In 
other  words,  the  veinlet  ah  has  gone  amissing  in  the  one,  and  ac  in 
the  other,  and  in  each  case  the  remaining  veinlet  has  sprung  into  a 
position  of  equihbrium.  This  is  one  of  more  than  two  hundred 
variations  which  have  been  recorded  in  the  wing-veins  of  this  one 
insect;  it  might  seem  superfluous  to  look  for  more. 

The  lower  algae  shew  us  many  beautiful  patterns  or  collocations 
of  cells,  sometimes  very  compUcated,  as  in  Volvox  or  Hydrodictyon. 
A  simpler  case  is  that  of  Gonium.  Of  its  sixteen  cells  four  commonly 
form  a  square,  being  so  thrust  apart  out  of  closer  packing  by  accu- 
mulated intercellular  substance;  the  other  twelve  are  grouped 
around  these  four,  and  obey  the  rule  that  three  cells  and  no  more 
meet  at  each  node  or  point  of  contact  (Fig.  264).     The  twelve  cells 

*  From  Arthur  Willey,  Graded  mutations  in  wings  of  a  stonefly  {Allocapnia 
pygmaea  Burm.),  Nature,  July  17,  1937. 


614 


THE  FORMS  OF  TISSUES 


[CH. 


Fig.  263.     Forewing  of  a  stonefly,  Allocapnia  sp. 
1-3  from  A.  WiUey. 


Fig.  204.     The  four-sided,  16-celled  disc  of  Cotilum, 
a  minute  algae.     After  Harper,  diagrammatic. 


VIII]  OF  THE  PERIDINIANS  6-15 

are  thus  in  three  series,  four  in  touch  with  six  neighbours  each, 
four  with  four,  and  four  with  three;  and  this  arrangement  leads 
to  curved  contour-lines  which  may  be  seen  in  some  of  Cohn's  figures. 
Sometimes  the  interstitial  substance  is  not  enough  to  keep  the  four 
cells  apart ;  then  they  come  together  in  the  usual  rhomb  or  lozenge, 
and  the  rest  group  themselves  around  in  the  simplest  and  most  sym- 
metrical way.  That  the  whole  arrangement  is  as  compact  as  possible 
under  the  conditions,  that  it  is  in  accord  with  the  principle  of  minimal 
areas,  and  is  "such  as-would  result  from  surface-tension  and  adhesion 
between  viscous  colloidal  globules"  is  now  well  known  to  botanists*. 

The  tiny  plates  which  form  the  microscopic  shell  of  a  Peridinian 
illustrate  over  again  by  their  various  collocations  the  principles 
which  we  have  been  studying  in  the  partition- walls  of  the  segmenting 
egg;  and. if  the  one  case  has  shewn  us  pitfalls  in  the  way  of  the 
embryologist,  the  other  shews  how  the  systematist,  in  his  endless 
task  of  describing  the  forms  and  patterij^s  of  things,  may  sometimes 
base  distinctions  on  what  seem  trivial  differences  from  the  physical 
or  mathematical  point  of  view. 

On  the  upper  half  (or  epitheca)  of  the  globular  test  of  a  Peridinium 
we  have  fourteen  Httle  plates,  or  fourteen  "cells,"  to  use  the  word 
in  a  mathematical  rather  than  a  histological  sense,  whose  boundary- 
walls  always  meet  in  three-way  nodes.  We  may  reproduce  the 
identical  arrangement  of  the  Peridinial  plates  by  blowing  bubbles 
in  a  saucer;  but  to  deal  with  so  many  bubbles  at  once  needs  more 
patience  than  do  the  other  similar  experiments  which  we  have 
described.  That  the  cells  are  fourteen  in  number  is,  from  the 
physical  point  of  view,  the  merest  accident,  but  from  the  zoologist's 
it  is  a  criterion  of  the  genus;  when  there  are  more  cells  or  fewer, 
the  organism  is  called  by  another  name.  The  number  of  possible 
arrangements  of  fourteen  polygonal  cells,  linked  by  three-way  nodes, 
is  very  large;  but  the  "characters  of  the  genus"  exclude  many  of 
the  variants.  Many  of  them  occur — it  is  quite  possible  that  all 
occur — in  Nature;  but  they  are  not  called  Peridinium.  The  fol- 
lowing arrangement  defines  the  genus.  There  is  a  central  or  apical 
cell,  around  which  are  grouped  six  others ;  of  these  six,  one  extends 
to  the  boundary  of  the  figure,  that  is,  to  the  equator  of  the  globular 

♦  R.  A.  Harper,  The  colony  in  Gonium,  Trans.  Anier.  Microsc.  Soc.  xxxi,  pp.  65- 
84,  1912;   cf.  F.  Cohn,  in  Nova  Acta  Acad.  C.L.C.,  xxiv,  p.  lOJ,  1854. 


616 


THE  FORMS  OF  TISSUES 


[CH. 


shell,  and  seven  other  "equatorial  cells"  complete  the  boundary. 
In  other  words,  the  apical  hexagon  is  surrounded  by  two  concentric 
rows,  originally  of  six  cells  each,  whereof  one  cell  of  the  inner  row 
has  (as  it  were)  burst  through  a  cell  of  the  outer  row,  so  reaching 
the  boundary  itself,  and  so  dividing  into  two  the  equatorial  cell 
which  it  encroached  on  and  bisected. 

In  any  such  collocation  as  this,  the  number  of  sides  and  the 
number  of  nodes  or  corners  are  strictly  determined ;  there  are  here 
fourteen  cells,  all  conjoined  by  three-way  nodes,  and  it  follows  that 
there  are  just  39  separate  walls  or  edges,  and  just  26  nodes  or 
corners.  Many  of  these  last  are  already  defined  for  us;  for  six  of 
them  are  the  corners  of  the  central  hexagon,  eight  lie  on  the  equator, 
and  six  more  are  at  the  inner  ends  of  the  radial  partitions  which 
separate  the  equatorial  cells  from  one  another.  Six  remain  to  be 
determined,  those,  namely,  where  the 
partitions  running  outwards  from  the 
apical  cell  meet  the  walls  of  the 
equatorial  cells.  The  diagram  (Fig. 
265)  shews  us  two  sets  of  radiating 
partitions,  six  running  inwards  from 
the  equator  (a,  6,  c,  d,  e,  /)  and  six 
running  outwards  from  the  central 
hexagon  (A. .  .F),  those  of  the  one 
set  being  nearly  opposite  to  those  of 
the  other;  but  near  as  they  may  be 
they  never  meet,  for  to  do  so  would 
be  to  make  a  four-way  node,  which 
theory  forbids  and  which  observation  tells  us  does  not  occur.  In 
every  case,  one  partition  must  be  slewed  a  little  to  one  side  or 
other  of  its  opposite  neighbour;  and  the  whole  range  of  possible 
variations  depends  on  whether  the  shift  be  to  the  one  side  or  to  the 
other.  We  have  six  pairs  of  partitions,  and  in  each  of  the  six  there 
is  this  possible  alternative  of  right  or  left ;  there  are  therefore  2®  or 
64  possible  variations  in  all.  Whether  all  six  may  vary,  I  do  not 
know;  there  is  no  obvious  reason  why  they  should  not.  But 
alternative  variation  does  occur  in  the  two  anterior  and  two  posterior 
pairs  of  partitions;  and  these  four  give  us  2^  or  16  possible  arrange- 
ments. 


Fig.  265.     Dorsal  view  of  a 
Peridinium :  diagrammatic. 


VIII]  OF  CERTAIN  DIATOMS  617 

When  the  partitions  A,  F  meet  the  equatorial  cells  on  the  hither 
side  of  the  partitions  a,  /,  then,  obviously,  the  large  cell  R  is  an 
irregular  hexagon,  and  is  in  contact  with  two  equatorial  cells  only; 
such  an  arrangement  is  said  to  define  the  genus  Orthoperidinium. 
When  A,  F  happen  to  fall  on  the  farther  sides  of  a,/,  the  large  cell 
has  eight  sides,  and  is  in  contact  with  four  equatorial  cells ;  we  have 
the  new  genus  Paraperidinium.  When  A  falls  within  a,  but  F  falls 
beyond  /,  we  have  the  genus  Metaperidinium.  There  remains  one 
alternative  case,  the  converse  of  the  last,  to  which  the  systematists 
have  not  given  a  name. 

The  same  physical  phenomenon  occurs  at  the  opposite  pole  of  the 
disc,  where  the  partitions  C,  D  may  fall  within  or  without,  or  one 
within  and  one  without  the  positions  of  c,  d\  where,  in  other  words, 
the  intercalated  cell  CD  is  in  contact  with  one,  with  three,  or  with 
two  equatorial  cells.  Jorgensen,  seeing  these  three  types  occurring 
both  among  the  Orthoperidinia  and  the  Metaperidinia,  draws  the 
conclusion  that  this  character  is  more  primitive,  or  more  ancestral, 
than  that  by  which  Ortho-  and  Metaperidinia  are  separated  from 
one  another,  a  phylogenetic  deduction  concerning  which  topology 
has  nothing  to  say*.  Within  the  restricted  genus  Peridinium  we 
have  at  present  two  sub-genera  and  seven  sub-groups  of  these,  this 
being  the  number  of  the  64  possible  arrangements  so  far  recognised 
and  named.  These  may  have  a  certain  constancy  or  stability ;  and 
trivial  as  their  differences  may  seem  to  the  physicist,  they  may  still 
be  worth  the  naturalist's  while  to  study  and  record. 

Ano'ther  case,  geometrically  akin  but  biologically  very  different, 
is  to  be  found  in  the  httle  diatoms  of  the  genus  Aster olampra,  and 
their  immediate  congenersf.  In  Asterolampra  we  have  a  little  disc, 
in  which  we  see  (as  it  were)  radiating  spokes  of  one  material  alter- 
nating with  intervals  occupied  on  the  flattened  wheel-like  disc  by 
another  (Fig.  266).  The  spokes  vary  in  number,  but  the  general 
appearance  is  in  a  high  degree  suggestive  of  the  Chladni  figures 
produced  by  the  vibration  of  a  circular  plate.  The  spokes  broaden 
out  towards  the  centre,  and  interlock  by  visible  junctions,  which 

*  E.  Jorgensen,  Ueber  Planktonproben,  Svenska  Hydrogr.  Biol.  Komm.  Skrifter, 
IV,  1913. 

t  See  K.  R.  Greville,  Monograph  of  the  genus  Aster  olampra  ^  Q.J. M.S.  viii, 
(Trans.),  pp.  102-124,  1860;    cf.  ibid.  (n.s.).  ii,  pp.  41-55,  1862. 


618 


THE  FORMS  OF  TISSUES 


[CH. 


generally  obey  the  rule  of  triple  intersection,  and  accordingly 
exemplify  the  partition-figures  with  which  we  have  been  dealing. 
But  whereas  we  have  found  the  particular  arrangement  in  which 
one  cell  is  in  contact  with  all  the  rest  to  be  unstable,  according  to 
Roux's  oil-drop  experiments,  and  to  be  conspicuous  by  its  absence 
from  our  diagrams  of  segmenting  eggs,  here  in  Asterolampra,  on  the 
other  hand,  it  occurs  frequently,  and  is  indeed  the  commonest 
arrangement  (Fig,  266,  B).  In  all  probability,  we  are  entitled  to 
consider  this  marked  difference  natural  enough.  For  we  may 
suppose  that  in  Asterolampra  (unlike  the 'case  of  the  segmenting 
egg)  the  tendency  is  to  perfect  radial  sjnnmetry,  all  the  spokes 
emanating  from  a  point  in  the  centre:   such  a  condition  would  be 


-IB  c 

Fig.  266.     (A)  Asterolampra  marylandica  Ehr.; 
(B,  C)  A.  variabilis  Grev.     After  Greville. 


eminently  unstable,  and  would  break  down  under  the  least  asym- 
metry. A  very  simple,  perhaps  the  simplest  case,  would  be  that 
one  single  spoke  should  differ  slightly  from  the  rest,  and  should  so 
tend  to  be  drawn  in  amid  the  others,  these  latter  remaining  similar 
and  symmetrical  among  themselves.  Such  a  configuration  would 
be  vastly  less  unstable  than  the  original  one  in  which  all  the 
boundaries  meet  in  a  point;  and  the  fact  that  further  progress  is 
not  made  towards  other  configurations' of  still  greater  stability  may 
be  sufficiently  accounted  for  by  viscosity,  rapid  solidification,  or 
other  conditions  of  restraint.  A  perfectly  stable  condition  would  of 
course  be  obtained  if,  as  in  the  case  of  Roux's  oil-drop  (Fig.  257,  6), 
one  of  the  cellular  spaces  passed  into  the  centre  of  the  system,  the 
other  partitions  radiating  outwards  from  its  circular  wall  to  the 
periphery  of  the  whole  system.     Precisely  such  a  condition  occurs 


VIIl] 


OF  RADIATE  PATTERNS 


619 


among  our  diatoms;   but  when  it  does  so,  it  is  looked  upon  as  the 
mark  and  characterisation  of  the  allied  genus  Arachnoidiscus. 

A  simple  case,  introductory  to  others  of  a  more  complex  kind, 
is  that  of  the  radial  canals  of  the  Medusae.  Here,  in  certain  cases 
(e.g.  Eleutheria),  the  usual  arrangement  of  eight  radial  canals  is  not 
seldom  modified,  as  for  example,  when  two  or  more  of  them  arise 


14  -< ^  15  --<.^-^l6 

Fig.  207,  Variations  observed  in  the  catial-system  of  a  raedusoid  {Eleutheria);  after 
Hans  Lengerich.  1-8,  the  eight  possible  arrangements  of  eight  radial  canals; 
9-10,  some  observed  instances  of  nine  radial  canals. 

not  separately  but  by  bifurcation*.  We  then  have  just  eight 
possible  arrangements,  as  shewn  in  Fig.  267,  1-8,  and  of  these  eight 
no  less  than  six  have  been  actually  observed.  The  other  two  are 
just  as  hkely  to  occur,  and  we  may  take  it  that  they  also  will  in 
due  time  be  recorded.     It  is  yet  another  simple  illustration  of  the 


*  Hans    Lengerich.  Verzweigungsarten    der   Radialkanale   bei   Eleutheria,  Zool. 
Jahrbuch,  1922,  p.  325. 


620  THE  FORMS  OF  TISSUES  [ch. 

aphorism  that  whatsoever  is  possible,  that  Nature  does,  all  in  her 
own  time  and  way;  what  things  Nature  does  not  do  are  the  things 
which  are  mathematically  impossible  or  are  barred  by  physical 
conditions.  It  is  possible  for  our  little  Medusa  to  develop  nine 
radial  canals  instead  of  eight,  and  so  at  times  she  does ;  again  they 
may  be  simple  or  branched,  and  trifurcations  as  well  as  bifurcations 
may  appear.  So  we  may  extend  our  Hst  of  possible  permutations 
and  combinations,  and  find  as  before  that  a  fair  proportion  of  these 
possible  arrangements  have  been  observed  already.  There  are 
many  other  Medusae '  (e.g.   Willsia),  where  the  number  of  radial 


III  ^<^JU^  IV 

Fig.  268.  A  medusa  {Willsia  ornnta) 
shewing,  diagrammatically,  the  order 
of  development  of  the  numerous  radial 
canals.     After  Mayor. 


Fig.  269.     Section  of  Alcyonarian 
polyp. 


canals  much  exceeds  the  simple  symmetry  of  four  or  eight ;  and  in 
these  we  may  sometimes  see,  very  beautifully,  how  the  successive 
canals  arrange  themselves  according  to  the  same  principles  which 
we  have  now  studied  in  so  many  diverse  cases  of  partitioning 
(Fig.  268)*. 

In  a  diagrammatic  section  of  an  Alcyonarian  polyp  (Fig.  269), 
we  have  eight  chambers  set,  symmetrically,  about  a  ninth,  which 
constitutes  the  "stomach."  In  this  arrangement  there  is  no  diffi- 
culty, for  it  is  obvious  that,  throughout  the  system,  three  boundaries 
meet  (in  plane  section)  in  a  point.     In  many  corals  we  have  as 

*  Such  branching  canals  are  characteristic  of  the  Dendrostaurinae,  a  subfamily 
of  the  Oceanidae,  a  family  of  Anthomedusae ;  and  very  much  the  same  occur  in 
a  certain  subfamily  of  Leptomedusae.  See  Mayor's  Medusae  of  the  World,  i, 
p.  190. 


VIII]  OF  THE  SEPTA  OF  CORALS  621 

simple  or  even  simpler  conditions,  for  the  radiating  calcified 
partitions  either  converge  upon  a  central  chamber,  or  fail  to  meet 
it  and  end  freely.  But  in  a  few  cases,  the  partitions  or  "septa" 
converge  to  meet  one  another,  there  being  no  central  chamber  on 
which  they  may  impinge;  and  here  the  manner  in  which  contact 
is  effected  becomes  comphcated,  and  involves  problems  identical 
with  those  which  we  are  now  studying. 

In  the  great  majority  of  corals  we  have  as  simple  or  even  simpler 
conditions  than  those  of  Alcyonium;  for  as  a  rule  the  calcified 
partitions  or  septa  of  the  coral  either  con- 
verge upon  a  central  chamber  (or  central 
"columella"),  or  else  fail  to  meet  it  and  end 
freely.  In  the  latter  case  the  problem  of 
space-partitioning  does  not  arise;  in  the 
former,  however  numerous  the  septa  be, 
their  separate  contacts  with  the  wall  of  the 
central  chamber  comply  with  our  funda- 
mental rule  according  to  which  three  lines  ^.    ^^      ^^         ,  „. 

^  .  Fig.  270.    Heterophylha  angu- 

and  no  more  meet  in  a  pomt,  and  from      ^^j^.    After  Nicholson., 
this  simple  and  symmetrical  arrangement 

there  is  little  tendency  to  variation.  But  in  a  few  cases,  the  septal 
partitions  converge  to  meet  one  another,  there  being  no  central 
chamber  on  which  they  may  impinge ;  and  here  the  manner  in  which 
contact  is  effected  becomes  complicated,  and  involves  problems  of 
space-partitioning  identical  with  those  which  we  are  now  studying. 
In  the  genus  Heterophyllia  and  in  a  few  alhed  forms  we  have  such 
conditions,  and  students  of  the  Coelenterata  have  found  them  very 
puzzling.  McCoy*,  their  first  discoverer,  pronounced  these  corals 
to  be  "totally  unhke"  any  other  group,  recent  or  fossil;  and 
Professor  Martin  Duncan,  writing  a  memoir  on  Heterophyllia  and 
its  allies  I,  described  them  as  "paradoxical  in  their  anatomy." 

The  simplest  or  youngest  Heterophylliae  known  have  six  septa 
(as  in  Fig.  271,  A);  in  the  case  figured,  four  of  these  septa  are 
conjoined  two  and  two,  thus  forming  the  usual  triple  junctions 
together  with  their  intermediate  partition-walls;  and  in  the  case 
of  the  other  two  we  may  fairly  assume  that  their  proper  and  original 

*  Ann.  Mag.  N.H.  (2),  iii,  p.  126,  1849. 
t  Phil.  Trans.  cLvii,  pp.  643-656,  1867. 


622 


THE  FORMS  OF  TISSUES 


[CH. 


arrangement  was  that  of  our  type  6  6  (Fig.  245),  though  the  central 
intermediate  partition  has  been  crowded  out  by  partial  coalescence. 
When  with  increasing  age  the  septa  become  more  numerous,  their 
arrangement  becomes  exceedingly  variable;  for  the  simple  reason 
that,  from  the  mathematical  point  of  view,  the  number  of  possible 
arrangements,  of  10,  12  or  more  cellular  partitions  in  triple  contact, 
tends  to  increase  with  great  rapidity,  and  there  is  little  to  choose 
among  many  of  them  in  regard  to  symmetry  and  equihbrium.  But 
while,  mathematically  speaking,  each  particular  case  among  the 


Fig.  271.    Heterophyllia  sp.     After  Martin  Duncan. 

multitude  of  possible  cases  is  an  orderly  and  definite  arrangement, 
from  the  purely  biological  point  of  view  on  the  other  hand  no  law 
or  order  is  recognisable;  and  so  McCoy  described  the  genus  as 
being  characterised  by  the  possession  of  septa  'destitute  of  any 
order  of  arrangement,  but  irregularly  branching  and  coalescing  in 
their  passage  from  the  sohd  external  walls  towards  some  indefinite 
point  near  the  centre  where  the  few  main  lamellae  irregularly 
anastomose." 

In  the  two  examples  figured  (Fig.  271  B,  C),  both  comparatively 
simple  ones,  it  will  be  seen  that,  of  the  main  chambers,  one  is  in  each 


VIII]  OF  THE  SEPTA  OF  CORALS  623 

case  an  unsymmetrical  one ;  that  is  to  say,  there  is  one  chamber  which 
is  in  contact  with  a  greater  number  of  its  neighbours  than  any 
other,  and  which  at  an  earher  stage  must  have  had  contact  with 
them  all ;  this  was  the  case  of  our  type  i,  in  the  eight-celled  system 
(p.  598).  Such  an  asymmetrical  chamber  (which  may  occur  in 
a  system  of  any  number  of  cells  greater  than  six)  constitutes  what 
is  known  to  students  of  the  Coelenterata  as  a  "fossula";  and  we 
may  recognise  it  not  only  here,  but  also  in  Zaphrentis  and  its  allies, 
and  in  a  good  many  other  corals  besides.  Moreover,  certain  corals 
are  described  as  having  more  than  one  fossula:  this  appearance 
being  naturally  produced  under  certain  of  the  other  asymmetrical 
variations  of  normal  space-partitioning.  Where  a  single  fossula 
occurs,  we  are  usually  told  that  it  is  a  symptom  of  "  bilateraUty " ; 
and  this  is  in  turn  interpreted  as  an  indication  of  a  higher  grade  of 
organisation  than  is  implied  in  the  purely  "radial  symmetry"  of  the 
commoner  types  of  coral.  The  mathematical  aspect  of  the  case 
gives  no  warrant  for  this  interpretation. 

Let  us  carefully  notice  (lest  we  run  the  risk  of  confusing  two 
distinct  problems)  that  the  space-partitioning  of  Heterophyllia  by 
no  means  agrees  with  the  details  of  that  which  we  have  studied 
in  (for  instance)  the  case  of  the  developing  disc  of  Erytkrotrichia: 
the  difference  simply  being  that  Heterophyllia  illustrates  the  general 
case  of  cell-partitioning  as  Plateau  and  Van  Rees  studied  it,  whil^ 
in  Erythrotrichia,  and  in  our  other  embryological  and  histological 
instances,  we  have  found  ourselves  justified  in  making  the  additional 
assumption  that  each  new  partition  divided  a  cell  into  co-equal 
parts.  No  such  law  holds  in  Heterophyllia,  whose  case  is  essentially 
different  from  the  others:  inasmuch  as  the  chambers  whose  parti- 
tion we  are  discussing  in  the  coral  are  mere  empty  spaces  (empty 
save  for  the  mere  access  of  sea-water);  while  in  our  histological 
and  embryological  instances,  we  were  speaking  of  the  division  of 
a  cellular  unit  of  living  protoplasm.  Accordingly,  among  other 
differences,  the  "transverse"  or  "periclinal"  partitions,  which  were 
bound  to  appear  at  regular  intervals  and  in  definite  positions,  when 
co-equal  bisection  was  a  feature  of  the  case,  are  comparatively  few 
and  irregular  in  the  earlier  stages  of  Heterophyllia,  though  they 
begin  to  appear  in  numbers  after  the  main,  more  or  less  radial, 
partitions  have   become  numerous,   and  when  accordingly  these 


624  THE  FORMS  OF  TISSUES  [ch. 

radiating  partitions  come  to  bound  narrow  and  almost  parallel-sided 
interspaces;  then  it  is  that  the  transverse  or  perichnal  partitions 
begin  to  come  in,  and  form  what  the  student  of  the  Coelenterata  calls 
the  ''dissepiments"  of  the  coral.  We  need  go  no  further  into  the 
configuration  and  anatomy  of  the  corals ;  but  it  seems  to  me  beyond 
a  doubt  that  the  whole  question  of  the  complicated  arrangement 
of  septa  and  dissepiments  throughout  the  group  (including  the 
curious,  vesicular  or  bubble-like  tissue  of  the  Cyathophyllidae  and 
the  general  structural  plan  of  the  Tetracoralla,  such  as  Streptoplasma 
and  its  allies)  is  well  worth  investigation  from  the  physical  and 
mathematical  point  of  view,  after  the  fashion  which  is  here  shghtly 
adumbrated. 

The  method  of  dividing  a  circular,  or  spherical,  system  into 
eight  parts,  equal  as  to  their  areas  but  unequal  in  their  peripheral 
boundaries,  is  probably  of  wide  biological  application;  that  is  to 
say,  without  necessarily  supposing  it  to  be  rigorously  followed,  the 
typical  configuration  which  it  yields  seems  to  recur  again  and 


Fig.  272.     Diagrammatic  section  of  a  Ctenophore  (^wcAan's). 

again,  with  more  or  less  approximation  to  precision,  and  under 
widely  different  circumstances.  I  am  inclined  to  think,  for  instance, 
that  the  unequal  division  of  the  surface  of  a  Ctenophore  by  its 
meridian-like  cihated  bands  is  a  case  in  point  (Fig.  272).  Here,  if  we 
imagine  each  quadrant  to  be  twice  bisected  by  a  curved  anticline, 


VIII]  OF  SUNDRY  PARTITIONS  625 

we  shall  get  what  is  apparently  a  close  approximation  to  the  actual 
position  of  the  cihated  bands.  The  case  however  is  comphcated 
by  the  fact  that  the  sectional  plan  of  the  organism  is  never  quite 
circular,  but  always  more  or  less  elhptical.  One  point,  at  least,  is 
clearly  seen  in  the  symmetry  of  the  Ctenophores;  and  that  is  that 
the  radiating  canals  which  pass  outwards  to  correspond  in  position 
with  the  cihated  bands  have  no  common  centre,  but  diverge  from 
one  another  by  repeated  bifurcations,  in  a  manner  comparable  to 
the  conjunctions  of  our  cell- walls. 

In  the  early  development  of  the  shell  (or  "test")  of  a  sea-urchin*, 
each  interambulacral  area'consists  of  a  lozenge  of  four  plates,  in  the 
famihar  configuration  assumed  by  four  cells  or  bubbles,  the  polar 
furrow  lying  in  the  direction  of  a  "  radius  " 
of  the  shell.     A  fifth  plate,   or   "cell," 
presently  fits  itself  in  between  the  third 
and  fourth,  that  is  to  say  between  the 
terminal  plate  and  one  of  the  lateral  ones, 
and  in  doing  so  thrusts  the  former  to 
one  side;  a  sixth  intercalates  itself  be-  ^'f'  ^^^-  /  ''^"^^"  .r"!,  ^ 

lozenge  :  stages  in  the  de- 
tween  the  fourth  and  fifth,  and  so  on  velopment  of  the  ambulacral 
alternately.      An  ambulacrum  consists  of      and  interambulacral  plates  of 

two  columns  of  calcareous  plates,  which  ^  J^'^^^^i^-  ^^'  I-  ^^'■ 
fit  into  one  another  in  the  usual  way  by 

sutures  set  at  angles  of  120°.  Each  plate  consists  of  three  platelets; 
a  "primary"  plate  (a)  is  succeeded  by  a  smaller  and  narrower 
secondary  plate  (b) ;  the  squarish  primary  has  one  corner  cut  off  by 
a  curved  partition,  to  form  a  "demiplate",  and  the  whole  is  called  by 
students  of  this  group  an  "  echinoid  triad  ".  Though  we  do  not  know 
precisely  how  the  partitions  arise,  nor  can  we  prove  by  measurement 
their  obedience  to  the  laws  of  maxima  and  minima,  yet  their 
general  analogy  to  the  principles  we  have  explained  is  sufficiently 
obvious. 

I  am  even  inchned  to  think  that  the  same  principle  helps  us  to 
understand  the  arrangement  of  the  skeletal  rods  of  a  larval 
Echinoderm,  and  the  complex  conformation  of  the  larva  which  is 
brought  about  by  the    presence   of  these   long,    slender   skeletal 

*  Isabella  Gordon,  The  development  of  the  calcareous  test  of  Echinus  miliaris, 
Phil.  Tram.'iB),  No.  214,  p.  282,  1926;   etc. 


626 
radii* 


THE  FORMS  OF  TISSUES 


[CH. 


In  Fig.  274  I  have  divided  a  circle  into  its  four  quadrants, 
and  have  bisected  each  quadrant  by  a  circular  arc  {BC),  passing  from 
radius  to  periphery,  as  in  the  foregoing  cases  of  cell-division  (e.g. 
p.  590);  and  I  have  again  bisected,  in  a  similar  way,  the  triangular 


B^  \B 

Fig.  274.     Diagrammatic  arrangement  of  partitions,  represented  by  skeletal 
rods,  in  a  larval  Echinoderm  (Ophiura). 

halves  of  each  quadrant  (D,  D).  I  have  also  inserted  a  small  circle  in 
the  middle  of  the  figure,  concentric  with  the  large  one.  If  now  we 
imagine  the  partition-hnes  in  the  figure  to  be  replaced  by  solid 
rods,  we  shall  have  at  once  the  frame-work  of  an  Ophiurid  (Pluteus) 


Fig.  275.     Pluteus-larva  of  Ophiurid. 

larva.  Let  us  imagine  all  these  arms  to  be  bent  symmetrically 
downwards,  so  that  the  plane  of  the  paper  is  transformed  into 
a  conical  surface  with  curved  sides;  let  a  membrane  be  spread, 
umbrella-Hke,    between   the    outstretched    skeletal   rods,    and    let 

*  J.  Loeb  has  shewn  (Amer.  Journ.  Physiol,  vii,  p.  441,  1900)  that  the  sea-urchin's 
egg  can  be  reared  for  a  time  in  a  balanced  solution  of  sodium,  potassium  and 
calcium  chloride,  developing  no  spicules  and  so  forming  no  pluteus  larva;  on 
adding  sodium  carbonate  the  spicules  are  laid  down  and  the  pluteus  larva  takes 
shape  accordingly. 


VIII]  OF  STOMATA  627 

its  margin  loop  from  rod  to  rod  in  curves  which  are  possibly 
catenaries  but  are  more  probably  portions  of  an  "elastic  curve," 
and  the  outward  resemblance  to  a  Pluteus  larva  is  now  complete. 
By  various  slight  modifications,  by  altering  the  relative  lengths  of 
the  rods,  by  modifying  their  curvature  or  by  replacing  the  curved 
rod  by  a  tangent  to  itself,  we  can  ring  the  changes  which  lead  us 
from  one  known  type  of  Pluteus  to  another.  The  case  of  the 
Bipinnaria  larvae  of  Echinids  is  certainly  analogous,  but  it  becomes 
very  much  more  compHcated;  we  have  to  do  with  a  more  complex 
partitioning  of  space,  and  I  confess  that  I  am  not  yet  able  to 
represent  the  more  compUcated  forms  in  so  simple  a  way. 

There  are  a  few  notable  exceptions  (besides  the  various  unequally 
segmenting  eggs)  to  the  general  rule  that  in  cell-division  the  mother- 
cell  tends  to  divide  into  equal  halves ;  and  one  of  these  exceptional 
cases  is  to  be  found  in  connection  with  the  development  of 
"stomata"  in  the  leaves  of  plants*.  The  epidermal  cells  by  which 
the  leaf  is  covered  may  be  of  various  shapes;  sometimes,  as  in  a 
hyacinth,  they  are  oblong,  but  more  often  they  have  an  irregular 
shape  in  which  we  can  recognise,  more  or  less  clearly,  a  distorted 
or  imperfect  hexagon.  In  the  case  of  the  oblong  cells,  a  transverse 
partition  will  be  the  least  possible,  whether  the  cell  be  equally  or 
unequally  divided,  unless  (as  we  have  already  seen)  the  space  to 
be  cut  off  be  a  very  small  one,  not  more  than  about  three-tenths 
the  area  of  a  square  based  on  the  short  side  of  the  original  rectangular 
cell.  As  the  portion  usually  cut  off  is  not  nearly  so  small  as  this, 
we  get  the  form  of  partition  shewn  in  Fig.  276,  and  the  cell  so  cut 
off  is  next  bisected  by  a  partition  at  right  angles  to  the  first;  this 
latter  partition  splits,  and  the  two  last-formed  cells  constitute  the 
so-called  "guard-cells"  of  the  stoma.  In  other  cases,  as  in  Fig.  277, 
there  will  come  a  point  where  the  minimal  partition  necessary  to 
cut  off  the  required  fraction  of,  the  cell-content  is  no  longer  a 

*  We  know  more  about  the  physical  activities  of  the  storaata  than  about  the 
mechanics  of  their  development.  It  is  known  that  the  rate  of  gaseous  diffusion 
through  apertures  of  their  order  of  magnitude  is  inversely  proportional  to  the 
diameters  of  the  apertures;  and  this  law,  by  which  the  sufficient  entry  of  carbonic 
acid  through  the  stomata  is  fully  accounted  for,  is  (like  Pfeffer's  work  on  natural 
semi-permeable  membranes)  one  of  the  notable  cases  where  physiology  has  enlarged 
the  boundaries  of  physical  science.  Cf.  Horace  T.  Brown,  Some  recent  work  on 
diffusion,  Proc.  Roy.  Instit.  March,  1901. 


628 


THE  FORMS  OF  TISSUES 


[CH. 


transverse  one,  but  is  a  portion  of  a  cylindrical  wall  (2)  cutting  off 
one  corner  of  the  mother-cell.  The  cell  so  cut  off  is  now  a  certain 
segment  of  a  circle,  with  an  arc  of  approximately  120°;  and  its 
next  division  will  be  by  means  of  a  curved  wall  cutting  it  into  a 


-A^ 


Fig.  276.     Diagrammatic  development  of  stomata  in  hyacinth. 

triangular  and  a  quadrangular  portion  (3).  The  triangular  portion 
will  continue  to  divide  in  a  similar  way  (4,  5),  and  at  length  (for 
a  reason  which  is  not  yet  clear)  the  partition  wall  between  the 
new-formed  cells  splits,  and  again  we  have  the  phenomenon  of  a 


Diagrammatic  development  of  stomata  in  Seduni. 
(Cf.  fig.  in  Sachs's  Botany,  1882,  p.  103.) 

"stoma"  with  its  attendant  guard-cells.  In  Fig.  277  are  shewn  the 
successive  stages  of  division,  and  the  changing  curvatures  of  the 
various  walls  which  ensue  as  each  subsequent  partition  appears, 
and  introduces  a  new  tension  into  the  system.     Among  the  oblong 


viii]  OF  STOMATA  629 

cells  of  the  epidermis  in  the  hyacinth  the  stomata  will  be  found 
arranged  in  regular  rows,  while  they  will  be  irregularly  distributed 
over  the  surface  of  the  leaf  in  such  a  case  as  we  have  depicted  in 
Sedum. 

As  I  have  said,  the  mechanical  cause  of  the  split  which  constitutes 
the  orifice  of  the  stoma  is  not  quite  clear.  It  may  be  directly  due 
to  the  subepidermal  air-space  which  the  stoma  communicates  with, 
for  an  air-surface  on  both  sides  of  the  dehcate  epidermis  might 
well  cause  such  an  alteration  of  tensions  that  the  two  halves  of 
the  dividing  cell  would  tend  to  part  company.  In  Professor 
Macallum's  experiments,  which  we  have  briefly  discussed  in  our 
short  chapter  on  Adsorption,  it  was  found  that  large  quantities  of 
potassium  gathered  together  along  the  outer  walls  of  the  guard-cells 
of  the  stoma,  thereby  indicating  a  low  surface-tension  along  these 
outer  walls.  The  tendency  of  the  guard-cells  to  bulge  outwards 
is  so  far  explained,  and  it  is  possible  that,  under  the  existing 
conditions  of  restraint,  we  may  have  here  a  force  tending,  or  helping, 
to  spht  the  two  cells  asunder.  It  is  clear  enough,  however,  that 
the  last  stage  in  the  development  of  a  stoma  is,  from  the  physical 
point  of  view,  not  yet  properly  understood*.  It  is  noteworthy, 
and  Nageli  took  note  of  it  wellnigh  a  hundred  years  ago,  that  the 
stomatal  mother-cells  remain  small  while  the  others  grow,  and  also 
that  they  only  divide  once  for  all,  while  their  neighbours  divide 
and  divide  again,  4»  produce  the  lateral  or  accessory  guard-cells. 

In  all  our  foregoing  examples  of  the  development  of  a  "tissue" 
we  have  seen  that  the  process  consists  in  the  successive  division 
of  cells,  each  act  of  division  being  accompanied  by  the  formation 
of  a  boundary-surface,  which,  whether  it  become  at  once  a.  soHd 
or  semi-sohd  partition  or  whether  it  remain  semi-fluid,  exercises 
in  all  cases  an  effect  on  the  position  and  the  form  of  the  boundary 
which  comes  into  being  with  the  next  act  of  division.  In  contrast 
to  this  general  process  stands  the  phenomenon  known  as  "free 
cell-formation,"  in  which,  out  of  a  common  mass  of  protoplasm, 
a  number  of  separate  cells  are  simultaneously,  or  all  but  simul- 
taneously, differentiated;    and  the  case  is  all  the  more  interesting 

*  Botanische  Beitrage,  Linnaea,  xvi,  p.  238,  1842.     Cf.  Garreau,  Mem.  sur  les 
stomates,  Ann.  Sc.  Nat.,  Bot.  (4),  i,  p.  213,  1854. 


630  THE  FORMS  OF  TISSUES  [ch. 

when  the  daughter-cells  remain,  for  a  time  at  least,  within  the 
envelope  of  the  mother-cell.  It  sometimes  happens,  to  begin  with, 
that  a  number  of  mother-cells  are  formed  simultaneously,  and 
that  the  content  of  each  divides,  by  successive  divisions,  into  four 
"daughter-cells."  These  daughter-cells  tend  to  groi^p  themselves, 
just  as  would  four  soap-bubbles,  into  a  "tetrad,"  the  four  cells 
forming  a  spherical  tetrahedron.  For  the  system  of  four  bodies 
is  in  perfect  symmetry.  The  four"  cells  are  closely  packed  within 
the  cell-wall  of  the  mother-cell ;  their  outer  walls  divide  the 
sphere  into  four  equiangular  triangles;  their  inner  walls  meet 
three-by-three  in  an  edge,  and  the  four  edges  converge  in  the 
geometrical  centre  of  the  system;  and  these  partition  walls  and 
their  respective  edges  meet  one  another  everywhere  at  co-equal 
angles.  This  is  the  typical  mode  of  development  of  pbllen-grains, 
common  among  monocotyledons  and  all  but  universal  among 
dicotyledonous  plants.  By  a  loosening  of  the  surrounding  tissue 
and  an  expansion  of  the  cavity,  or  anther-cell,  in  which  they  He, 
the.  pollen-grains  afterwards  fall  apart,  and  their  individual  form 
will  depend  upon  whether  or  no  their  walls  have  soKdified  before 
this  liberation  takes  place.  For  if  not,  then  the  separate  grains  will 
be  free  to  assume  a  spherical  form  as  a  consequence  of  their  own 
individual  and  unrestricted  growth ;  but  if  they  become  set  or  rigid 
prior  to  the  separation  of  the  tetrad,  then  they  will  conserve  more  or 
less  completely  the  plane  interfaces  and  sharp  angles  of  the  elements 
of  the  tetrahedron.  The  latter  is  apparently  the  case  in  the  pollen- 
grains  of  Epilobium  (Fig.  278, 1 )  and  in  many  others.  In  the  passion- 
flower (2)  we  have  an  intermediate  condition :  in  which  we  can  still  see 
an  indication  of  the  facets  where  the  grains  abutted  on  one  another 
in  the  tetrad,  but  the  plane  faces  have  been  swollen  by  growth  into 
spheroidal  or  spherical  surfaces.  In  heaths  and  in  azaleas  the  four 
cells  of  the  tetrad  remain  attached  together,  and  form  a  compound 
tetrahedral  pollen-grain.  Six  furrows  correspond  to  the  six  edges 
of  the  tetrahedron,  and  each  is  continued  across  a  pair  of  cells;  they 
are  formed  (I  take  it)  along  lines  of  weakness  at  the  edges  of  the 
tetrahedron,  and  they  make  three  furrows  upon  each  one  of  the 
four  coherent  grains,  just  as  we  see  them  on  a  large  number  of 
ordinary  separate  and  non-coherent  pollen-grains.  On  the  other 
hand,  there  may  easily  be  cases  where  the  tetrads  of  daughter-cells 


VIII]  OF  SPORES  AND  POLLEN  631 

fail  to  assume,  even  temporarily,  the  tetrahedral  form:  cases,  in  a 
general  way,  where  the  four  cells  escape  from  the  confinement  of 
their  envelope,  and  fall  into  a  looser,  less  close-packed  arrangement*. 
The  figures  given  by  Goebel  of  the  development  of  the  pollen* of 
Neottia  (3,  a-e\  all  the  figures  referring  to  grains  taken  from  a 
single  anther)  illustrate  this  to  perfection,  and  it  will  be  seen  that, 


Fig.  278.  Various  pollen-grains  and  spores  (after  Berthold,  Campbell,  Goebe 
and  others).  (1)  Epilobium;  (2)  Passiflora;  (3)  Neottia;  (4)  Periploca 
graeca;  (5)  Apocynum;  (6)  Erica;  (7)  spore  of  Osmuyida;  (8)  tetraspore  of 
CnUithavinion. 


Fig.  271).     I'ollon  of  bulrush  (Tifpha).     After  Wodehouse. 

when  the  four  cells  lie  in  a  plane,  they  conform  exactly  to  our 
typical  diagram  of  the  first  four  .cells  in  a  segmenting  ovum; 
physically,  as  well  as  biologically,  the  tetrads  a-d  and  the  tetrad  e 
are  "allelomorphs"  of  one  another.     Again  in  the  bulrush  (Fig.  279), 

*  Cf.  C.  Nageli,  Znr  Entwickhingsgeschichte  des  Pollens  bei  den  Phanerogamen, 
36  pp.,  Zurich,  1842;  Hugo  Fischer,  Vergleichende  Morphologie  der  Pollenkorner, 
Berlin,  1890;  see  also,  for  many  and  varied  illustrations,  R.  P.  Wodehouse's 
beautiful  book  on  Pollen,  574  pp..  New  York,  193o,  and  earlier  papers. 


632  THE  FORMS  OF  TISSUES  [ch. 

the  four  cells  remain  attached  to  one  another,  and  lie  upon  a  level 
with  a  "  polar  furrow  "  well  displayed.  Occasionally,  though  the  four 
cells  lie  in  a  plane,  the  diagran^^seenis  to  fail  us,  for  the  cells  appear 
to  meet  in  a  simple  cross  (as  in  5);  but  here  we  soon  perceive  that 
the  cells  are  not  in  complete  interfacial  contact,  but  are  kept  apart 
by  a  httle  intervening  drop  of  fluid  or  bubble  of  air.  The  spores  of 
ferns  (7)  for  the  most  part  develop  in  much  the  same  way  as  pollen- 
grains;  they  also  very  often  retain  traces  of  the  shape  which  they 
assumed  as  members  of  a  tetrahedral  figure,  and  the  same  is  equally 
true  of  liverworts.  Among  the  "  tetraspores  "  (8)  of  the  Florideae, 
or  red  seaweeds,  we  have  a  condition  which  is  in  every  respect 
analogous.  The  same  thing  happens  in  certain  simple  algae  alHed  to 
Protococcus:  where  four  daughter-cells,  confined  within  a  mother- 
cell,  form  a  spherical  tetrahedron,  much  hke  a  spore  of  Osmunda  on 
a  smaller  scale*. 

Here  again  it  is  obvious  that,  apart  from  differences  in  actual 
magnitude,  and  apart  from  superficial  or  "accidental"  differences 
(referable  to  other  .physical  phenomena)  in  the  way  of  colour, 
texture  and  minute  sculpture  or  pattern,  a  very  small  number  of 
diagrammatic  figures  will  sufficiently  represent  the  outward  forms 
of  all  the  tetraspores,  four- celled  pollen-grains,  and  other  four- 
celled  aggregates  which  are  known  or  are  even  capable  of  existence. 
And  it  is  equally  obvious  that  the  resemblance  of  these  things,  to 
this  extent,  is  a  matter  of  physical  and  mathematical  symmetry,  and 
carries  no  proof  of  near  relationship  or  common  ancestry. 

We  have  been  dealing  hitherto  (save  for  some  slight  exceptions) 
with  the  partitioning  of  cells  on  the  assumption  that  the  system 
either  remains  unaltered  in  size  or  else  that  growth  has  proceeded 
uniformly  in  all  directions.  But  we  extend  the  scope  of  our  enquiry 
greatly  when  we  begin  to  deal  with  unequal  growth,  with  cells  so 
growing  and  dividing  as  to  produce  a  greater  extension  along 
some  one  axis  than  another.  And  here  we  come  close  in  touch 
with  that  great  and  still  (as  I  think)  insufficiently  appreciated 
generahsation  of  Sachs,  that  the  manner  in  which  the  cells  divide 
is  the  result,  and  not  the  cause,  of  the  form  of  the  dividing  structure : 
that  the  form  of  the  mass  is  caused  by  its  growth  as  a  whole,  and 

*  Cf.  A.  Pascher,  Arch.  f.  Protozoenk.  lxxvi,  p.  409;  lxxvii,  p.  195,  1932. 


VIII]  OF  THE  GROWING  POINT  633 

is  not  a  resultant  of  the  growth  of  the  cells  individually  considered*. 
Such  asymmetry  of  growth  may  be  easily  imagined,  and  may 
conceivably  arise  from  a  variety  of  causes.  In  any  individual  cell, 
for  instance,  it  may  arise  from  molecular  asymmetry  of  the  structure 
of  the  cell-wall,  giving  it  greater  rigidity  in  one  direction  than 
another,  while  all  the  while  the  hydrostatic  pressure  within  the 
cell  remains  constant  and  uniform.  In  an  aggregate  of  cells,  it 
may  very  well  arise  from  a  greater  chemical,  or  osmotic,  activity 
in  one  than  another,  leading  to  a  locahsed  increase  in  the  fluid 
pressure,  and  to  a  corresponding  bulge  over  a  certain  area  of  the 
external  surface.  It  might  conceivably  occur  as  a  direct  result  of 
preceding  cell-divisions,  when  these  are  such  as  to  produce  many 
peripheral  or  concentric  walls  in  one  part  and  few  or  none  in  another, 
with  the  obvious  result  of  strengthening  the  boundary  wall  here 
and  weakening  it  there ;  that  is  to  say,  in  our  dividing  quadrant, 
if  its  quadrangular  portion  subdivide  by  perichnes,  and  the 
triangular  portion  by  oblique  antichnes  (as  we  have  seen  to  be 
the  natural  tendency),  then  we  might  expect  that  external  growth 
would  be  more  manifest  over  the  latter  than  over  the  former  areas. 
As  a  direct  and  immediate  consequence  of  this  we  might  expect 
a  tendency  for  special  outgrowths,  or  "buds,"  to  arise  from  the 
triangular  rather  than  from  the  quadrangular  cells;  and  this  turns 
out  to  be  not  merely  a  tendency  towards  which  theoretical  con- 
siderations point,  but  a  widespread  and  important  factor  in  the 
morphology  of  the  cryptogams.  But  meanwhile,  without  enquiring 
further  into  this  complicated  question,  let  us  simply  take  it  that, 
if  we  start  from  such  a  simple  case  as  a  round  cell  which  has  divided 
into  two  halves  or  four  quarters  (as  the  case  may  be),  we  shall  at 
once  get  bilateral  symmetry  about  a  main  axis,  and  other  secondary 
results  arising  therefrom,  as  soon  as  one  of  the  halves,  or  one  of 
the  quarters,  begins  to  shew  a  rate  of  growth  in  advance  of  the 
others;  for  the  more  rapidly  growing  cell,  or  the  peripheral  wall 
common  to  two  or  more  such  rapidly  growing  cells,  will  bulge  out, 
and  may  finally  extend  into  a  cylinder  with  rounded  end.  This 
latter  very  simple  case   is   illustrated   in   the   development    of   a 

*  Sachs,  Pflanzenphysiologie  {Vorlesung  xxiv),  1882;  cf.  Rauber,  Neue  Grund- 
legungen  zur  Kenntniss  der  Zelle,  Morphol.  Jahrb.  vni,  p.  303  seq.,  1883; 
E.  B.  Wilson,  Cell-lineage  of  Nereis,  Journ.  Morph.  vi,  p.  448,  1892;   etc. 


634  THE  FORMS  OF  TISSUES  [ch. 

pollen-tube,  where  the  rapidly  growing  cell  develops  into  the 
elongated  cyUndrical  tube,  and  the  slow-growing  or  quiescent  part 
remains  behind  as  the  so-called  "vegetative"  cell  qf  cells. 

Just  as  we  have  found  it  easier  to  study  the  segmentation  of 
a  circular  disc  than  that  of  a  spherical  cell,  so  let  us  begin  in  the 
same  way,  by  enquiring  into  the  divisions  which  will  ensue  if  the 
disc  tend  to  grow,  or  elongate,  in  some  one  particular  direction 
instead  of  in  radial  symmetry.  The  figures  which  we  shall  then 
obtain  will  not  only  apply  to  the  disc,  but  will  also  represent,  in 
all  essential  features,  a  projection  or  longitudinal  section  of  a  solid 
body,  spherical  to  begin  with,  preserving  its  symmetry  as  a  solid 
of  revolution,  and  subject  to  the  same  general  laws  as  we  study 
in  the  disc*. 

(1)  Suppose,  in  the  first  place,  that  the  axis  of  growth  lies 
symmetrically  in  one  of  the  original  quadrantal  cells  of  a  segmenting 
disc;  and  let  this  growing  cell  elongate  with  comparative  rapidity 
before  it  subdivides.  When  it  does  divide,  it  will  necessarily  do 
so  by  a  transverse  partition,  concave  towards  the  apex  of  the  cell : 
and,  as  further  elongation  takes  place,  the  cylindrical  structure 
which  will  be  developed. thereby  will  tend  to  be  again  and  again 
subdivided  by  similar  transverse  partitions  (Fig.  280).  If  at  any 
time,  through  this  process  of  concurrent  elongation  and  subdivision, 
the  apical  cell  become  equivalent  to,  or  less  than,  a  hemisphere, 
it  will  next  divide  by  means  of  a  longitudinal,  or  vertical  partition; 
and  similar  longitudinal  partitions  will  arise  in  the  other  segments 
of  the  cylinder,  as  soon  as  it  comes  about  that  their  length  (in  the 
direction  of  the  axis)  is  less  than  their  breadth. 

But  when  we  think  of  this  structure  in  the  solid,  we  at  once 
perceive  that  each  of  these  flattened  segments,  into  which  our 
cyUnder  divided  to  begin  with,  is  equivalent  to  a  flattened  circular 
disc;    and  its  further  division  will  accordingly  tend  to  proceed  like 

*  In  the  following  account  I  follow  closely  on  the  lines  laid  down  by  Berthold; 
Protoplasmamechanik,  cap.^  vii.  Many  botanical  phenomena  identical  and  similar 
to  those  here  dealt  with  are  elaborately  discussed  by  Sachs  in  his  Physiology  of 
Plants  (chap,  xxvii,  pp.  431-459,  Oxford,  1887),  and  in  his  earlier  papers,  Ueber 
die  Anordnung  der  Zellen  in  jiingsten  Pflanzentheilen,  and  Ueber  Zellenanordnung 
und  Wachsthum  {Arh.  d.  botav.  Inst.  Wurzbnrg,  1877/78).  But  Sachs's  treatment 
differs  entirely  from  that  which  I  adopt  and  advocate  here:  his  explanations  being 
based  on  his  "law"  of  rectangular  succession,  and  involving  complicated  systems 
of  confocal  conies,  with  their  orthogonally  intersecting  ellipses  and  hyperbolas. 


VIIl] 


OF  THE  GROWING  POINT 


635 


any  other  flattened  disc,  namely  into  four  quadrants,  and  afterwards 
by  anticlines  and  periclines  in  the  usual  way.  A  section  across  the 
cyhnder,  then,  will  tend  to  shew  us  precisely  the  same  arrangements 
as  we  have  already  so  fully  studied  in  connection  with  the  typical 
division  of  a  circular  cell  into  quadrants,  and  of  these  quadrants 
into  triangular  and  quadrangular  portions,  and  so  on. 

But  there  are  other  possibilities  to  be  considered,  in  regard  to 
the  mode  of  division  of  the  elongating  quasi-cylindrical  portion,  as 
it  gradually  develops  out  of  the  growing  and  bulging  quadrantal 
cell;  for  the  manner  in  which  this  latter  cell  divides  will  simply 
depend  upon  the  form  it  has  assumed  before  each  successive  act 


Fig.  280.     Diagrammatic,  or  hypothetical,  result  of  asymmetrical  growth. 

of  division  takes  place,  that  is  to  say  upon  the  ratio  between  its 
rate  of  growth  and  the  frequency  of  its  successive  divisions.  For, 
as  we  have  already  seen,  if  the  growing  cell  attain  a  markedly 
oblong  or  cylindrical  form  before  division  ensues,  then  the  partition 
will  arise  transversely  to  the  long  axis;  if  it  be  but  a  httle  more 
than  a  hemisphere,  it  will  divide  by  an  oblique  partition;  and  if 
it  be  less  than  a  hemisphere  (as  it  may  come  to  be  after  successive 
transverse  divisions)  it  will  divide  by  a  vertical  partition,  that  is 
to  say  by  one  coinciding  with  its  axis  of  growth.  An  immense 
number  of  permutations  and  combinations  may  arise  in  this  way, 
and  we  must  confine  our  illustrations  to  a  small  number  of  cases. 
The  important  thing  is  not  so  much  to  trace  out  the  various 
conformations  which  may  arise,  but  to  grasp  the  fundamental 
principle:    which  is,  that  the  forces  which  dominate  the  form  of 


636  THE  FORMS  OF  TISSUES  [ch. 

each  cell  regulate  the  manner  of  its  subdivision,  that  is  to  say  the 
form  of  the  new  cells  into  which  it  subdivides;  or  in  other  words, 
the  form  of  the  growing  organism  regulates  the  form  and  number 
of  the  cells  which  eventually  constitute  it.  The  complex  cell- 
network  is  not  the  cause  but  the  result  of  the  general  configuration, 
which  latter  has  its  essential  cause  in  whatsoever  physical  and 
chemical  processes  have  led  to  a  varying  velocity  of  growth  in  one 
direction  as  compared  with  another. 

In  the  annexed  figure  of  an  embryo  of  Sphagnum  we  see  a  mode 
of  development  almost  precisely  corresponding  to  the  hypothetical 
case  which  we  have  just  de'scribed — the  case, 
that  is  to  say,  where  one  of  the  four  original 
quadrants  of  the  mother-cell  is  the  chief 
agent  in  future  growth  and  development. 
We  see  at  the  base  of  our  first  figure  (a), 
the  three  stationary,  or  undivided  quadrants, 
one  of  which  has  further  slowly  divided  in 
the  stage  b.  The  active  quadrant  has  grown 
quickly  into  a  cyHndrical  structure,  which 
inevitably  divides,  in  the  next  place,  into  a 
series  of  transverse  partitions;  and  accord- 
ingly, this  mode  of  development  carries  with 
it  the  presence  of  a  single  "  apical  cell,"  whose 
lower  wall  is  a  spherical  surface  with  its  p^g  ^gi.  Development  of 
convexity  downwards.  Each  cell  of  the  Sphagnum.  After  Camp- 
subdivided  cylinder  now  appears  as  a  more  ^^^• 
or  less  flattened  disc,  whose  mode  of  further  subdivision  we  may 
prognosticate  according  to  our  former  investigation,  to  which  subject 
we  shall  presently  return. 

(2)  In  the  next  place,  still  keeping  to  the  case  where  only  one 
of  the  original  quadrant-cells  continues  to  grow  and  develop,  let  us 
suppose  that  this  growing  cell  falls  to  be  divided  when  by  growth 
it  has  become  just  a  little  greater  than  a  hemisphere;  it  will  then 
divide,  as  in  Fig.  282,  2,  by  an  oblique  partition,  in  the  usual  way, 
whose  precise  position  and  inclination  to  the  base  will  depend 
entirely  on  the  configuration  of  the  cell  itself,  save  only,  of  course, 
that  we  may  have  also  to  take  into  account  the  possibility  of  the 
division  being  into  two  unequal  halves.     By  our  hypothesis,  the 


VIIl] 


OF  THE  GROWING  POINT 


637 


growth  of  the  whole  system  is  mainly  in  a  vertical  direction,  which 
is  as  much  as  to  say  that  the  more  actively  growing  pro'toplasm, 
or  at  least  the  strongest  osmotic  force,  will  be  found  near  the  apex; 
where  indeed  there  is  obviously  more  external  surface  for  osmotic 
action.  It  will  therefore  be  that  one  of  the  two  cells  which  contains, 
or  constitutes,  the  apex  which  will  grow  more  rapidly  than  the 
other,  and  which  therefore  will  be  the  first  to  divide;  and  indeed 
in  any  case,  it  will,  usually  be  this  one  of  the  two  which  will  tend 
to  divide  first,  inasmuch  as  the  triangular  and  not  the  quadrangular 
half  is  bound  to  constitute  the  apex*.     It  is  obvious  that  (unless 


4  5  6  7 

Fig.  282.     Development  of  antheridium  of  liverwort  (diagrammatic). 

the  act  of  division  be  so  long  postponed  that  the  cell  has  become 
quasi-cylindrical)  it  will  divide  by  another  obhque  partition,  starting 
from,  and  running  at  right  angles  to,  the  first.  And  so  division 
will  proceed  by  oblique  alternate  partitions,  each  one  tending  to 
be,  at  first,  perpendicular  to  that  on  which  it  is  based  and  also  to 
the  peripheral  wall;  but  all  these  points  of  contact  soon  tending, 
by  reason  of  the  equal  tensions  of  the  three  films  or  surfaces  which 
meet  there,  to  form  angles  of  120°.^  There  will  always  be  a  single 
apical  cell,  of  a  triangular  form.  The  developing  antheridium  of  a 
liverwort  (Riccia)  is  a  typical  example  of  such  a  case.  In  Fig.  283 
which  represents  a  "gemma"  of  a  moss,  we  see  just  the  same  thing; 
with  this  addition,  that  here  the  lower  of  the  two  original  cells  has 
grown  even  more  quickly  than  the  other,  constituting  a  long  cyhn- 

*  Cf.  p.  590. 


638 


THE  FORMS  OF  TISSUES 


[CH. 


drical  stalk,  and  dividing  in  accordance  with  its  shape  by  means  of 
transverse  septa.  In  all  such  cases  the  cells  may  continue  to  sub- 
divide, and  the  manner  in  which  they  do  so  must  depend  upon 
their  own  proportions;  and  in  all  cases  there  will  sooner  or  later  be 
a  tendency  to  the  formation  of  perichnal  walls,  cutting  off  an 
epidermal  layer  of  cells,  as  Fig.  284  illustrates  very  well. 

The  method  of  division  by  means  of  oblique  partitions  is  a 
common  one  in  the  case  of  "growing  points";  for  it  evidently 
includes  all  cases  in  which  the  act  of  cell-division  does  not  lag  far 
behind  that  elongation  which  is  determined  by  the  specific  rate  of 


Fig.  283.  Gemma 
of  moss.  After 
Campbell. 


Fig.  284.     Development  of  antheridium 
of  Riccia.     After  Campbell. 


growth.  And  it  is  also  obvious  that,  under  a  common  type,  there 
must  here  be  included  a  variety  of  cases  which  will,  at  first  sight, 
present  a  very  different  appearance  one  from  another.  For  instance, 
in  Fig.  285  which  represents  a  growing  shoot  of  Selaginella,  and 
somewhat  less  diagrammatically  in  the  young  embryo  of  Junger- 
mannia  (Fig.  286),  we  have  the  appearance  of  an  almost  straight 
vertical  partition  running  up  in  the  axis  of  the  system,  and  the 
primary  cell-walls  are  set  almost  at  right  angles  to  it — almost 
transversely,  that  is  to  say,  to  the  outer  walls  and  to  the  long  axis 
of  the  structure.  We  soon  recognise,  however,  that  the  difference 
is  merely  a  difference  of  degree.  The  more  remote  the  partitions 
are,  that  is  to  say  the  greater  the  velocity  of  growth  relatively  to 


VIIl] 


OF  THE  GROWING  POINT 


639 


that  of  division,  the  less  abrupt  will  be  the  alternate  kinks  or 
curvatures  of  the  portions  which  lie  along  the  axis,  and  the  more  will 
these  portions  appear  to  constitute  a  single  unbroken  wall. 

(3)  But  an  appearance  nearly,  if  not  quite,  indistinguishable 
from  this  may  be  got  in  another  way,  namely,  when  the  original 
growing  cell  is  so  nearly  hemispherical  that  it  is  actually  divided 


Fig.  285.     Section  of  growing  shoot 
of  Selaginella,  diagrammatic. 


Fig.  286.     Embryo  of  Jungermannia. 
After  Kienitz-Gerloif. 


by  a  vertical  partition  into  two  quadrants,  and  when  from  this  vertical 
partition,  as  it  elongates,  lateral  partition-walls  arise  on  either 
side.    Then,  by  the  tensions  exercised  by  these,  the  vertical  partition 


Fig.  287. 

will  be  bent  into  little  portions  set  at  120°  one  to  another,  and  the 
whole  w^ill  come  to  look  just  like  that  which,  in  the  former  case,  was 
made  up  of  parts  of  many  successive  obhque  partitions  (Fig.  287). 

Let  us  now,  in  one  or  two  cases,  follow  out  a  httle  further  the 
stages  of  cell-division  whose  beginnings  we  have  studied  in  the 
last  paragraphs.  In  the  antheridium  of  Riccia,  after  successive 
obhque  partitions  have  produced  the  longitudinal  series  of  cells 
shewn  in  Fig.  284,  4,  it  is  plain  that  the  next  partitions  will  arise 
periclinally,  that  is  to  say  parallel  to  the  outer  wall,  which  coin- 
cides with  the  short  axis  of  the  oblong  cells.     The  effect  is  to  produce 


640  THE  FORMS  OF  TISSUES  [ch. 

an  epidermal  layer,  whose  cells  subdivide  further  by  partitions 
perpendicular  to  the  surface,  that  is  to  say  crossing  the  flattened 
cells  by  their  shortest  diameter.  The  inner  mass  consists  of  cells 
which  are  still  more  or  less  oblong,  or  which  become  so  in  process  of 
growth;  and  these  again  divide,  parallel  to  their  short  axes,  into 
squarish  cells,  which  as  usual,  by  the  mutual  tension  of  their  walls, 
become  hexagonal  as  seen  in  a  plane  section.  There  is  a  clear  dis- 
tinction, then,  in  form  as  well  as  in  position,  between  the  outer 
covering-cells  and  those  which  he  within  this  envelope;  the  latter 
are  reduced  to  a  condition  which  fulfils  the  mechanical  function  of  a 
protective  coat,  while  the  former  undergo  less  modification,  and 
become  the  actively  Uving,  reproductive  elements. 


Fig.  288.     Development  of  sporangium  of  Osmuuda.     After  Bower. 

In  Fig.  288  is  shewn  the  development  of  the  sporangium  of  a 
fern  (Osmunda).  We  may  trace  here  the  common  phenomenon  of 
a  series  of  oblique  partitions,  built  alternately  on  one  another,  and 
cutting  off  a  conspicuous  triangular  apical  cell.  Over  the  whole 
system  an  epidermal  layer ,  is  formed,  in  the  manner  we  have 
described;  and  in  this  case  it  covers  the  apical  cell  also,  owing  to 
the  fact  that  it  was  of  such  dimensions  that,  at  one  stage  of  growth, 
a  pericKnal  partition  wall,  cutting  off  its  outer  end,  was  indicated 
as  of  less  area  than  an  anticlinal  one.  This  periclinal  wall  cuts 
down  the  apical  cell  to  the  proportions,  very  nearly,  of  an  equi- 
lateral triangle,  but  the  solid  form  of  the  cell  is  obviously  that  of 
a  tetrahedron  with  curved  faces ;  and  accordingly,  the  least  possible 
partitions  by  which  further  subdivision  can  be  effected  will  run 
successively  parallel  to  its  four  sides  (or  its  three  sides  when  we 


VIII]  OR  CELL-AGGJIEGATES  641 

confine  ourselves  to  the  appearances  as  seen  in  section).  The  effect 
,  is  to  cut  off  on  each  side  of  the  apical  cell  a  characteristically 
flattened  cell,  oblong  as  seen  in  section,  still  leaving  a  triangular 
(or  strictly  speaking,  a  tetrahedral)  one  in  the  centre.  The  oblong 
cells,  which  constitute  no  specific  structure  and  perform  no  specific 
physiological  function*,  but  which  merely  represent  certain  direc- 
tions in  space  towards  which  the  whole  system  of  partitioning  has 
gradually  led,  are  called  by  botanists  the  "tapetum."  The  active 
growing  tetrahedral  cell  which  lies  between  them,  and  from  which 
in  a  sense  every  other  cell  in  the  system  has  been  either  directly 
or  indirectly  segmented  off,  still  manifests  its  vigour  and  activity, 
and  becomes,  by  internal  subdivision,  the  mother-cell  of  the  spores. 

In  all  these  cases,  for  simplicity's  sake,  we  have  merely  con- 
sidered the  appearances  presented  in  a  single  longitudinal  plane 
of  optical  section.  But  it  is  not  difficult  to  interpret  from  these 
appearances  what  would  be  seen  in  another  plane,  for  instance  in 
a  transverse  section.  In  our  first  example,  for  instance,  that  oi 
the  developing  embryo  of  Sphagnum  (Fig.  281  c,  d),  we  see  that, 
at  appropriate  levels,  the  cells  of  the  original  cyhndrical  row  have 
divided  into  transverse  rows  of  four,  and  then  of  eight  cells.  We 
may  be  sure  that  the  four  cells  represent,  approximately,  quadrants 
of  a  cylindrical  disc,  the  four  cells,  as  usual,  not  meeting  in  a  point, 
but  intercepted  by  a  small  intermediate  partition.  Again,  where 
we  have  a  plate  of  eight  cells,  we  may  well  imagine  that  the  eight 
octants  are  arranged  in  what  we  have  found  to  be  the  way  naturally 
resulting  from  the  division  of  four  quadrants,  that  is  to  say  into 
alternately  triangular  and  quadrangular  portions ;  and  this  is  found 
by  means  of  sections  to  be  the  case.  The  figure  is  precisely  com- 
parable to  our  previous  diagrams  of  the  arrangement  of  eight  pells  m 
a  dividing  disc,  save  only  that,  in  two  cases,  the  cells  have  already 
undergone  a  further  subdivision. 

It  follows  that  we  are  apt  to  meet  with  this  characteristic  figure,  in 
one  or  other  of  its  possible  and  strictly  limited  variations,  in  the 
cross-sections  of  many  growing  structures,  just  as  we  have  already 

*  This  is  not  to  say  that  Nature  makes  no  use  of  the  tapetal  cells.  In  the  end 
they  break  down  and  contribute  to  the  growth  of  the  spore-mother-cell:  very 
much  as  the  "superfluous"  eggs  in  a  fly's  ovary  contribute  yolk-material  to  the 
developing  ovum. 


642  THE  FORMS  OF  TISSUES  [en. 

seen  it  appear  in  cases  where  the  entire  system  consists  of  eight 
cells  only.  For  example,  we  have  it  in  a  section  of  a  young  embryo 
of  a  moss  (Phascum),  and  again,  in  a  section  of  an  embryo  of  a 
fern  (Adiantum).  In  Fig,  290,  shewing  a  section  through  a  growing 
frond  of  a  sea-weed  (Girardia),  we  have  a  case  where  the  partitions 
forming  the  eight  octants  have  conformed  to  the  usual  type;  but 
instead  of  the  usual  division  by  periclines  of  the  four  quadrangular 
spaces,  these  latter  are  dividing  by  means  of  oblique  septa,  apparently 
owing  to  the  fact  that  the  cell  is  not  dividing  into  two  equal,  but 
into  two  unequal  portions.  In  this  last  figure  we  have  a  peculiar 
look  of  stiffness  or  formahty,  such  that  it  appears  at  first  to  bear 
little  resemblance  to  the  rest.  The  explanation  is  of  the  simplest. 
The  mode  of  partitioning  differs  little  (except  to  some  slight  extent 


Fig.  280.     (A,  B)  Sections  of  younger  and  older  embryos  oi  Phascum; 
(C)  do.  of  Adiantum.     After  Kienitz-Gerloff. 

in  the  way  already  mentioned)  from  the  normal  type;  but  in  this 
case  the  partition  walls  are  so  thick  and  become  so  soon  com- 
paratively solid  and  rigid,  that  the  secondary  curvatures  due  to  their 
successive  mutual  tractions  are  here  imperceptible. 

A  curious  „and  beautiful  case,  apparently  aberrant  but  which 
would  doubtless  be  found  conforming  strictly  to  physical  laws  if 
only  we  'clearly  understood  the  actual  conditions,  is  indicated  in 
the  development  of  the  antheridium  of  a  fern,  as  described  by 
Strasblirger.  Here  the  antheridium  develops  from  a  single  cell, 
which  (Fig.  291)  has  grown  to  something  more  than  a  hemisphere; 
and  the  first  partition,  instead  of  stretching  transversely  across  the 
cell,  as  we  should  expect  it  to  do  if  the  cell  were  actually  spherical, 
has  as  it  were  sagged  down  to  come  in  contact  with  the  base,  and 
so  to  develop  into  an  annular  partition,  running  round  the  lower 


vm]«  OR  CELL-AGGREGATES  643 

margin  of  the  cell.  The  phenomenon  is  precisely  identical  to  that 
bisection  of  a  quadrant  by  means  of  a  circular  arc,  of  which  we 
spoke  on  p.  581,  and  the  annular  film  is  very  easy  to  reproduce 
by  means  of  a  soap-bubble  in  the  bottom  of  a  cyUndrical  dish  or 
beaker.  The  next  partition  is  a  perichnal  one,  concentric  with  the 
outer  surface  of  the  young  antheridium ;  and  this  in  turn  is  followed 
by  a  concave  partition  which  cuts  off  the  apex  of  the  original  cell : 
but  which  becomes  connected  with  the  second,  or  perichnal  partition 
in  precisely  the  same  annular  fashion  as  the  first  partition  did  with 
the  base  of  the  httle  antheridium.  The  result  is  that,  at  this  stage, 
we  have  four  cell-cavities  in  the  little  antheridium-    (1)  a  central 


Fig.  291.     Development  of  anthe- 

11-     c,n,.      o    ^-       ^L         L  r      J     r  ridium  of  Pteris.     After  Stras- 

J^ig.  290.     Section  through  frond  of 

Girardia  sphacelaria.    After  Goebel.  ^ 

cavity;  (2)  an  annular  space  around  the  lower  margin;  (3)  a  narrow 
annular  or  cylindrical  space  around  the  sides  of  the  antheridium; 
and  (4)  a  small  terminal  or  apical  cell.  It  is  evident  that  the 
tendency,  in  the  next  place,  will  be  to  subdivide  the  flattened 
external  cells  by  means  of  anticlinal  partitions,  and  so  to  convert 
the  whole  structure  into  a  single  layer  of  epidermal  cells,  surrounding 
a  central  cell  within  which,  in  course  of  time,  the  antherozoids  are 
developed. 

The  foregoing  account  deals  only  with  a  few  elementary  pheno- 
mena, and  may  seem  to  fall  far  short  of  an  attempt  to  deal  in  general 
with  "the  forms  of  tissues."  But  it  is  the  principle  involved,  and 
not  its  ultimate  and  very  complex  results,  that  we  can  alone  attempt 
to  grapple  with.  The  stock-in-trade  of  mathematical  physics,  in 
all  the  subjects  with  which  that  science  "deals,  is  for  the  most  part 


644  THE  FORMS  OF  TISSUES  [ch.  viii 

made  up  of  simple,  or  simpKfied,  cases  of  phenomena  which  in  their 
actual  and  concrete  manifestations  are  usually  too  complex  for 
matheniatical  analysis;  hence,  even  in  physics,  the  full  mechanical 
explanation  of  a  phenomenon  is  seldom  if  ever  more  than  the 
''cadre  ideal"  towards  which  our  never-finished  picture  extends. 
When  we  attempt  to  apply  the  same  methods  of  mathematical 
physics  to  our  biological  and  histological  phenomena,  we  need 
not  wonder  if  we  be  Hmited  to  illustrations  of  a  simple  kind, 
which  cover  but  a  small  part  of  the  phenomena  with  which 
histology  has  to  do.  But  yet  it  is  only  relatively  that  these  pheno- 
mena to  which  we  have  found  the  method  appUcable  are  to  be 
deemed  simple  and  few..  They  go  already  far  beyond  the  simplest 
phenomena  of  all,  such  as  we  see  in  the  dividing  Protococcus,  and 
in  the  first  stages,  two-celled  or  four-celled,  of  the  segmenting  egg., 
They  carry  us  into  stages  where  the  cells  are  already  numerous, 
and  where  the  whole  conformation  has  become  by  no  means  easy 
to  depict  or  visuahse,  without  the  help  and  guidance  which  the 
phenomena  of  surface-tension,  the  laws  of  equihbrium  and  the 
principle  of  minimal  areas  are  at  hand  to  supply.  And  so  far  as 
we  have  gone,  and  so  far  as  we  can  discern,  we  see  no  sign  of  the 
guiding  principles  faihng  us,  or  of  the  simple  laws  ceasing  to  hold 
good. 


CHAPTER  IX 

ON  CONCRETIONS,  SPICULES,  AND 
SPICULAR  SKELETONS 

The  deposition  of  inorganic  material  in  the  living  body,  usually 
in  the  form  of  calcium  salts  or  of  silica,  is  a  common  phenomenon. 
It  begins  by  the  appearance  of  small  isolated  particles,  crystalline 
or  non-crystalline,  whose  form  has  little  relation  or  none  to  the 
structure  of  the  organism;  it  culminates  in  the  complex  skeletons 
of  the  vertebrate  animals,  in  the  massive  skeletons  of  the  corals,  or 
in  the  polished,  sculptured  and  mathematically  regular  molluscan 
shells.  Even  among  very  simple  organisms,  such  as  diatoms, 
radiolarians,  foraminifera  or  sponges,  the  skeleton  displays  extra- 
ordinary variety  and  beauty,  whether  by  reason  of  the  intrinsic 
form  of  its  elementary  constituents  or  the  geometric  symmetry  with 
which  these  are  interconnected  and  arranged. 

With  regard  to  the  form  of  these  various  structures  (and  this  is 
all  that  immediately  concerns  us  here),  we  have  to  do  with  two 
distinct  problems,  which  merge  with  one  another  though  they  are 
theoretically  distinct.  For  the  form  of  the  spicule  or  other  skeletal 
element  may  depend  solely  on  its  chemical  nature,  as  for  instance, 
to  take  a  simple  but  not  the  only  case,  when  it  is  purely  crystalline ; 
or  the  inorganic  material  may  be  laid  down  in  conformity  with  the 
shapes  assumed  by  cells,  tissues  or  organs,  and  so  be,  as  it  were, 
moulded  to  the  living  organism ;  and  there  may  well  be  intermediate 
stages  in  which  both  phenomena  are  simultaneously  at  work,  the 
molecular  forces  playing  their  part  in  conjunction  with  the  other 
forces  inherent  in  the  system. 

So  far  as  the  problem  is  a  purely  chemical  one  we  must  deal 
with  it  very  briefly  indeed:  all  the  more  because  special  investiga- 
tions regarding  it  have  as  yet  been  few,  and  even  the  main  facts 
of  the  case  are  very  imperfectly  known.  This  at  least  is  clear,  that 
the  phenomena  with  which  we  are  about  to  deal  go  deep  into  the 
subject  of  colloid  chemistry,  and  especially  that  part  of  the  science 


646  ON  CONCRETIONS,  SPICULES  [ch. 

which  deals  with  colloids  in  connection  with  surface  phenomena. 
It  is  to  the  special  student  of  the  chemistry  and  physics  of  the 
colloids  that  we  must  look  for  the  elucidation  of  our  problem*. 

In  the  first  and  simplest  part  of  our  subject,  the  essential  problem 
is  the  problem  of  crystallisation  in  presence  of  colloids.  In  the  cells 
of  plants  true  crystals  are  found  in  comparative  abundance,  and 
consist,  in  the  majority  of  cases,  of  calcium  oxalate.  In  the  stem 
and  root  of  the  rhubarb  for  instance;  in  the'  leaf-stalk  of  Begonia 
and  in  countless  other  cases,  sometimes  within  the  cell,  sometimes 
in  the  substance  of  the  cell-wall,  we  find  large  and  well-formed 
crystals  of  this  salt;  their  varieties  of  form,  which  are  extremely 
numerous,  are  simply  the  crystalline  forms  proper  to  the  salt  itself, 
and  belong  to  the  two  systems,  cubic  and  monocHnic,  in  one  or  other 
of  which,  according  to  the  amount  of  water  of  crystallisation,  this 
salt  is  known  to  crystalhse.  When  calcium  oxalate  crystalhses 
according  to  the  latter  system  (as  it  does  when  its  molecule  is  com- 
bined with  two  molecules  of  water),  the  microscopic  crystals  have 
the  form  of  fine  needles,  or  "raphides";  these  are  very  common  in 
plants,  and  may  be  artificially  produced  when  the  salt  is  crystallised 
out  in  presence  of  glucose  or  of  dextrinf . 

Calcium  carbonate,  on  the  other  hand,  when  it  occurs  in  plant- 
cells,  as  it  does  abundantly  (for  instance  in  the  "cystoliths"  of  the 
Urticaceae  and  Acanthaceae,  and  in  great  quantities  in  Melobesia 
and  the  other  calcareous  or  "stony"  algae),  appears  in  the  form 
of  fine  rounded  granules,  whose  inherent  crystalline  structure  is 
only  revealed  (like  that  of  a  molluscan  shell)  under  polarised  light. 
Among  animals,  a  skeleton  of  carbonate  of  lime  occurs  under  a 
multitude  of  forms,  of  which  we  need  only  mention  a  few  of  the 
most  conspicuous.  The  spicules  of  the  calcareous  sponges  are 
triradiate,  occasionally  quadriradiate,  bodies,  with  pointed  rays,  not 
crystalline  in  outward  form  but  with  a  definitely  crystalline  internal 

*  There  is  much  information  regarding  the  chemical  composition  and  minera- 
logical  structure  of  shells  and  other  organic  products  in  H.  C.  .Sorby's  Presidential 
Address  to  the  Geological  Society  {Proc.  G'eol.  Soc.  1879,  pp.  56-93);  but  Sorby 
failed  to  recognise  that  association  with  "organic"  matter,  or  with  colloid  matter 
whether  living  or  dead,  introduced  a  new  series  of  purely  physical  phenomena. 

f  Jidien  \'esque,  Sur  la  production  artificielle  de  cristaux  d"oxalate  de  chaux 
semblables  a  ceux  qui  se  forment  dans  les  plantcs.  Ann.  Sc.  Xat.  (Bot.)  (5)  xix. 
pp.  300-313.  1874. 


IX 


AND  SPICULAR  SKELETONS 


647 


structure;  we  shall  return  again  to  these,  and  find  for  them  what 
would  seem  to  be  a  satisfactory  explanation  of  their  form.  Among 
the  Alcyonarian  zoophytes  we  have  a  great  variety  of  spicules*, 
which  are  sometimes  straight  and  slender  rods,  sometimes  flattened 
and  more  or  less  striated  plates,  and  still  more  often  disorderly 
aggregations  of  micro-crystals,  in  the  form  of  rounded  or  branched 
concretions  with  rough  or  knobby  surfaces t  (Figs.  292,  298).  A 
third  type,  presented  by  several  very  different  things,  such  as  a 


Fio;.  292.    Alcyonarian  sijicules:   Siphonogorgia  Sind  AntJioyorgia.    After  Studer. 

})earl  or  the  ear-bone  of  a  bony  fish,  consists  of  a  more  or  less  rounded 
body,  sometimes  spherical,  sometimes  flattened,  in  which  the  cal- 
careous matter  is  laid  down  in  concentric  zones,  denser  and  clearer 
layers  alternating  with  one  another.  In  the  development  of  the 
molluscan  shell  and  in  the  calcification  of  a  bird's  egg  or  a  crab's 
shell,  small  spheroidal  bodies  with  similar  concentric  stria tion  make 
their  appearance;    but  instead  of  remaining  separate  they  become 

*  Cf.  Kolliker,  Icones  Histologicae,  1804,  p.  119,  etc. 

t  In  rare  cases,  these  shew  a  single  optic  axis  and  behave  as  individual  crystals: 
W.- J.  Schmidt,  Arch.f.  Knhr.  Mech.  lt,  pp.  .")00-.V)  1 .  1922. 


648  ON  CONCRETIONS,  SPICULES  [ch. 

crowded  together,  and  in  doing  so  are  apt  to  form  a  pattern  of 
hexagons.  In  some  cases  the  carbonate  of  lime,  on  being  dissolved 
away  by  acid,  leaves  behind  it  a  certain  small  amount  of  organic 
residue;  in  many  cases  other  salts,  such  as  phosphates  of  lime, 
ammonia  or  magnesia,  are  present  in  small  quantities;  and  in  most 
cases,  if  not  all,  the  developing  spicule  or  concretion  is  somehow  so 
associated  with  Hving  cells  that  we  are  apt  to  take  it  for  granted 
that  it'  owes  its  form  to  the  constructive  or  plastic  agency  of  these. 

The  appearance  of  direct  association  with  hving  cells,  however, 
is  apt  to  be  fallacious;  for  the  actual  precipitation  takes  place,  as 
a  rule,  not  in  actively  hving,  but  in  dead  or  at  least  inactive  tissue  * ; 
that  is  to  say  in  the  "formed  material"  or  matrix  which  accumulates 
round  the  living  cells,  or  in  the  interspaces  between  these  latter, 
or,  as  often  happens,  in  the  cell-wall  or  cell-membrane  rather  than 
within  the  substance  of  the  protoplasm  itself.  We  need  not  go  the 
length  of  asserting  that  this  is  a  rule  without  exception;  but,  so 
far  as  it  goes,  it  is  of  great  importance  and  to  its  consideration  we 
shall  presently  return  f. 

Cognate  with  this  is  the  fact  that,  at  least  in  some  cases,  the 
organism  can  go  on,  in  apparently  unimpaired  health,  when  stinted 
or  even  wholly  deprived  of  the  material  of  which  it  is  wont  to  make 
its  spicules  or  its  shell.  Thus  the  eggs  of  sea-urchins  reared  in  lime- 
free  water  develop,  in  apparent  health  and  comfort,  into  larvae 
which  lack  the  usual  skeleton  of  calcareous  rods:  and  in  which, 
accordingly,  the  long  arms  of  the  Pluteus  larva,  which  the  rods 
should  support  and  extend,  are  entirely  absent  {.  Again,  when 
foraminifera  are  kept  for  generations  in  water  from  which  they 
gradually  exhaust  the  lime,  their  shells  grow  hyaline  and  trans- 
parent, and  dwindle  to  a  mere  chitinous  peUicle;  on  the  other  hand, 

♦  In  an  interesting  paper  by  Robert  Irvine  and  Sims  Woodhead  on  the  Secretion  of 
carbonate  of  lime  by  animals  (Proc.  R.S.E.  xv,  pp.  308-316;  xvi,  pp.  324-351, 
1889-90)  it  is  asserted  (p.  351)  that  "Hme  salts,  of  whatever  form,  are  deposited 
only  in  vitally  inactive  tissue." 

t  The  tube  of  Teredo  shews  no  trace  of  organic  matter,  but  consists  of  irregular 
prismatic  crystals:  the  whole  structure  "being  identical  with  that  of  small  veins 
of  calcite,  such  as  are  seen  in  thin  sections  of  rocks"  (Sorby,  Proc.  Geol.  Soc.  1879, 
p.  58).  This,  then,  would  seem  to  be  a  somewhat  exceptional  case  of  a  shell  laid 
down  completely  outside  of  the  animal's  external  layer  of  organic  substance. 

t  Cf.  Pouchet  and  Chabry,  C.R.  Soc.  Biol.  Paris  (9),  i,  pp.  17-20,  1889;  C.R. 
Acad.  Sci.  cvm,  pp.  196-198,  1889. 


IX]  AND  SPICULAR  SKELETONS  649 

in  the  presence  of  excess  of  lime  their  shells  become  much  altered, 
are  strengthened  with  various  ridges  or  "ornaments,"  and  come  to 
resemble  other  varieties  and  even  "species*." 

The  crucial  experiment,  then,  is  to  attempt  the  formation  of 
similar  spicules  or  concretions  apart  from  the  living  organism.  But 
however  feasible  the  attempt  may  be  in  theory,  we  must  be  prepared 
to  encounter  many  difficulties;  and  to  realise  that,  though  the 
reactions  involved  may  be  well  within  the  range  of  physical  chemistry, 
yet  the  actual  conditions  of  the  case  may  be  so  complex,  subtle  and 
dehcate  that  only  now  and  then,  and  only  in  the  simplest  of  cases, 
has  it  been  found  possible  to  imitate  the  natural  objects  successfully. 
Such  an  attempt  is  part  of  that  wide  field  of  enquiry  through  which 
Stephane  Leduc  and  other  workers  have  sought  to  produce,  by 
synthetic  means,  forms  similar  to  those  of  living  things;  but  it  is 
a  circumscribed  and  well-defined  part  of  that  wider  investigationf. 

When  we  find  ourselves  investigating  the  forms  assumed  by 
chemical  compounds  under  the  pecuhar  circumstances  of  association 
with  a  Hving  body,  and  when  we  find  these  forms  to  be  characteristic 
or  recognisable,  and  somehow  different  from  those  which  the  same 
substance  is  wont  to  assume  under  other  circumstances,  an  analogy, 
captivating  though  perhaps  remote,  presents  itself  to  our  minds 
between  this  subject  of  ours  and  certain  synthetic  problems  of  the 
organic  chemist.  There  is  doubtless  an  essential  difference,  as  well 
as  a  difference  of  scale,  between  the  visible  form  of  a  spicule  or  con- 

*  Cf.  Heron-Allen,  Phil.  Trans.  (B),  ccvi,  p.  262,  1915. 

f  Leduc's  artificial  growths  were  mostly  obtained  by  introducing  salts  of  the 
heavy  metals  or  alkaline  earths  into  solutions  which  form  with  them  a  "precipitation- 
membrane" — as  when  we  introduce  copper  sulphate  into  a  ferrocyanide  solution. 
See  his  Mechanism  of  Life,  1911,  ch.  x,  for  copious  references  to  other  works  on 
the  "artificial  production  of  organic  forms."  Closely  related  to  Leduc's  experi- 
ments are  those  of  Denis  Monnier  and  Carl  Vogt,  Sur  la  fabrication  artificielle 
des  formes  des  elements  organiques,  Journ.  de  VAnaf.  xviii,  pp.  117-123,  1882; 
cf.  Moritz  Traube,  Zur  Geschichte  der  mechanischen  Theorie  des  Wachstums  der 
organischen  Zelle,  Botan.  Ztg.  xxxvi,  1878.  Cf.  also  A.  L.  Herrera,  Sur  les 
phenomenes  de  vie  apparente  observes  dans  les  emulsions  de  carbonate  de  chaux 
dans  la  silice  gelatineuse,  Mem.  Soc.  Alzaje,  Mexico,  xxvi,  1908;  Los  Protobios, 
Boll,  de  la  Dir.  de  Estud.  Biolog.,  Mexico,  i,  pp.  607-631,  and  other  papers.  Also 
{int.  al.)  R.  S.  Lillie  and  E.  N.  Johnston,  Precipitation-structures  simulating  organic 
growth,  Biol.  Bull,  xxxiii,  p.  135,  1917;  xxxvi,  pp.  225-272,  1919;  Scientific 
Monthly,  Feb.  1922,  p.  125;  H.  W.  Morse,  C.  H.  Warren  and  J.  D.  H.  Donnay, 
Artificial  spherulites,  etc.,  Ainer.  .11.  of  Sci.  (5)  xxiii,  pp.  421-439,  1932. 


650  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

cretion  and  the  hypothetical  form  of  an  individual  molecule.  But 
molecular  form  is  a  very  important  concept;  and  the  chemist  has 
not  only  succeeded,  since  the  days  of  Wohler,  in  synthetising  many 
substances  which  are  characteristically  associated  with  living  matter, 
but  his  task  has  included  the  attempt  to  account  for  the  molecular 
forms  of  certain  "asymmetric"  substances — glucose,  malic  acid  and 
many  more — as  they  occur  in  Nature.  These  are  bodies  which,  when 
artificially  synthetised,  have  no  optical  activity,  but  which,  as  we 
actually  find  them  in  organisms,  turn  (when  in  solution)  the  plane 
of  polarised  hght  in  one  direction  rather  than  the  other;  thus 
dextroglucose  and  laevomahc  acid  are  common  products  of  plant 
metabohsm,  but  dextromalic  acid  and  laevoglucose  do  not  occur  in 
Nature  at  all.  The  optical  activity  of  these  bodies  depends,  as 
Pasteur  shewed  eighty  years  ago*,  upon  the  form,  right-handed  or 
left-handed,  of  their  molecules,  which  molecular  asymmetry  further 
gives  rise  to  a  corresponding  right-  or  left-handedness  (or  enantio- 
morphism)  in  the  crystalline  aggregates.  It  is  a  distinct  problem 
in  organic  or  physiological  chemistry,  and  by  no  means  without  its 
interest  for  the  norpholonrist,  to  discover  how  it  is  that  Nature,  for 
each  particular  substance,  habitually  builds  up,  or  at  least  selects, 
its  molecules  in  a  one-sided  fashion,  right-handed  or  left-handed  as 
the  case  may  be.  It  will  serve  us  no  better  to  assert  that  this  pheno- 
menon has  its  origin  in  "fortuity"  than  to  repeat  the  Abbe  Galiani's 
saying,  ''les  des  de  la  nature  sont  pipes.'' 

The  problem  is  not  so  closely  related  to  our  immediate  subject 
that  we  need  discuss  it  at  length;  but  it  has  its  relation,  such  as  it 
is,  to  the  general  question  oi  form  in  relation  to  vital  phenomena, 
and  it  has  its  historic  interest  as  a  theme  of  long-continued  discussion. 
According  to  Pasteur,  there  lay  in  the  molecular  asymmetry  of 
the  natural  bodies  and  their  symmetry  when  artificially  produced, 
one  of  the  most  deep-seated  differences  between  vital  and  non- vital 
phenomena:  he  went  further,  and  declared  that  "this  was  perhaps 
the  only  well-marked  line  of  demarcation  th'at  can  at  present  [1860] 
be  drawn  between  the  chemistry  of  dead  and  of  living  matter." 
Nearly  forty  years  afterwards  the  same  theme  was  pursued  and 

*  Lectures  on  the  molecular  asymmetry  of  natural  organic  compounds,  Chemical 
Soc.  of  Paris,  1860;  also  in  Ostwald's  Klassiker  d.  exact.  Wiss.  No.  28,  and  in 
Alembic  Club  Reprints,  No.  14,  Edinburgh,  1897;  cf.  G.  M.  Richardson,  Foundations 
of  Stereochemistry,  New  York,  1901. 


IX]  OF  MOLECULAR  ASYMMETRY  651 

elaborated  by  Japp  in  a  celebrated  lecture*,  and  the  distinction 
still  Has  its  weight,  I  believe,  in  the  minds  of  many  chemists. 
"We  arrive  at  the  conclusion,"  said  Professor  Japp,  "that  the 
production  of  single  asymmetric  compounds,  or  their  isolation  from 
the  mixture  of  their  enantiomorphs,  is,  as  Pasteur  firmly  held,  the 
prerogative  of  life.  Only  the  living  organism,  or  the  Hving  intelH- 
gence  with  its  conception  of  asymmetry,  can  produce  this  result. 
Only  asymmetry  can  beget  asymmetry."  In  these  last  words 
(which,  so  far  as  the  chemist  and  the  biologist  are  concerned,  we 
may  acknowledge  to  be  truef)  lies  the  crux  of  the  difficulty. 

Observe  that  it  is  only  the  first  beginnings  of  chemical  asymmetry 
that  we  need  discover;  for  when  asymmetry  is  once  manifested, 
it  is  not  disputed  that  it  will  continue  "to  beget  asymmetry." 
A  plausible  suggestion  is  at  hand,  which  if  it  were  confirmed  and 
extended  would  supply  or  at  least  sufficiently  illustrate  the  kind  of 
explanation  that  is  required.  We  know  that  when  ordinary  non- 
polarised light  acts  upon  a  chemical  substance,  the  amount  of 
chemical  action  is  proportionate  to  the  amount  of  light  absorbed. 
We  know  in  the  second  place  J  that  light  circularly  polarised  is 
absorbed  in  certain  cases  in  different  amounts  by  the  right-handed 
or  left-handed  varieties  of  an  asymmetric  substance.  And  thirdly, 
we  know  that  a  portion  of  the  light  which  comes  to  us  from  the  sun 
is  already  plane-polarised  hght,  which  becomes  in  part  circularly 
polarised,  by  reflection  (according  to  Jamin)  at  the  surface  of  the 
sea,  and  then  rotated*  in  a  particular  direction  under  the  influence 
of  terrestrial  magnetism.  We  only  require  to  be  assured  that  the 
relation  between  absorption  of  light  and  chemical  activity,  will 
continue  to  hold  good  in  the  case  of  circularly  polarised  light; 
that  is  to  say  that  the  formation  of  some  new  substance  or  other, 
under  the  influence  of  light  so  polarised,  will  proceed  asymmetrically 
in  consonance  with  the  asymmetry  of  the  light  itself;   or  conversely, 

*  F.  R.  Japp,  Stereochemistry  and  vitalism,  Brit.  Ass.  Rep.  (Bristol),  1898,  p.  813; 
cf.  also  a  voluminous  discussion  in  Nature,  1^98-91). 

.  t  They  represent  the  general  theorem  of  which  particular  cases  are  found,  for 
instance,  in  the  asymmetry  of  the  ferments  (or  enzymes)  which  act  upon 
asymmetrical  bodies,  the  one  fitting  the  other,  according  to  Emil  Fischer's  well- 
known  phrase,  as  lock  and  key.  Cf.  his  Bedeutung  der  Stereochemie  fiir  die 
Physiologie,  Z.  f.  physiol.  Chemie,  v,  p.  60,  1899,  and  various  papers  in  the  Ber. 
d.  d.  cheni.  (Jes.  from  1894. 

+  Cf.  Cotton.  Ann.  de  Chim.  el  de  Phys.  (7),  viii,  pp.  347-432  (cf.  p.  373),  1896. 


652  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

that  the  asymmetrically  polarised  light  will  tend  to  more  rapid 
decomposition  of  those^  molecules  by  which  it  is  chiefly  absorbed. 
This  latter  proof  is  said  to  be  furnished  by  Byk*.  who  asserts 
that  certain  tartrates  become  unsymmetrical  under  the  continued 
influence  of  the  asymmetric  rays.  Here  then  we  seem  to  have 
an  example,  of  a  particular  kind  and  in  a  particular  instance,  an 
example  hmited  but  yet  crucial  if  confirmed,  of  an  asymmetric 
force,  non-vital  in  its  origin,  which  might  conceivably  be  the 
starting-point  of  that  asymmetry  which  is  characteristic  of  so  many 
organic  products. 

The  mysteries  of  organic  chemistry  are  great,  and  the  differences 
between  its  processes  or  reactions  as  they  are  carried  out  in  the 
organism  and  in  the  laboratory  are  manyt;  the  actions,  catalytic 
and  other,  which  go  on  in  the  living  cell  are  of  extraordinary 
complexity.  But  the  contention  that  they  are  different  in  kind 
from  ordinary  chemical  operations,  or  that  in  the  production  of 
single  asymmetric  compounds  there  is  actually,  as  Pasteur  main- 
tained, a  "prerogative  of  life,"  would  seem  to  be  no  longer  tenable. 
Oiir  historic  interest  in  the  whole  question  is  increased  by  the 
fact,  or  the  great  probability,  that  "the  tenacity  with  which  Pasteur 
fought  against  the  doctrine  of  spontaneous  generation  was  not 
unconnected  with  his  belief  that  chemical  compounds  of  x)ne-sided 
symmetry  could  not  arise  save  under  the  influence  of  hfej."  But 
the  question  whether  spontaneous  generation  be  a  fact  or  not  does 
not  depend  upon  theoretical  considerations;  our  negative  response 
is  based,  and  is  soundly  based,  on  repeated  failures  to  demonstrate 
its  occurrence.  Many  a  great  law  of  physical  science,  not  excepting 
gravitation  itself,  has  no  higher  claim  on  our  acceptance. 

Let  us  return  from  this  digression  to  the  general  subject  of  the 
forms  assumed  by  certain  chemical  bodies  when  deposited  or 
precipitated  within  the  organism,  and  to  the  question  of  how  far 
these  forms  may  be  artificially  imitated  or  theoretically  explained. 

*  A,  Byk,  Zur  Frage  der  8paltbarkeit  von  Racemverbindungen  durch  zirkular- 
polarisiertes  Licht,  ein  Beitrag  zur  primaren  Entstehung  optisch-activer  Substanzen, 
Zeitsch.  f.  physikal.  Chetnie,  xlix,  pp.  641-687,  1904.  It  must  be  admitted  that 
positive  evidence  on  these  lines  is  still  awanting. 

f  Cf,  (int.  al.)  Emil  Fischer,  Untersuchungen  iiber  Aminosduren,  Proteine,  etc. 
Berlin,  1906. 

J  Japp,  loc.  cit.  p.  828. 


IX]  HARTING'S  MORPHOLOGIE  SYNTHETIQUE  653 

Mr  George  Rainey,  of  St  Thomas's  Hospital  (of  whom  we  have 
spoken  before),  and  Professor  P.  Harting,  of  Utrecht,  were  the 
first  to  deal  with  this  specific  problem.  Rainey  published,  between 
1857  and  1861,  a  series  of  valuable  and  thoughtful  papers  to  shew 
that  shell  and  bone  and  certain  other  organic  structures  were  formed 
"by  a  process  of  molecular  coalescence,  demonstrable  in  certain 
artificially  formed  products*."  Harting,  after  thirty  years  of 
experimental  work,  published  in  1872  a  paper,  which  has  become 
classical,  entitled  Recherches  de  morphologie  synthetique,  sur  la  pro- 
duction artificielle  de  quelques  fonnations  calcaires  organiques  f ;  his 
aim  was  to  pave  the  way  for  a  "morphologie  synthetique,"  as 
Wohler  had  laid  the  foundations  of  a  "chimie  synthetique"  by  his 
classical  discovery  forty  years  before. 

Rainey  and  Harting  used  similar  methods — and  these  were  such 
as  other  workers  have  continued  to  employ — partly  with  the  direct 
object  of  explaining  the  genesis  of  organic  forms  and  partly  as  an 
integral  part  of  what  is  now  known  as  Colloid  Chemistry.  The  gist 
of  the  method  was  to  bring  some  soluble  salt  of  hme,  such  as  the 
chloride  or  nitrate,  into  solution  within  a  colloid  medium,  such  as 
gum,  gelatine  or  albumin;  and  then  to  precipitate  it  out  in  the 
form  of  some  insc/luble  compound,  such  as  the  carbonate  or  oxalate. 
Harting  found  that,  when  he  added  a  httle  sodium  or  potassium 
carbonate  to  a  concentrated  solution  of  calcium  chloride  in  albumin, 
he  got  at  first  a  gelatinous  mass,  or  "colloid  precipitate":  which 
slowly  transformed  by  the  appearance  of  tiny  microscopic  particles, 

*  George  Rainey,  On  the-  elementary  formation  of  the  skeletons  of  animals,  and 
other  hard  structures  formed  in  connection  with  living  tissue,  Brit,  and  For.  Med. 
Ch.  Rev.  XX,  pp.  451-476,  1857;  published  separately  with  additions,  8vo,  London, 
1858.  For  other  papers  by  Rainey  on  kindred  subjects  see  Q.J. M.S.  vi  (Tr. 
Microsc.  Soc),  pp.  41-50,  1858;  vii,  pp.  212-225,  1859;  vm,  pp.  1-10,  1860; 
I  (n.s.),  pp.  23-32,  1861.  Cf.  also  W.  Miller  Ord,  On  the  influence  exercised  by  colloids 
upon  crystalline  form,  pp.  x,  179,  1874;  cf.  also  Q.J.M.S.  xii,  pp.  219-239,  1872; 
also  the  early  but  still  interesting  observations  of  Mr  Charles  Hatchett,  Chemical 
experiments  on  zoophytes;  with  some  observations  on  the  component  parts  of 
membrane,  Phil.  Trans.  1800,  pp.  327-402.  For  early  references  to  sclerites  formed 
in  cells,  see  (e.g.)  L.  Selenka,  Z.J.w.Z.  xxxiii,  p.  45,  1879  and  R.  Semon,  Mitth. 
Zool.  St.  Neapel,  vii,  p.  288,  1886  (both  in  holothurians) ;  Blochmann,  Die 
Epithelfrage  hei  Cestodenu.  Trcmatoden,  Hamburg,  1896;  also  Leger's  Observations 
on  crystals  of  calcium  oxalate  in  tlie  cysts  , of  Lithocystis  Schneider i,  A. M.N. II.  (6), 
xviii^  p.  479,  1895. 

t  Cf.  Q.J.M.S.  XII,  pp.  118-123,  1872. 


654  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

shewing,  as  they  grew  larger,  the  typical  Brownian  movement.  So 
far,  much  the  same  phenomena  were  witnessed  whether  the  solution 
were  albuminous  or  not,  and  similar  appearances  indeed  had  been 
witnessed  and  recorded  by  Gustav  Rose,  so  far  back  as  1837*;  but 
in  the  later  stages  the  presence  of  albuminoid  matter  made  a  great 
difference.  Now,  after  a  few  days,  the  calcium  carbonate  was  seen 
to  be  deposited  in  the  form  of  large  rounded  concretions,  each  with 
a  more  or  less  distinct  central  nucleus  and  with  a  surrounding 
structure  at  once  radiate  and  concentric;  the  presence  of  concentric 
zones  or  lamellae,  alternately  dark  and  clear,  was  especially  charac- 
teristic. These  round  "  calcospherites "  shewed  a  tendency  to 
aggregate  in  layers,  and  then  to  assume  polyhedral,  often  regularly 
hexagonal,  outhnes.  In  this  latter  condition  they  closely  resemble 
the  early  stages  of  calcification  in  a  molluscan  (Fig.  296),  or  still 
more  in  a  crustacean  shell  f;   while  in  their  isolated  condition  they 

*  Cf.  Quincke,   Ueber  unsichtbare  Flussigkeitsschichten,  etc.,  Ann.  der  Physik 
(4),  VII,  pp.  631-682,  701-744,  1902. 

t  See  for  instance  other  excellent  illustrations  in  Carpenter's  article  "Shell," 
in  Todd's  Cyclopcedia,  iv,  pp.  556-571,  1847-49.  According  to  Carpenter,  the 
shells  of  the  mollusca  (and  also  of  the  Crustacea)  are  "essentially  composed  of 
cells,  consolidated  by  a  deposit  of  carbonate  of  lime  in  their  interior."  That  is 
to  say,  Carpenter  supposed  that  the  spherulites  or  calcospherites  of  Harting  were, 
to  begin  with,  just  so  many  living  protoplasmic  cells.  Soon  afterwards,  however, 
Huxley  pointed  out  that  the  mode  of  formation,  while  at  first  sight  "irresistibly 
suggesting  a  cellular  structure. .  .is  in  reality  nothing  of  the  kind,"  but  "is  simply 
the  result  of  the  concretionary  manner  in  which  the  calcareous  matter  is  deposited"; 
ibid.  art.  "Tegumentary  organs,"  v,  p.  487,  1859.  Quekett  (Lectures  on  Histology, 
II,  p.  393,  1854,  and  Q.J. M.S.  xi,  pp.  95-104,  1863)  supported  Carpenter;  but^ 
Williamson  (Histological  features  in  the  shells  of  the  Crustacea,  Q.J. M.S.  viii, 
pp.  35-47,  1860)  amply  confirmed  Huxley's  view,  which  in  the  end  Carpenter 
himself  adopted  {The  Microscope,  1862,  p.  604).  A  like  controversy  arose  later 
in  regard  to  corals.  Mrs  Gordon  (M.  M.  Ogilvie)  asserted  that  the  coral  was  built 
up  "of  successive  layers  of  calcified  cells,  which  hang  together  at  first  by  their 
cell-walls,  and  ultimately,  as  crystalline  changes  continue,  form  the  individual 
laminae  of  the  skeletal  structures"  {Phil.  Trans,  clxxxvii,  p.  102,  1896):  whereas 
von  Koch  had  figured  the  coral  as  formed  out  of  a  mass  of  "  Kalkconcremente " 
or  "crystalline  spheroids,"  laid  down  outside  the  ectoderm,  and  precisely  similar 
both  in  their  early  rounded  and  later  polygonal  stages  (though  von  Koch  was  not 
aware  of  the  fact)  to  the  calcospherites  of  Harting  (Entw.  d.  Kalkskelettes  von 
Astroides,  Mitth.  Zool.  St.  Neapel,  iii,  pp.  284-290,  pi.  xx,  1882).  Lastly,  W.  H. 
Bryan  finds  all  ordinary  corals  {Hexacoralla)  to  be  mineral  aggregates  formed  by 
"spherulitic  crystallisation,"  due  in  turn  to  the  presence  of  a  colloid  matrix  secreted 
by  certain  areas  of  ectoderm;  see  Prof .  R.S.  Queensland,  Lu,  pp.  41-53,  1940;  Univ. 
of  Queensland  Papers,  Geology,  u,  4  and  5,  1941.  Cf.  J.  E.  Duerden,  On  Siderastraea . 
Carnegie  Inst.  Washington,  1904,  p.  34. 


IX]        OF  SPHERULITES  OR  CALCOSPHERITES     655 


Fig.  293.  Calcospherites,  or  con- 
cretions of  calcium  carbonate, 
deposited  in  white  of  egg. 
^fter  Harting. 


Fig.  294.  A  single 
calcospherite,  with 
central  "nucleus," 
and  striated,  iride- 
scent border.  After 
Harting. 


Fig.  295.     Later  stages  in  the  same  experiment- 


A  B 

Fig.  2y(),     A,  Section  of  shell;    B,  Section  of  hinge-tooth  of  Mya. 
After  Carpenter. 


656  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

closely  resemble  the  little  calcareous  bodies  in  the  tissues  of  a 
trematode  or  a  cestode  worm,  or  in  the  oesophageal  glands  of  an 
earthworm*. 

When  the  albumin  was  somewhat  scanty,  or  when  it  was  mixed 
with  gelatine,  and  especially  when  a  little  phosphate  of  lime  was 
added  to  the  mixture,  the  spheroidal  globules  tended  to  become 
rough,  by  an  outgrowth  of  spinous  or  digitiform  projections;    and 


Fig.  297.     Large  irregular  calcareous  concretions,  or  spicules,  deposited  in  a  piece 
of  dead  cartilage,  in  presence  of  calcium  phosphate.     After  Harting. 

in  some  cases,  but  not  without  the  presence  of  the  phosphatef,  the 
result  was  an  irregularly  shaped  knobJ3y  spicule,  precisely  similar 
to  those  which  are  characteristic  of  the  Alcyonaria%. 

The  rough  spicules  of  the  Alcyonaria  are  extraordinarily  variable  in  shape 
and  size,  as,  looking  at  them  from  the  chemist's  or  the  physicist's  point  of 
view,  we  should  expect  them  to  be.  Partly  upon  the  form  of  these  spicules, 
and  partly  on  the  general  form  or  mode  of  branching  of  the  entire  colony  of 
polyps,  a  vast  number  of  separate  "species"  have  been  based  by  systematic 

*  Cf.  Claparede,  Z.f.w.Z.  xix,  p.  604,  1869.  On  the  structure  of  the  moUuscan 
shell,  see  O.  B.  Boggild,  K.  Vidensk.  Selsk.  Skr.,  Kjobenh.,  (9)  ii,  1930.  On  nacre, 
or  mother-of-pearl,  see  Brewster,  Treatise  on  Optics,  1853,  p.  137;  Schmidt,  Die 
Bausteine  der  Tierkorper  in  polarisirtem  Licht,  Bonn,  1924.  Also  S.  Ruma  Swamy, 
Proc.  Ind.  Acad.  Sci.  (A),  i,  p.  871,  1935;  P.  S.  Srinivasam,  ibid,  v,  pp.  464-483, 
1937;  and,  on  the  specific  qualities  of  the  nacre  in  the  several  divisions  of  the 
Mollusca,  Sir  C.  V.  Raman,  ibid.  pp.  559,  etc.,  1935. 

t  On  the  deposition  of  phosphates  in  organisms,  cf.  Pauli  u.  Samec,  Biochem. 
/Aschr.  XVII,  p.  235,  1909;    Wiener  mediz.  Wochenschr.  1910,  pp.  2287-2292. 

X  Spicules  much  like  those  of  the  Alcyonaria  occur  also  in  a  few  sponges;  cf.  (e.g.), 
V'aughan  Jennings,  Journ.  Linn.  Soc.  xxiii,  p.  .")31,  pi.  13,  fig.  8,  1891. 


ixj      OF  SPHERULITES  OR  CALCOSPHERITES     657 

zoologists./  But  it  is  now  admitted  that  even  in  specimens  of  a  single  species, 
from  one  and  the  same  locality,  the  spicules  may  vary  immensely  in  shape 
and  size:  and  Professor  S.  J.  Hickson  declared  that  after  many  years  of 
laborious  work  in  striving  to  determine  species  of  these  animal  colonies,  he 
felt  "'quite  convinced  that  we  have  been  engaged  in  a  more  or  less  fruitless 
task*." 

The  formation  of  a  tooth  is  a  phenomenon  of  the  same  order.  That  is  to 
say,  '"calcification  in  both  dentine  and  enamel  is  in  great  part  a  physical 
phenomenon;  the  actual  deposit  in  both  tissues  occurs  in  the  form  of  calco- 
spherites,  and  the  process  in  mammalian  tissue  is  identical  in  every  point  with 


Fig^  298.     Additional  illustrations  of  alcyonarian  spicules:   Eunicea.     After  Studer. 

the  same  process  occurrmg  m  lower  organisms!."  The  ossification  of  bone, 
we  may  be  sure,  is  in  the  same  sense  and  to  the  same  extent  a  physical 
phenomenon. 

The  typical  structure  of  a  calcospherite  is  no  other  than  that  of 
a  pearl,  nor  does  it  differ  essentially  from  tha^  of  the  otohth  of  a 
mollusc  or  of  a  bony  fish.  (The  otoliths  of  the  elasmobranch  fishes, 
like  those  of  reptiles  and  birds,  are  not  developed  after  this  fashion, 
but  are  true  crystals  of  calc-spar.) 

The  effect  of  surface-tension  is  manifest  throughout  these  pheno- 
mena. It  is  by  surface-tension  that  ultra-microscopic  particles  are 
brought  together  in  the  first  floccular  precipitate  or  coagulum;    by 

*  Mem.  Manchester  Lit.  and  Phil.  Soc.  lx,  p.  II,  1916. 

t  J.  H.  Mummery.  On  calcification  in  enamel  and  dentine,  Phil.  Trans.  (B), 
rev,  pp.  95-111.  1914. 


658  ON  CONCRETIONS,  SPICULES,  ETC.  [en. 

the  same  agency  the  coarser  particles  are  in  turn  agglutinated  into 
visible  lumps;  and  the  form  of  the  calcospherites,  whether  it  be 
that  of  the  solitary  spheres  or  that  assumed  in  various  stages  of 
aggregation  (e.g.  Fig.  300)*,  is  hkewise  due  to  the  same  agency. 

From  the  point  of  view  of  colloid  chemistry  the  whole  pheno- 
menon is  important  and  significant;  and  not  the  least  significant 
part  is  this  tendency  of  the  solidified  deposits  to  assume  the  form 
of  "spheruhtes"  and  other  rounded  contours.     In  the  phraseology 


Fig.  299.     A   "crust"   of  close-packed 

calcareous  concretions,  precipitated  Fig.  300.     Aggregated  calco- 

at  the   surface  of  an   albuminous  spherites.     After  Harting. 

solution.     After  Harting. 

of  that  science,  we  are  dealing  with  a  iwo-phase  system,  which 
finally  consists  df  solid  particles  in  suspension  in  a  liquid — a  disperse 
phase  ir^  a  dispersion  medium.  In  accordance  with  a  rule  first 
recognised  by  Ostwald,  when  a  substance  begins  to  separate  out 
from  a  solution,  so  making  its  appearance  as  a  new  phase,  it  always 
makes  its  appearance  first  as  a  liquid  f.  Here  is  a  case  in  point. 
The  minute  quantities  of  material,  on  th^ir  way  from  a  state  of 
solution  to  a  state  of  "suspension,"  pass  through  a  Hquid  to  a  sohd 
form;  their  temporary  sojourn  in  the  former  leaves  its  impress  in 
the  rounded  contours  which  surface-tension  brought  about  while  the 
httle  aggregate  was  still  labile  or  fluid :  while  coincidently  with  this 
surface-tension  effect,  crystallisation  tends  to  take  place  throughout 
the  httle  liquid  mass,  or  in  such  portions  of  it  as  have  not  yet  con- 
solidated and  crystallised. 

*  The  artificial  concretion  represented  in  Fig.  300  is  identical  in  appearance 
with  the  concretions  found  in  the  kidney  oi  Nautilus,  as  figured  by  Willey  (Zoological 
Results,  p.  Ixxvi,  Fig.  2,  1902). 

t  This  rule,  undreamed  of  by  Errera,  supports  and  justifies  his  cardinal 
assumption  (of  which  we  have  had  so  much  to  say  in  discussing  the  forms  of  cells 
and  tissues)  that  the  incipient  cell-wall  behaves  as,  and  indeed  actually  is,  a  liquid 
film  (cf.  p.  482). 


IX]  ON  CONOSTATS  659 

Where  we  have  simple  aggregates  of  two  or  three  calcospherites 
the  resulting  figure  is  that  of  so  many  contiguous  soap-bubbles. 
In  other  cases  composite  forms  result  which  are  not  so  easily  ex- 
plained, but  which,  if  we  could  only  account  for  them,  would  be 
of  very  great  interest  to  the  biologist.  For  instance,  when  smaller 
calcospheres  seem,  as  it  were,  to  invade  the  substance  of  a  larger  one, 
we  get  curious  conformations  which  somewhat  resemble  the  outlines 
of  certain  diatoms  (Fig.  301).  Another  curious  formation,  which 
Harting  calls  a  '"conostat,"  is  of  frequent  occurrence,  and  in  it  we 


Fig.  .'iUl.     Composite  calcospheres.     Alter  Harting. 

see  at  least  a  suggestion  of  analogy  with  the  configuration  which, 
in  a  protoplasmic  structure,  we  have  spoken  of  as  a  "collar-cell." 
The  conostats,  which  are  formed  in  the  surface  layef  of  the  solution, 
consist  of  a  portion  of  a  spheroidal  calcospherite,  whose  upper  part 
is  continued  into  a  thin  spheroidal  collar  of  somewhat  larger  radius 
than  the  solid  sphere;  but  the  precise  manner  in  which  the  collar, 
is  formed,  possibly  around  a  bubble  of  gas,  possibly  about  a  vortex- 
like diffusion-current,  is  not  obvious. 

Among  these  various  phenomena,  the  concentric  striation  of  the 
calcospherite  has  acquired  a  special  interest  and  importance*.  It 
is  part  of  a  phenomenon  now  widely  known  under  the  name  of 
"Liesegang's  Ringst." 

*  Cf.  Harting,  op.  cit.  pp.  22,  50:  "J'avais  cru  d'abord  que  ces  couches 
concentriques  etaient  produites  par  I'alternance  de  la  chaleur  ou  de  la  lumiere, 
pendant  le  jour  et  la  nuit.  Mais  I'experience,  expressement  instituee  pour 
examiner  cette  question,  y  a  repondu  negativement." 

f  R.  E.  Liesegang,  Ueber  die  Schichtungen  hei  Diffusionen,  Leipzig,  1907,  and 
earlier  papers.  A  periodic  precipitate  is  said  to  have  been  first  noticed  (on  filter- 
paper)  by  Runge,  in  1885;  cf.  Quincke,  Ueber  unsichtbare  Fliissigkeitsschichten, 
Ann.  d.  Physik  (4),  vii,  pp.  643-7,  1902.  On  a  very  minute  periodicity  in  the 
so-called  Hookham's  crystals,  formed  by  crystallising  copper  sulphate  and  salicin 
in  strong  syrup,  see  Rayleigh,  Collected  Papers,  vi,  p.  661:  "There  is  much  here,"' 
says  Rayleigh,  '"to  excite  admiration  and  perj)lexity." 


660  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

If  we  dissolve,  for  instance,  a  little  potassium  bichromate  in 
gelatine,  pour  it  on  to  a  glass  plate,  and  after  it  is  set  pour  upon  it  a 
drop  of  silver  nitrate  solution,  there  appears  in  the  course  of  a  few 
hours  the  phenomenon  of  Liesegang's  rings.     At  first  the  silver 


Fig.  302.     Conostats.  *  After  Harting. 

forms  a  central  patch  of  abundant  reddish-brown  chromate  pre- 
cipitate; but  around  this,  as  the  silver  nitrate  diffuses  slowly 
through  the  gelatine,  the  precipitate  no  longer  comes  down  con- 
tinuously, but  forms  a  series  of  concentric  rings  or  zones,  beautifully 


Fig.  .303.     Liesegang'.s  rings.     After  Leduc. 

regular,  which  alternate  with  clear  interspaces  of  jelly  and  stand 
farther  and  farther  apart  in  a  definite  ratio  as  they  recede  from 
the  centre*.  For  a  discussion  of  the  raison  d'etre  of  this  phenomenon, 
the  student  will  consult  the  textbooks  of  physical  and  colloid 
chemistry.     But,  speaking  generally,  we  may  say  that  the  appearance 

*  It  is  now  known  that  periodic  precipitation  may  be  exhibited  even  in  aqueous 
solutions,  and  that  what  the  gel  does  is  to  enlarge  the  intervals,  and  to  enhance 
the  phenomenon,  by  affecting  the  rate  or  relative  rates  of  diffusion.  Cf.  H.  W. 
Morse,  Journ.  Phys.  Chem.  1931. 


IX]  OF  LIESEGANG'S  RINGS  661 

of  Liesegang's  rings  is  but  a  particular  case  of  a  more  general 
phenomenon,  namely  the  influence  on  crystalhsation  of  the  presence 
of  foreign  bodies  or  "impurities,"  represented  in  this  case  by  the 
gel  or  colloid  matrix.  F.  S.  Beudant  had  shewn  in  a  fine  paper, 
more  than  a  hundred  years  ago,  that  impurities  were  the  chief  cause 
of  variation  of  crystal  habit*.  Faraday  proved  that  to  diffusion 
in  presence  of  slight  impurities,  not  to  actual  stratification  or 
altei'nate  deposition,  could  be  ascribed  the  banded  structure  of  ice, 


Fig.  304.     The  Liesegang  phenomena.     After  Emil  Hatschek. 

of  agate  or  of  onyx;  and  Quincke  and  Tomhnson  added  to  our 
scanty  knowledge  of  this  remarkable  phenomenon  f.  Ruskin,  who 
knew  a  great  deal  about  agates,  spoke  of  the  perpetual  difficulty  of 
distinguishing  "between  concretionary  separation  and  successive 
deposition."     And  Rayleigh  shewed  how  to  such  a  periodic,  but 

*  F.  S.  Beudant,  Recherehes  sur  les  causes  qui  peuvent  varier  les  formes  crystal- 
lines d'une  meme  substance  minerale,  Ann.  de  Chimie,  viii,  pp.  5-52,  1818.  See 
also  his  Memoire  sur  les  parties  solides  des  MoUusques,  Mem.  du  Museum,  xv, 
pp.  66-75,  1810. 

t  Cf.  Faraday,  On  ice  of  irregular  fusibility,  Phil.  Trans.  1858,  p.  228;  Researches 
in  Chemistry,  etc.,  1859,  p.  374;  Canon  Moseley,  On  the  veined  structure  of  the 
ice  of  glaciers,  Phil.  Mag.  (4),  xxxix,  p.  241,  1870;  R.  Weber,  in  Poggend.  Ann. 
cix,  p.  379,  1860;  Tyndall,  Forms  of  Water,  1872,  p.  178;  C.  Tomlinson,  On  some 
eflFects  of  small  quantities  of  foreign  matter  on  crystallisation,  Phil.  Mag.  (5) 
XXXI,  p.  393,  1891,  and  other  papers.  Cf.  Liesegang,  Centralbl.  f.  Mineralogie, 
XVI,  p.  497,  1911;  E.  S.  Hedges  and  J.  E.  Myers,  The  ^problem,  of  physico-chemical 
periodicity,  London,  1926;  W.  F.  Berg,  Crystal  growth  from  solutions,  Proc.  R.S.  (A), 
CLxi\',  pp.  79-95,  1938. 


662  ON  CONCRETIONS,  SPICULES,  ETC.  L^^h. 

unstratified  structure  all  the  colours  of  the  opal  and  the  iridescence 
of  ancient  glass  are  alike  due. 

Besides  the  tendency  to  rhythmic  action,  as  manifested  in 
Liesegang's  rings,  the  association  of  colloid  matter  with  a  crystalloid 
in  solution  may  lead  to  other  well-marked  effects.  These  include*: 
(1)  the  total  prevention  of  crystallisation;  (2)  suppression  of  certain 
of  the  lines  of  crystal  growth;  (3)  extension  of  the  crystal  to 
abnormal  proportions,  with  a  tendency  to  become  compound; 
(4)  a  curving  or  gyrating  of  the  crystal  or  its  parts. 

It  would  seem  that,  if  the  supply  of  material  to  the  growing 
crystal  begin  to  run  short  (as  may  well  happen  in  a  colloid  medium 
for  lack  of  convection-currents),  then  growth  will  follow  only  the 
strongest  lines  of  crystallising  force,  and  will  be  suppressed  or 
partially  suppressed  along  other  axes.  The  crystal  will  have  a 
tendency  to  become  filiform,  or  ''fibrous";  and  the  raphides  of  our 
plant-cells,  and  the  needle-hke  "oxyotes"  of  sponges,  are  cases  in 
point.  Again,  the  long  slender  crystal  so  formed,  pushing  its  way 
into  new  material,  may  start  a  new  centre  of  crystallisation: 
whereby  we  get  the  phenomenon  known  as  a  "relay,"  along  the 
principal  lines  of  force  and  sometimes  along  subordinate  axes  as 
well.  This  phenomenon  is  illustrated  in  the  accompanying  figure 
of  common  salt  crystaUising  in  a  colloid  medium;  and  it  may  be 
that  we  have  here  an  explanation,  or  part  of  an  explanation,  of 
the  compound  siliceous  spicules  of  the  Hexactinellid  sponges. 
Lastly,  when  the  crystaUising  force  is  nearly  equalled  by  the 
resistance  of  the  viscous  medium,  the  crystal  takes  the  fine  of  least 
resistance,  with  very  various  results.  One  of  these  results  would 
seem  to  be  a  gyratory  course,  giving  to  the  crystal  a  curious  wheel- 
like shape,  as  in  Fig.  306;  and  other  results  are  the  feathery, 
fern-like  or  arborescent  shapes  so  frequently  seen  in  microscopic 
crystallisation. 

To  return  to  Liesegang's  rings,  the  typical  appearance  of  con- 
centric rings  upon  a  plate  of  gelatine  may  be  modified  in  various 
experimental  ways.  For  instance,  if  our  gelatinous  medium  be  placed 
in  a  capillary  tube  immersed  in  a  solution  of  the  precipitating  salt, 
we  obtain  (Fig.  304)  a  vertical  succession  of  bands  or  zones  regularly 

*  Cf.  J.  H.  Bowman,  A  study  in  crystallisation,  Journ.  Soc.  of  Chem.  Industry, 
XXV,  p.  143,  1906. 


IX] 


OF  LIESEGANG'S  RINGS 


663 


interspaced :  the  result  being  very  closely  comparable  to  the  banded 
pigmentation  which  we  see  in  the  hair  of  a  rabbit  or  a  rat.  In  the 
ordinary  plate  preparation,  the  free  surface  of  the  gelatine  is  under 
different  conditions  to  the  layers  below  and  especially  to  the  lowest 
layer  of  all  in  contact  with  the  glass;    and  so  we  often  obtain  a 


Fig.  305.     Relay- crystals  of  common  salt.     After  Bowman. 

double  series  of  rings,  one  deep  and  the  other  superficial,  whiclj.  by 
occasional  blending  or  interlacing  may  produce  a  netted  pattern. 
Sometimes,  when  only  the  inner  surface  of  our  capillary  tube  is 
covered  with  a  layer  of  gelatine,  there  is  a  tendency  for  the  deposit 


Fig.  306.     Wheel-like  crystals  in  a  colloid.     After  Bowman. 

to  take  place  in  a  continuous  spiral,  rather  than  in  concentric  and 
separate  zones.  By  such  means,  -according  to  Kiister*,  various 
forms  of  annular,  spiral  and  reticulated  thickenings  in  the  vascular 
tissue  of  plants  may  be  closely  imitated ;  and  he  and  certain  other 
writers   have   been   incHned   to   carry  the   same   chemico-physical 

*  E.  Kiister,  Ueber  die  Schichtung  der  Starkekorner,  Ber.  d.  botan.  Gesellsch. 
XXXI,  pp.  339-346,  1913;  Ueber  Zonenbildung  in  kolloidalen  Medien,  Roll.  Ztachr. 
xiii,  pp.  192-194;   XIV,  pp.  307-319,  1913-14. 


664  ON  CONCKETIONS,  SPICULES,  ETC.  [en. 

phenomenon  a  very  long  way,  in  the  explanation  of  various  banded, 
striped,  and  other  rhythmically  successional  types  of  structure  or 
pigmentation.  The  striped  leaves  of  many  plants  (such  as  Eulalia 
jajponica),  the  striped  or  clouded  colouring  of  many  feathers  or  of 
a  cat's  skin,  the  patterns  of  many  fishes,  such  for  instance  as  the 
brightly  coloured  tropical  Chaetodonts  and  the  like,  are  all  regarded 
by  him  as  so  many  instances  of  "diffusion-figures"  closely  related 
to  the  typical  Liesegang  phenomenon.  Gebhardt*  declares  that  the 
banded  wings  of  Papilio  podalirius  are  analogous  to  or  even  closely 
imitated  in  Liesegang's  experiments;  that  the  finer  markings  on 
the  wings  of  the  goatmoth  shew  a  double  rhythm,  alternately 
coarse  and  fine,  such  as 'is  manifested  in  certain  experimental  cases 
of  the  same  kind ;  that  the  alternate  banding  of  the  antennae  (for 
instance  in  Sesia  spheciformis),  a  pigmentation  not  concurrent  with 
the  antennal  joints,  is  exphcable  in  the  same  way;  and  that  the 
ocelli  on  the  wings  of  the  Emperor  moth  are  typical  illustrations 
of  the  common  concentric  type.  Darwin's  well-known  disquisition 
on  the  ocellar  pattern  of  the  feathers  of  the  Argus  pheasant,  as  a 
result  of  sexual  selection,  will  occur  to  the  reader's  mind,  in  striking 
contrast  to  this  or  to  any  other  direct  physical  explanationf. 

To  turn  from  the  distribution  of  pigment  to  more  deeply  seated 
structural  characters,  Leduc  has  argued,  for  instance,  that  the 
laminar  structure  of  the  cornea  or  the  lens  is,  or  may  be,  a  similar 
phenomenon.  In  the  lens  of  the  fish's  eye,  we  have  a  very  curious 
appearance,  the  consecutive  lamellae  being  roughened  or  notched 
by  close-set,  interlocking  sinuosities;  and  the  same  appearance, 
save  that  it  is  not  quite  so  regular,  is  presented  in  one  of  Klister's 
figures  as  the  effect  of  precipitating  a  little  sodium  phosphate  in 
a  gelatinous  medium.     Biedermann  has  studied,   from  the  same 

*    Verh.  d.  d.  zool.  Gesellsch.  p.  179,  1912. 

t  As  a  matter  of  fact,  the  phenomena  associated  with  the  development  of  an 
"ocellus"  are  or  may  be  of  great  complexity,  inasmuch  as  they  involve  not  only 
a  graded  distribution  of  pigment,  but  also,  in  "optical"  coloration,  a  symmetrical 
distribution  of  structure  or  form.  The  subject  therefore  deserves  very  careful 
discussion,  such  as  Bateson  gives  to  it  ( Variation,  chap.  xii).  This,  by  the  way, 
is  one  of  the  very  rare  cases  in  which  Bateson  appears  inclined  to  suggest  a  purely 
physical  explanation  of  an  organic  phenomenon:  "The  suggestion  is  strong  that 
the  whole  series  of  rings  (in  Morpho)  may  have  been  formed  by  some  one  central 
disturban(»e,  somewhat  as  a  series  of  concentric  waves  may  be  formed  by  the  splash 
of  a  stone  thrown  into  a  pool."     Cf.  Darwin,  Descent  of  Man,  ii,  p.  132,  1871. 


IX]  OF  SCALES  AND  OTOLITHS  665 

point  of  view,  the  structure  and  development  of  the  molluscan  shell, 
the  problem  which  Rainey  had  first  attacked  more  than  fifty  years 
before*;  and  Liesegang  himself  has  applied  his  results  to  the 
formation  of  pearls,  and,  as  Bechhold  has  also  done,  to  the  develop- 
ment of  bonef. 

The  presence  of  concentric  rings  or  zones  in  slow-growing 
structures  is  evidently  after  some  fashion  a  function  of  the  time, 
and  an  indication  of  periodic  acceleration  or  variation  of  growth; 
it  is  apt  to  be  referred,  rightly  or  wrongly,  to  the  seasons  of  the 
year,  and  to  be  interpreted  (with  or  without  confirmation  and  proof) 
as  a  sure  mark  and  measure  of  the  creature's  age.  This  is  the  case, 
for  instance,  with  the  scales,  bones  and  otoliths  of  fishes;  and  a 
kindred  phenomenon  in  starch-grains  has  given  rise,  in  like  manner, 
to  the  belief  that  they  indicate  a  diurnal  and  nocturnal  periodicity 
of  activity  and  rest  J  on  the  part  of  the  cell  wherein  they  grew. 

That  this  is  actually  the  case  in  growing  starch-grains  is  often 
if  not  generally  beheved,  on  the  authority  of  Meyer  §;  but  while 
under  certain  circumstances  a  marked  alternation  of  growing  and 
resting  periods  may  occur,  and  may  leave  its  impress  on  the  structure 
of  the  grain,  there  is  now  more  reason  to  beheve  that,  apart  from 
such  external  influences,  the  internal  phenomena  of  diffusion  may, 
just  as  in  the  typical  Liesegang  experiment,  produce  the  well-known 
concentric  rings.  The  spherocrystals  of  inulin,  in  like  manner, 
shew,  like  the  calcospherites  of  Harting  (Fig.  307),  a  concentric 
structure  which  in  all  likelihood  has  had  no  causative  impulse  save 
from  within. 

The  striation,  or  concentric  lamella tion,  of  the  scales  and  otoHths 
of  fishes  has  been  much  employed,  not  as  a  mere  indication,  but 

*  Cf.  also  Sir  D.  Brewster,  On  optical  properties  of  mother  of  pearl,  Phil.  Trans. 
1814,  p.  397;   and  J.  F.  W.  Hersehel,  in  Edin.  Phil.  Journ.  ii,  p.  116,  1819. 

f  VV.  Biedermann,  Ueber  die  Bedeutung  von  KristaUisationsprozessen  der 
Skelette  wirbelloser  Thiere,  naraentlich  der  Molluskenschalen,  Z.  f.  allg.  Physiol. 
I,  p.  154,  1902;  Ueber  Bau  und  Entstehung  der  Molluskenschale,  Jen.  Zeitschr. 
XXXVI,  pp.  1-164,  1902.  Cf.  also  Steinmann,  Ueber  Schale  und  Kalksteinbildungen, 
Ber.  Naturf.  Ges.  Freiburg  i.  Br.  iv,  1889;  Liesegang,  Naturw.  Wochenschr.  1910. 
p.  641;  Arch.  f.  Entw.  Mech.  xxxiv,  p.  452,  1912;  H.  Bechhold,  Ztschr.  f.  phys. 
Chem.  Lii,  p.  185,  1905. 

X  Cf.  Biitschli,  Ueber  die  Herstellung  kiinstlicher  Starkekorner  oder  von 
Spharokrystallen  der  Starke,  Verh.  nat.  med.  Ver.  Heidelberg,  v,  pp.  457-472,  1896. 

§    U ntersuchiLngen  ilber  die  Stdrkekorner,  Jena,  1905. 


666  ON  CONCRETIONS,  SPICDLES,  ETC.  [ch. 

as  a  trustworthy  and  unmistakeable  measure  of  the-  fish's  age  (see 
ante,  p.  180).  There  are  some  difficulties  in  the  way  of  accepting 
this  hypothesis,  not  the  least  of  which  is  the  fact  that  the  otolith- 
zones,  for  instance,  are  extremely  well  marked  even  in  the  case  of 
some  fishes  which  spend  their  lives  in  deep  water,  where  temperature 
and  other  physical  conditions  shew  little  or  no  appreciable  fluctuation 
with  the  seasons  of  the  year.  There  are,  on  the  other  hand,  pheno- 
mena which  seem  strongly  confirmatory  of  the  hypothesis:  for 
instance,  the  fact  (if  it  be  fully  established)  that  in  such  a  fish  as 
the  cod,  zones  of  growth^  identical  in  number,  are  found  both  on 


Fig.  307.     A  sphero- 
crystal  of"  inulin. 

Fig.  .*i08.  Otoliths  of  plaice,  shewing 
four  zones  or  "age-rings."  After 
Wallace. 

the  scales  and  in  the  otoliths*.  The  subject  is  as  difficult  as  it  is 
important,  but  it  is  at  least  certain,  with  the  Liesegang  pheno- 
menon in  view,  that  we  have  no  right  to  assume,  without  proof 
and  confirmation,  that  rhythm  and  periodicity  in  structure  and 
growth  are  necessarily  bound  up  with,  and  indubitably  brought 
about  by,  a  periodic  or  seasonal  recurrence  of  particular  external 
conditions^. 

But   while    in    the    ordinary   Liesegang    phenomenon   rhythmic 

*  Cf.  Winge,  Meddel.  fra  Komni,  for  Hav under sogelse  {Fiskeri),  iv,  p.  20,  Copen- 
hagen, 1915. 

t  A.  VV.  Morosow  strongly  supports  the  view — uncertain  as  it  seems  to  be — 
that  the  concentric  pattern  of  a  fish's  scale  is  due  to  the  Liesegang  phenomenon; 
he  produces  an  "artificial  scale,"  with  its  "summer  and  winter  rings,"  by 
precipitating  sodium  carbonate  and  calcium  chloride  in  gelatin:  Zur  Frage  iiber 
die  Natur  des  Schuppenwachstums  bei  Fischen  (and  in  Russian),  Nation.  Comm. 
Agriculture:  Rep.  Sci.  Inst.  Fisheries,  i,  Moscow,  1924;  abstract  in  Michael 
Graham's  Studies  of  age-determination  in  fish,  Rep.  Ministry  of  Agr.  and  Fisheries, 
Fishery  Investigations,  (2)  xi,  no.  3,  p.  28,  1928. 


IX]  OF  CERTAIN  RHYTHMIC  PHENOMENA        667 

precipitation  depends  only  on  forces  intrinsic  to  the  system,  and 
is  independent  of  any  corresponding  rhythmic  changes  in  external 
conditions,  we  have  not  far  to  seek  for  analogous  chemico-physical 
phenomena  where  rhythmic  alternations  of  structure  are  produced 
in  close  relation  to  periodic  fluctuations  of  temperature.  The 
banding,  or  "varving,"  of  Swedish  and  Irish  glacial  clays  is  a  re- 
markable instance.  A  well-known  and  a  simple  case  is  that  of  the 
Stassfurt  deposits,  where  the  rock-salt  alternates  with  thin  layers  of 
"anhydrite,"  or  (in  another  series  of  beds)  with  " polyhahte * " : 
and  where  these  zones  are  commonly  regarded  as  marking  years, 
and  their  alternate  bands  as  due  to  the  seasons.  A  discussion, 
however,  of  this  remarkable  and  significant  phenomenon,  and  of 
how  the  chemist  explains  it,  by  help  of  the  "phase-rule,"  in  con- 
nection with  temperature  conditions,  would  lead  us  far  beyond  our 
scope. 

We  may  turn  aside  to  touch,  for  a  single  moment,  on  certain 
forms  and  patterns  not  easy  to  classify:  some  of  which  depend  on 
the  molecular  structure  of  a  colloid  matrix,  while  others  are  of  a 
coarser  and  more  mechanical  grade.  So  many  organic  forms  and 
patterns  await  explanation  that  we  cannot  seek  too  widely  for 
examples,  nor  for  explanations,  of  such  things.  For  instance,  a 
drop  of  dried  egg-albumin  shews  beautiful  radial  cracks,  with  cross- 
lines  here  and  there;  and  a  drop  of  blood  drying  on  a  glass  plate 
shews  a  complete  system  of  radial  fissures,  in  series  after  series, 
sometimes  with  and  sometimes  without  a  clear  central  space.  The 
general  resemblance  to  the  cross-section  of  a  stem,  with  its  pith  and 
its  primary  and  secondary  medullary  rays,  is  striking  enough  to 
have  led  some  even  to  look  upon  a  tree  as  one  great  complicated 
but  symmetrical  colloid  massf.  We  may  compare  also  the  beautiful 
radiating  structure  which  Biitschli  observed  long  ago  around  small 

*  The  anhydrite  is  sulphate  of  lime  (Ca>S04) ;  the  polyhalite  is  a  triple  sulphate 
of  lime,  magnesia  and  potash  (2CaSO4.MgSO4.K2.SO4 +  2H2O). 

t  Cf.  H.  WisHcenus,  Ztschr.  f.  Chemie  u.  Kolloide,  vi,  1910;  A.  Lingelsheim, 
PHanzenanatomische  Strukturbilder  in  trocknenden  Kolloiden,  Arch.f.  Entw.  Mech. 
XLii,  pp.  117-125,  1917.  Cf.  also  Liesegang,  Trocknungserscheinungen  bei  Gelen, 
Ztschr.  f.  Ch.  u.  K.  x,  p.  229  sq.,  1912;  Biitschli,  Verh.  n.  h.  Ver.  Heidelberg,  vii, 
p.  653,  1904.  Also  {int.  al.)  Norman  Stuart,  on  Spiral  growths  in  silica  gel,  Nature, 
Oct.  2,  1937,  p.  .589. 


668  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

bubbles  in  chrome-gelatine,  and  which  he  usecT  in  one  of  his  early 
(and  none  too  fortunate)  speculations  on  the  nature  of  the  nuclear 
spindle. 

We  see  that  the  methods  by  which  we  attempt  to  study  the 
chemico-physical  characteristics  of  an  inorganic  concretion  or  spicule 
within  the  body  of  an  organism  soon  introduce  us  to  a  multitude 
of  phenomena  of  which  our  knowledge  is  extremely  scanty,  and 
which  we  must  not  attempt  to  discuss  at  greater  length.  As  regargls 
our  main  point,  namely  the  formation  of  spicules  and  other 
elementary  skeletal  forms,  we  have  seen  that  some  of  them  may 
be  safely  ascribed  to  precipitation  or  crystalhsation  of  inorganic 
materials  in  ways  modified  by  the  presence  of  albuminous  or  other 
colloid  substances.  The  effect  of  these  latter  is  found  to  be  much 
greater  in  the  case  of  some  crystalhsable  bodies  than  in  others. 
For  instance  Harting,  and  Rainey  also,  found  that  calcium  oxalate 
was  much  less  affected  by  a  colloid  medium  than  was  calcium 
carbonate;  it  shewed  in  their  hands  no  tendency  to  form  rounded 
concretions  or  "  calcospherites "  in  presence  of  a  colloid,  but  con- 
tinued to  crystallise,  either  normally  or  with  a  tendency  to  form 
needles  or  raphides.  It  is  doubtless  for  this  reason  that,  as  we  have 
seen,  crystals  of  calcium  oxalate  are  so  common  in  the  tissues  of 
plants,  while  those  of  other  calcium  salts  are  rare;  but  true  calco- 
spherites, or  spherocrystals,  even  of  the  oxalate  are  occasionally 
found,  for  instance  in  certain  Cacti,  and  Biitschli*  has  succeeded 
in  making  them  artificially  in  Harting's  usual  way,  that  is  to  say 
by  crystalhsation  in  a  colloid  medium.  If  the  nature  of  the  salt 
has  a  marked  specific  qffect,  so  also  has  the  gel:  silver  chromate 
is  thrown  down  in  rings  in  gelatin  but  not  in  agar;  replace  the 
silver  by  lead,  and  the  rings  come  in  agar  but  not  in  gelatin;  while 
neither  lead  nor  silver  produce  them  in  silicic- acid  gel. 

There  link  on  to  such  observations  as  Harting's,  and  to  the 
statement  already  quoted  that  calcareous  deposits  are  associated 
with  the  dead  residua,  or  "formed  materials,"  rather  than  with 
the  living  cells  of  the  organism,  certain  very  interesting  facts  in 
regard  to  the  solubility  of  salts  in  colloid  media,  which  go  far  to 
account  for  the  presence  (apart  from  the  form)  of  calcareous  pre- 

*  Spharocrystalle  von  Kalkoxalat  bei  Kakteen,  Ber.  d.  d.  Bot.  Gesellsch.  p.  178, 
1885. 


IX]  OF  SOLUBILITY  IN  COLLOID  MEDIA  669 

cipitates  within  the  organism*.  It  has  been  shewn,  in  the  first 
place,  that  the  presence  of  albumin  has  a  notable  effect  on  the 
solubility  in  a  watery  solution  of  calcium  salts,  increasing  the 
solubility  of  the  phosphate  in  a  marked  degree  and  that  of  the 
carbonate  in  still  greater  proportion;  but  the  sulphate  is  only  very 
little  more  soluble  in  presence  of  albumin  than  in  pure  water,  and 
the  rarity  of  its  occurrence  within  the  organism  is  accounted  for 
thereby.  On  the  other  hand,  the  bodies  derived  from  the  breaking 
down  of  the  albumins — their  "catabolic''  products,  such  as  the 
peptones,  etc. — dissolve  the  calcium  salts  to  a  much  less  degree  than 
albumin  itself;  and  phosphate  of  lime  is  scarcely  more  soluble  in 
them  than  in  water.  The  probabihty  is,  therefore,  that  the  actual 
precipitation  of  the  calcium  salts  is  not  due  to  the  direct  action  of 
carbonic  acid  on  a  more  soluble  salt  (as  was  at  one  time  believed); 
but  to  catabolic  changes  in  the  proteids  of  the  organism,  which 
throw  down  salts  that  had  been  already  formed,  but  had  remained 
hitherto  in  albuminous  solution.  The  very  shght  solubiHty  of 
calcium  phosphate  under  such  circumstances  accounts  for  its  pre- 
dominance in  mammalian  bonef;  and,  in  short,  wherever  a  supply 
of  this  salt  has  been  available  to  the  organism. 

To  sum  up,  we  see  that,  whether  from  food  or  from  sea-water, 
calcium  sulphate  will  tend  to  pass  but  little  into  solution  in  the 
albuminoid  substances  of  the  body:  that  calcium  carbonate  will 
enter  more  freely,  but  a  considerable  part  of  it  will  tend  to  remain 
in  solution:  while  calcium  phosphate  will  pass  into  solution  in 
considerable  amount,  but  will  be  almost  wholly  precipitated  again 
as  the  albumin  becomes  broken  down  in  the  normal  process  of 
metabolism.  We  have  still  to  wait  for  a  similar  and  equally 
illuminating  study  of  the  solution  and  precipitation  of  silica  in 
presence  of  organic  colloids. 

When  carbonate  of  hme  is  secreted  or  precipitated  by  hving 
organisms,  to  form  bone,  shell,  egg-shell,  coral  and  what  not,  its 
mineralogical  form  may  vary,  but  the  causes  which  determine  it 

*  W.  Pauli  u.  M.  Samec,  Ueber  Loslichkeitsbeeinfliissung  von  Elektrolyten 
durch  Eiweisskorper,  Biochem.  Zeitschr.  xvii,  p.  235,  1910.  Some  of  these  results 
were  known  much  earher;  cf.  Fokker  in  Pflilger's  Archiv,  vii,  p.  274,  1873;  also 
Robert  Irvine  and  Sims  Woodhead,  op.  cit.  p.  347. 

t  Which,  in  1000  parts  of  ash.  contains  about  840  parts  of  phosphate  and 
76  parts  of  calcium  carbonate. 


670  ON  CONCRETIONS,  SPICULES,  [ch. 

are  all  but  unknown.  It  is  amorphous  in  our  Ijones.  It  has  the 
form  of  calcite  in  an  oyster,  a  starfish,  a  Gorgonia,  a  Globigerina; 
but  of  aragonite  in  most  molluscs  and  in  all  ordinary  corals.  It  is 
of  calcite  in  a  bird's  egg,  of  aragonite  in  a  tortoise's;  of  the  one 
in  Argonauta,  of  the  other  in  Nautilus;  of  the  one  in  an  Ammonite, 
and  the  other  in  its  Aptychus-\id ;  of  the  one  in  Ostrea,  the  other  in 
Unio;  of  the  one  in  the  outer  and  the  other  in  the  inner  layers  of 
a  limpet  or  a  mussel-shell.  Physical  chemistry  has  little  to  say 
of  the  formation  of  these  two,  of  the  parts  played  by  temperature, 
by  the  presence  of  sulphate  of  hme,  or  of  magnesia  or  of  various 
impurities;  it  leaves  us  in  the  dark  as  to  what  brings  the  one  form 
or  the  other  into  being  in  the  organism*. 

Organic  fibres,  animal  and  vegetable,  proteid  and  non-pro teid, 
hair  and  wool,  silk,  cotton  and  the  rest,  may  be  mentioned  here 
in  passing:  because,  as  formed  material,  they  have  a  certain  analogy 
to  the  spicular  formations  with  which  we  are  concerned.  A  hair 
or  a  wool-fibre  may  shew  upon  its  surface  the  scaly  or  scurfy 
remnants  of  the  living  cells  among  which  its  substance  was  laid 
down;  but  the  wool  itself  is  by  no  means. living,  but  is  so  nearly 
crystalline  as  to  shew,  in  an  X-ray  photograph,  the  Laue  interference- 
figures  well  known  to  physicists.  Moreover,  the  same  identical 
figure  is  obtained  from  such  diverse  sources  as  human  hair,  merino- 
wool  and  porcupine's  quill.  But  if  we  stretch  the  thread,  whether 
of  hair  or  wool,  the  first  Laue  diagram  changes  to  another;  one 
crystalline  arrangement  has  shifted  over  into  a  new  form  of  molecular 
equihbrium.  We  are  deahng  with  a  crystalline,  or  crystal-hke, 
form  of  keratin,  the  substance  of  which  hoof  and  horn,  nail,  scale 
and  feather  are  made;  and  this  remarkable  substance  turns  out  to 
be  a  comparatively  simple  substance  after  all,  with  no  very  high 
or  protein-like  molecule  |. 

From  the  comparatively  small  groun  of  inorganic  formations 
which,  arising  within  living  organisms,  owe  their  form  to  precipita- 
tion or  to  crystallisation,  that  is  to  say  to  chemical  or  other  molecular 

*  Cf.  Marcel  Prenant,  Les  formes  mineralogiques  du  calcaire  chez  les  etres 
vivants,  Biol.  Reviews,  ii,  pp.  365-393,  1927. 

•f"  The  study  of  wool  and  other  fibres  has  much  technical  importance,  ancj  has 
gone  far  during  the  last  few  years;  cf.  W.  T.  Astbury,  in  Phil.  Trans.  (A),  ccxxx, 
pp.  75-100,  1931,  and  other  papers. 


IX]  AND  SPICULAR  SKELETONS  671 

forces,  we  shall  presently  pass  to  that  other  and  larger  group  which 
appears  to  be  conformed  in  direct  relation  to  the  forms  and  the 
arrangement  of  cells  or  other  protoplasmic  elements*.  The  two 
principles  of  conformation  are  both  illustrated  in  the  spicular 
skeletons  of  the  sponges. 

In  a  considerable  number  but  withal  a  minority  of  cases,  the 
form  of  the  sponge-spicule  may  be  deemed  sufficiently  explained 
on  the  Hnes  of  Harting's  and  Rainey's  experiments,  that  is  to  say 
as  the  direct  result  of  chemical  or  physical  phenomena  associated 
with  the  deposition  of  Ume  or  of  silica  in  presence  of  colloids  f. 
This  is  the  case,  for  instance,  with  various  small  spicules  of  a 
globular  or  spheroidal  form,  consisting  of  amorphous  silica,  con- 
centrically striated  within,  and  often  developing  irregular  knobs 
or  tiny  tubercles  over  their  surfaces.  In  the  aberrant  sponge 
AstroscleraX,  we  have,  to  begin  with,  rounded,  striated  discs  or 
globules,  which  in  hke  manner  are  nothing  more  nor  less  than  the 
calcospherites  of  Harting's  experiments;  and  as  these  grow 
they  become  closely  aggregated  together  (Fig.  309),  and  assume  an 
angular,  polyhedral  form,  once  more  in  complete  accordance  with 
the  results  of  experiment §.  Again,  in  many  monaxonid  sponges, 
we  have  irregularly  shaped,  or  branched  spicules,  roughened  or 
tuberculated  by  secondary  superficial  deposits,  and  reminding  one 
of  the  spicules  of  the  Alcyonaria.  These  also  must  be  looked 
upon  as  the  simple  result  of  chemical  deposition,  the  form  of  the 
deposit  being  somewhat  modified  in  conformity  with  the  surrounding 
tissues:  just  as  in  the  simple  experiment  the  form  of  the  con- 
cretionary precipitate  is  affected  by  the  heterogeneity,  visible  or 
invisible,  of  the  matrix.     Lastly,  the  simple  needles  of  amorphous 

*  Cf.  Fr.  Dreyer,  Die  Principien  der  Geriistbildung  bei  Rhizopoden,  Spongien 
und  Echinodermen,  Jen.  Zeitschr.  xxvi,  pp.  204-468,  1892. 

t  In  a  very  anomalous  Australian  sponge,  described  by  Professor  Dendy  {Nature, 
May  18,  1916,  p.  253)  under  the  name  of  Collosderophora,  the  spicules  are 
"gelatinous,"  consisting  of  a  gel  of  colloid  silica  with  a  high  percentage  of  water. 
It  is  not  stated  whether  an  organic  colloid  is  present  together  with  the  silica. 
These  gelatinous  spicules  arise  as  exudations  on  the  outer  surface  of  cells,  and 
come  to  lie  in  intercellular  spaces  or  vesicles. 

J  J.  J.  Lister,  in  Willey's  Zoological  Results,  pt  iv,  p.  459,  1900. 

§  The  peculiar  spicules  of  Astrosdera  are  said  to  consist  of  spherules,  or  calco- 
spherites, of  aragonite,  spores  of  a  certain  red  seaweed  forming  ^  the  nuclei  or 
starting-points  of  the  concretions  (R.  Kirkpatrick,  Proc.  E.S.  (B),  lxxxiv,  p.  579, 
1911). 


ON  CONCRETIONS,  SPICULES, 


[CH. 


672 

silica  which  constitute  one  of  the  commonest  types  of  spicule  call 
for  little  in  the  way  of  explanation ;  they  are  accretions  or  deposits 
about  a  linear  axis,  or  fine  thread  of  organic  material,  just  as  the 
ordinary  rounded  calcospherite  is  deposited  about  some  minute 
point  or  centre  of  crystallisation,  and  as  ordinary  crystallisation 
may  be  started  by  a  particle  of  dust;  in  some  cases  they  also,  like 
the  others,  are  apt  to  be  roughened  by  more  irregular  secondary 


NMNf 


Fig.  .*i09.     Close-packed  caleospherites,  or  so-called  '"spicules," 
oi  Astrosclera.     After  Lister. 

deposits,  which  probably,  as  in  Harting's  experiments,  assume  this 
irregular  form  when  material  runs  short. 


Our  few  foregoing  examples,  diverse  as  they  are  in  look  and  kind, 
from  the  spicules  of  Astrosclera  or  Alcyonium  to  the  otoliths  of  a 
fish,  seem  all  to  have  their  free  origin  in  some  larger  or  smaller 
fluid-containing  space  or  cavity  of  the  body:  pretty  much  as 
Harting's  calcospheres  made  their  appearance  in  the  albuminous 
content  of  a  dish.  But  we  come  at  last  to  a  much  larger  class  of 
spicular  and  skeletal  structures,  for  whose  regular  and  often  complex 
forms  some  other  explanation  than  the  intrinsic  forces  of  crystal- 
lisation or  molecular  adhesion  is  required.     As  we  enter  on  this 


IX]  ANL  SPICULAR  SKELETONS  673 

subject,  which  is  certainly  no  small  nor  easy  one,  it  may  conduce 
to  simplicity  and  to  brevity  if  we  make  a  rough  classification,  by 
way  of  forecast,  of  the  conditions  we  are  likely  to  meet  with. 

Just  as  we  look  upon  animals  as  constituted,  some  of  a  great 
number  of  cells,  others  of  a  single  cell  or  of  but  few,  and  just  as 
the  shape  of  the  former  has  no  longer  a  visible  relation  to  the 
individual  shapes  of  its  constituent  cells  while  in  the  latter  it  is 
cell-form  which  dominates  or  is  actually  equivalent  to  the  form  of 
the  organism,  so  shall  we  find  it  to  be,  with  more  or  less  exact 
analogy,  in  the  case  of  the  skeleton.  For  example,  our  own  skeleton 
consists  of  bones,  in  the  formation  of  each  of  which  a  vast  number 
of  minute  ll.ing  cellular  elements  are  necessarily  concerned;  but 
the  form  and  even  the  arrangement  of  these  bone-forming  cells  or 
corpuscles  are  monotonously  simple,  and  give  no  physical  explana- 
tion of  the  outward  and  visible  configuration  of  the  bone.  It  is  as 
part  of  a  far  larger  field  of  force — in  which  we  must  consider  gravity, 
the  action  of  various  muscles,  the  compressions,  tensions  and 
bending  moments  due  to  variously  distributed  loads,  the  whole 
interaction  of  a  very  complex  mechanical  system — that  we  must 
explain  (if  we  are  to  explain  at  all)  the  configuration  of  a  bone. 

In  contrast  to  these  massive  skeletons  we  have  other  skeletal 
elements  whose  whole  magnitude  is  commensurate  with  that  of  a 
living  cell,  or  (as  comes  to  very  much  the  same  thing)  is  comparable 
to  the  range  of  action  of  the  molecular  forces.  Such  is  the  case 
with  the  ordinary  spicules  of  a  sponge,  with  the  dehcate  skeleton 
of  a  radiolarian,  or  with  the  denser  and  robuster  shells  of  the 
foraminifera.  The  effect  of  scale',  then,  of  which  we  had  so  much 
to  say  in  our  introductory  chapter  on  Magnitude,  is  bound  to  be 
apparent  in  the  study  of  skeletal  fabrics,  and  to  lead  to  essential 
differences  between  the  big  and  the  Httle,  the  massive  and  the 
minute,  in  regard  to  their  controlling  forces  and  resultant  forms. 
And  if  all  this  be  so,  and  if  the  range  of  action  of  the  molecular 
forces  be  now  the  important  and  fundamental  thing,  then  we  may 
somewhat  extend  our  statement  of  the  case,  and  include  among  our 
directive  or  constructive  influences  not  only  association  with  the 
living  cellular  elements  of  the  body,  but  also  association  with  any 
bubbles,  drops,  vacuoles  or  vesicles  which  may  be  comprised  within 
the  bounds  of  the  organism,  and  which  are  (as  their  names  and 


674  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

characters  connote)  of  the  order  of  magnitude  of  which  we  are 
speaking. 

Proceeding  a  Httle  farther  in  our  classification,  we  may  conceive 
each  Httle  skeletal  element  to  be  associated  with,  and  developed  by, 
a  single  cell  or  vesicle,  or  alternatively  a  cluster  or  "system"  of 
consociat^d  cells.  In  either  case  there  are  various  possibilities. 
For  instance,  the  calcified  or  other  skeletal  material  may  tend  to 
overspread  the  entire  outer  surface  of  the  cell  or  cluster  of  cells, 
and  so  tend  to  assume  a  configuration  comparable  to  the  surface 
of  a  fluid  drop  or  aggregation  of  drops ;  this,  in  brief,  is  the  gist  and 
essence  of  our  story  of  the  foraminiferal  shell.  Another  common 
but  very  different  condition  will  arise  if,  in  the  case  of  the  cell- 
aggregates,  the  skeletal  material  tends  to  accumulate  in  the  inter- 
stices between  the  cells,  in  the  partition-walls  which  separate  them, 
or  in  the  still  more  restricted  edges,  or  junctions  between  these 
partition-walls;  conditions  such  as  these  will  go  a  long  way  to 
help  us  to  understand  many  sponge-spicules  and  an  immense 
variety  of  radiolarian  skeletons.  And  lastly  (for  the  present), 
there  is  a  possible  and  very  interesting  case  of  a  skeletal  element 
associated  with  the  surface  of  a  cell,  not  so  as  to  cover  it  like 
a  shell,  but  only  so  as  to  pursue  a  course  of  its  own  within  it, 
and  subject  to  the  restraints  imposed  by  such  confinement  to  a 
curved  and  limited  surface.  With  this  curious  condition  we  shall 
deal  immediately. 

This  preliminary  and  much  simplified  classification  of  the  lesser 
skeletal,  or  micro-skeletal,  forms  does  not  pretend  (as  is  evident 
enough)  to  completeness.  It  leaves  out  of  account  some  conforma- 
tions and  configurations  with  which  we  shall  attempt  to  deal,  and 
others  which  we  must  perforce  omit.  But  nevertheless  it  may  help 
to  clear  or  mark  our  way  towards  the  subjects  which  this  chapter 
has  to  consider,  and  the  con4itions  by  which  they  are  at  least 
partially  defined. 

Among  the  possible,  or  conceivable,  types  of  microscopic  skeletons 
let  us  begin  with  the  case  of  a  spicule,  more  or  less  simply 
linear  as  far  as  its  intrinsic  powers  of  growth  are  concerned,  but 
which  owes  its  more  complicated  form  to  a  restraint  imposed  by 
the  cell  to  which  it  is  confined,  and  within  whose  bounds  it  is  generated. 


IX]  OF  INTRACELLULAR  SPICULES  675 

The  conception  of  a  spicule  developed  under  such  conditions  came 
from  that  very  great  mathematical  physicist,  G.  F.  FitzGerald. 
Many  years  ago,  Sollas  pointed  out  that  if  a  spicule  begin  to  grow- 
in  some  particular  way,  presumably  under  the  control  or  constraint 
imposed  by  the  organism,  it  continues  to  grow  by  further  chemical 
deposition  in  the  same  form  or  direction  even  after  it  has  got  beyond 
the  boundaries  of  the  organism  or  its  cells.  This  phenomenon  is 
what  we  see  in,  and  this  imperfect  explanation  goes  so  far  to  account 
for,  the  continued  growth  in  straight  Hnes  of  the  long  calcareous 
spines  of  Globigerina  or  Hastigerina,  or  the  similarly  radiating  but 
siliceous  spicules  of  many  Radiolaria.  In  physical  language,  if  our 
crystalline  structure  has  once  begun  to  be  laid  down  in  a  definite 
orientation,  further  additions  tend  to  accrue  in  a  like  regular  fashion 
and  in  an  identical  direction:  corresponding  to  the  phenomenon 
of  so-called  "orientirte  Adsorption,"  as  described  by  Lehmann. 

In  Globigerina  or  in  Acanihocystis  the  long  needles  grow  out 
freely  into  the  surrounding  medium,  with  nothing  to  impede  their 
rectihnear  growth  and  approximately  radiate  symmetry.  But  let 
us  consider  some  simple  cases  to  illustrate  the  forms  which  a  spicule 
will  tend  to  assume  when,  striving  (as  it  were)  to  grow  straight, 
it  comes  under  some  simple  and  constant  restraint  or  compulsion. 

If  we  take  any  two  points  on  a  smooth  curved  surface,  such 
as  that  of  a  sphere  or  spheroid,  and  imagine  a  string  stretched 
between  them,  we  obtain  what  is  known  in  mathematics  as  a 
''geodesic"  curve.  It  is  the  shortest  Une  which  can  be  traced 
between  the  two  points  upon  the  surface  itself,  and  it  has  always 
the  same  direction  upon  the  surface  to  which  it  is  confined;  the 
most  famihar  of  all  cases,  from  which  the  name  is  derived,  is  that 
curve,  or  "rhumb-line,"  upon  the  earth's  surface  which  the  navi- 
gator learns  to  follow  in  the  practice  of  "great-circle  sailing,"  never 
altermg  his  direction  nor  departing  from  his  nearest  road.  Where 
the  surface  is  spherical,  the  geodesic  is  Hterally  a  "great  circle," 
a  circle,  that  is  to  say,  whose  centre  is  the  centre  of  the  sphere. 
If  instead  of  a  sphere  we  be  dealing  with  a  spheroid,  whether 
prolate  or  oblate  (that  is  to  say  a  figure  of  revolution  in  which  an 
ellipse  rotates  about  its  long  or  its  short  axis),  then  the  system  of 
geodesies  becomes  more  complicated.  For  in  it  the  elliptic  meridians 
are  all  geodesies,  and  so  is  the  circle  of  the  equator;  though  the 


676  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

circles  of  latitude  are  not  so,  any  more  than  in  the  sphere.  But 
a  line  which  crosses  the  equator  at  an  oblique  angle,  if  it  is  to-be 
geodesic,  will  go  on  so  far  and  then  turn  back  again,  winding  its 
way  in  a  continual  figure-of-eight  curve  between  two  extreme 
latitudes,  as  when  we  wind  a  ball  of  wool.  To  say,  as  we  have  done, 
that  the  geodesic  is  the  shortest  Une  between  two  points  upon  the 
surface,  is  as  much  as  to  say  that  it  is  a  trace  of  some  particular 
straight  line  upon  the  surface  in  question ;  and  it  follows  that,  if  any 
linear  body  be  confined  to  that  surface,  while  retaining  a  tendency  to 
grow  (save  only  for  its  confinement  to  that  surface)  in  a  straight  line, 
the  resultant  form  which  it  will  assume  will  be  that  of  a  geodesic. 

Let  us  now  imagine  a  spicule  whose  natural  tendency  is  to  grow 
into  a  straight  Hnear  element,  either  by  reason  of  its  own  molecular 
anisotropy  or  because  it  is  deposited  about  a  thread-Hke  axis,  and 
let  us  suppose  that  it  is  confined  either  within  a  cell-wall  or  in 
adhesion  thereto ;  its  fine  of  growth  will  be  a  geodesic  to  the  surface 
of  the  cell.  And  if  the  cell  be  an  imperfect  sphere,  or  a  more  or 
less  regular  elhpsoid,  the  spicule  will  tend  to  grow  into  one  or  other 
of  three  forms:  either  a  plane  curve  of  nearly  circular  arc;  or, 
more  commonly,  a  plane  curve  which  is  a  portion  of  an  eUipse; 
or,  most  commonly  of  all,  a  curve  which  is  a  portion  of  a  spiral  in 
space.  In  the  latter  case,  the  number  of  turns  of  the  spiral  will 
depend  not  only  on  the  length  of  the  spicule,  but  on  the  relative 
dimensions  of  the  ellipsoidal  cell,  as  well  as  on  the  angle  by  which 
the  spicule  is  inclined  to  the  elhpsoid  axes;  but  a  very  common 
case  will  probably  be  that  in  which  the  spicule  looks  at  first  sight 
to  be  a  plane  C-shaped  figure,  but  is  discovered,  on  more  careful 
inspection,  to  he  not  in  one  plane  but  in  a  more  comphcated  twist. 
This  investigation  includes  a  series  of  forms  which  are  abundantly 
represented  among  actual  sponge-spicules,  as  illustrated  in  Figs.  310 
and  311.  ^t^, 

Growth  or  motion,  when  confined  to  some  particular  curved 
surface,  may  appear  in  various  forms  and  in  unexpected  places. 
An  amoeba,  creeping  along  the  inside  or  the  outside  of  a  glass  tube, 
was  found  in  either  case  to  follow  a  winding,  spiral  path :  it  was 
really  doing  its  best  to  go  straight — in  other  words  it  was  following 
a  geodesic  or  loxodromic  path,  determined  by  whatsoever  angle  of 
obhquity  to  the  axis  of  the  tube  it  had  chanced  to  start  out  upon. 


:x] 


OF  INTRACELLULAR  SPICULES 


677 


The  spiral  bands  of  chlorophyll  in  Spirogyra,  set  at  varying  angles 
of  hehcoid  obhquity,  are  (I  take  it)  very  beautiful  examples  of  con- 
tinuous growth  under  the  restraint  of  a  cylindrical  surface. 


(Q) 


Fig.  310.     Sponge  and  holothurian  spicules. 

To  return  to  our  sponge-spicules.  If  the  spicule  be  not  restricted 
to  linear  growth,  but  have  a  tendency  to  expand,  or  to  branch  out 
from  a  main  axis,  we  shall  obtain  a  series  of  more  complex  figures, 
all  related  to  the  geodesic  system  of  curves.  A  notable  case  will 
arise  where  the  spicule  occupies,  in  the  first  instance,  the  axis  of  the 
containing  cell,  and  then,  on  reaching  its  boundary,  tends  to  branch 
or  spread  outwards.     We  shall  now  get  various  figures,  in  some  of 


Fig.  312.   An  "amphidisc" 
of  Ilyalonema. 

which  the  spicule  will  appear  as  an  axis  expanding  into  a  disc  or 
wheel  at  either  end;  and  in  other  cases,  the  terminal  disc  will  be 
replaced  by  rays  or  spokes  with  a  reflex  curvature,  corresponding 
to  the  spherical  or  elhpsoid  curvature  of  the  cell.  Such  spicules  as 
these  are  exceedingly  conmion  among  various  sponges  (Fig.  312). 


678  ON"  CONCRETIONS,  SPICULES,  ETC.  [ch. 

Furthermore,  if  these  mechanical  methods  of  conformation,  and 
others  hke  to  these,  be  the  true  cause  of  the  shapes  which  the 
spicules  assume,  it  is  plain  that  the  production  of  these  spicular 
shapes  is  not  a  specific  function  of  the  sponge,  but  that  we  should 
expect  the  same  or  similar  spicules  to  occur  in  other  organisms, 
wherever  the  conditions  of  inorganic  secretion  within  closed  cells 
were  very  much  the  same.  As  a  matter  of  fact,  in  the  sea-cucumbers, 
where  the  formation  of  intracellular  spicules  is  a  characteristic 
feature  of  the  group,  all  the  principal  types  of  conformation  which 
we  have  just  described  can  be  closely  paralleled;  indeed,  in  many 
cases,  the  forms  of  the  holothurian  spicules  are  identical  and  indis- 
tinguishable from  those  of  the  sponges*.  But  the  holothurian 
spicules  are  composed  of  calcium  carbonate  while  those  which  we 
have  just  described  in  the  case  of  sponges  are  siUceous:  this  being 
just  another  proof  of  the  fact  that  in  such  cases  as  these  the  form 
of  the  spicule  is  not  due  to  its  chemical  nature  or  molecular 
structure,  but  to  the  external  forces  to  which  it  is  subjected. 

The  broad  fact  that  the  skeleton  is  calcareous  in  certain  large 
groups  of  animals  and  calcareous  in  others  is  as  remarkable  as  its 
causes  are  obscure.  I  for  one  have  no  idea  why  some  sponge- 
skeletons  are  of  the  one  and  some  the  other,  with  never  the  least 
admixture  of  the  two;  or  why  the  diatoms  and  radiolarians  are  all 
the  one,  and  the  molluscs  and  corals  and  foraminifera  are  all  the 
other!. 

So  much  for  that  small  class  of  sponge-spicules  whose  forms  seem 
due  to  the  fact  that  they  are  developed  within,  or  under  the  restraint 
imposed  by,  the  surface  of  a  single  cell  or  vesicle.  Such  spicules 
are  usually  of  small  size  as  well  as  of  simple  form ;  and  they  are 
greatly  outstripped  in  number,  in  size,  and  in  supposed  importance 
as  guides  to  zoological  classification,  by  another  class  of  spicules. 
These  are  the  many  and  various  cases  which  we  explain  on  the 

*  See  for  instance  the  plates  in'  Theel's  Monograph  of  the  Challenger  Holo- 
thuroidea;  also  Sollas's  Telractinellida,  p.  Ixi.  Cf.  also  E.  Merke,  Studien  am  Skelet 
der  Echin^dernien,  Zool.  Jahrbiicher  (Abth.  f.  allgem.  Zoologie),  1916-19. 

t  The  particles  of  lime  and  silica  tend  to  bear  opposite  charges;  siliceous 
organisms  seem  to  flourish  in  the  colder  waters  as  the  calcareous  certainly  do  in 
warmer  seas.  And  such  facts,  or  tendencies,  as  these  may  help  some  day  to  explain 
the  phenomenon. 


IX]  OF'THE  SKELETON  OF  SPONGES  679 

assumption  that  they  develop  in  association  (of  some  sort  or  another) 
with  the  lines  of  junction,  or  boundary-edges,  of  contiguous  cells. 
They  include  the  triradiate  spicules  of  the  calcareous  sponges,  the 
quadriradiate  or  "  tetractinelHd "  spicules  which  occur  sometimes 
in  the  same  group  but  more  characteristically  in  certain  siHceous 
sponges  known  as  the  TetractineUidae,  and  perhaps  (though  these 
last  are  somewhat  harder  to  understand)  the  six-rayed  spicules  of 
the  HexactinelHds.  We  shall  come  later  on  to  more  compUcated 
skeletons  of  the  same  type  among  the  Radiolaria. 

The  spicules  of  the  calcareous  sponges  are  commonly  triradiate, 
and  the  three  radii  are  usually  inclined  to  one  another  at  nearly 
equal  angles;  in  certain  cases,  two  of  the  three  rays  are  nearly  in 
a  straight  line,  and  at  right  angles  to  the  third  *.  They  are  not  always 
in  a  plane,  but  are  often  inchned  to  one  another  in  a  trihedral 
angle,  not  easy  of  precise  measurement  under  the  microscope.  The 
three  rays  are  often  supplemented  by  a  fourth,  which  is  set  tetra- 
hedrally,  making  nearly  co-equal  angles  with  the  other  three.  The 
calcareous  spicule  consists  mainly  of  carbonate  of  lime  in  the  form 
of  calcite,  with  (according  to  von  Ebner)  some  admixture  of  soda 
and  magnesia,  of  sulphates  and  of  water.  According  to  the  sarne 
writer  there  is  no  organic  matter  in  the  spicule,  either  in  the  form 
of  an  axial  filament  or  otherwise,  and  the  appearance  of  stratifica- 
tion, often  simulating  the  presence  of  an  axial  fibre,  is  due  to  "mixed 
crystallisation"  of  the  various  constituents.  The  spicule  is  a  true 
crystal,  and  therefore  its  existence  and  its  form  are  primarily  due 
to  the  molecular  forces  of  crystallisation;  moreover  it  is  a  single 
crystal  and  not  a  group  of  crystals,  as  is  seen  by  its  behaviour  in 
polarised  fight.  But  its  axes  are  not  crystalline  axes,  its  angles  are 
variable  and  indefinite,  and  its  form  neither  agrees  with,  nor  in  any 
way  resembles,  any  one  of  the  countless,  polymorphic  forms  in 
which  calcite  is  capable  of  crystallising.  It  is  as  though  it  were 
carved  out  of  a  solid  crystal;  it  is,  in  fact,  a  crystal  under  restraint, 
a  crystal  growing,  as  it  were,  in  an  artificial  mould,  and  this  mould 
is  constituted  by  the  surrounding  cells  or  structural  vesicles  of  the 
sponge. 

*  For  very  numerous  illustrations  of  the  triradiate  and  quadriradiate  spicules 
of  the  calcareous  sponges,  see  (int.  at.),  papers  by  Dendy  {Q.J. M.S.  xxxv,  1893), 
Minchin  {P.Z.S.  1904),  Jenkin  {P.Z.S.  1908),  etc. 


680  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

We  have  already  studied  in  an  elementary  way,  but  enough  for 
our  purpose,  the  manner  in  which  three,  four  or  more  cells,  or 
bubbles,  meet  together  under  the  influence  of  surface-tension,  in 
configurations  geometrically  similar  to  what  may  be  brought  about 
by  a  uniform  distribution  of  mechanical  pressure.  And  we  have 
seen  how  surface-energy  leads  to  the  adsorption  of  certain  chemical 
substances,  first  at  the  corners,  then  at  the  edges,  lastly  in  the 
partition- walls,  of  such  an  assemblage  of  cells.  A  spicule  formed 
in  the  interior  of  such  a  mass,  starting  at  a  corner  where  four  cells 
meet  and  extending  along  the  adjacent  edges,  would  then  (in  theory) 
have  the  characteristic  form  which  the  geometry  of  the  bee's  cell 
has  taught  us,  of  four  rays  radiating  from  a  point,  and  set  at  co-equal 
angles  to  one  another  of  109°,  approximately.  Precisely  such 
"  tetractinellid "  spicules  are  often  formed. 

But  when  we  confine  ourselves  to  a  plane  assemblage  of  cells,  or 
to  the  outer  surface  of  a  mass,  we  need  only  deal  with  the  simpler 
geometry  of  the  hexagon.  In  such  a  plane  assemblage  we  find  the 
cells  meeting  one  another  in  threes ;  when  the  cells  are  uniform  in 
size  the  partitions  are  straight  lines,  and  combine  to  form  regular 
hexagons;  but  when  the  cells  are  unequal,  the  partitions  tend  to  be 
curved,  and  to'  combine  to  form  other  and  less  regular  polygons. 
Accordingly,  a  skeletal  secretion  originating  in  a  layer  or  surface 
of  cells  will  begin  at  the  corners  and  extend  to  the  edges  of  the  cells, 
and  will  thus  take  the  form  of  triradiate  spicules,  whose  rays  (in  a 
typical  case)  will  be  set  at  co-equal  angles  of  120°  (Fig.  313,  F).  This 
latter  condition  of  inequahty  will  be  open  to  modification  in  various 
ways.  It  will  be  modified  by  any  inequality  in  the  specific  tensions 
of  adjacent  cells ;  as  a  special  case,  it  will  be  apt  to  be  greatly  modified 
at  the  surface  of  the  system,  where  a  spicule  happens  to  be  formed 
in  a  plane  perpendicular  to  the  cell-layer,  so  that  one  of  its  three  rays 
lies  between  two  adjacent  cells  and  the  other  two  are  associated 
with  the  surface  of  contact  between  the  cells  and  the  surrounding 
medium;  in  such  a  case  (as  in  the  cases  considered  in  connection 
with  the  forms  of  the  cells  themselves  on  p.  494),  we  shall  tend  to 
obtain  a  spicule  with  two  equal  angles  and  one  unequal  (Fig.  313, 
A,  C) ;  in  the  last  case,  the  two  outer,  or  superficial  rays,  will  tend 
to  be  markedly  curved.  Again,  the  equiangular  condition  will  be 
departed  from,  and  more  or  less  curvature  will  be  imparted  to  the 


IX 


OF  THE  SKELETON  OF  SPONGES 


681 


rays,  wherever  the  cells  of  the  system  cease  to  be  uniform  in  size, 
and  when  the  hexagonal  symmetry  of  the  system  is  lost  accordingly. 
Lastly,  although  we  speak  of  the  rays  as  meeting  at  certain  definite 
angles,  this  statement  applies  to  their  axes  rather  than  to  the  rays 
themselves.  For  if  the  triradiate  spicule  be  developed  in  the 
interspace  between  three  juxtaposed  cells  it  is  obvious  that  its  sides 
will  tend  to  be  concave,  because  the  space  between  three  contiguous 
equal  circles  is  an  equilateral,  curvilinear  triangle ;   and  even  if  our 


Fig.  313.     Spicules  of  Granlia  and  other  calcareous  sponges. 
After  Haeckel. 

spicule  be  deposited,  not  in  the  space  between  our  three  cells,  but 
in  the  mere  thickness  of  an  intervening  wall,  then  we  may  recollect 
that  the  several  partitions  never  actually  meet  at  sharp  angles,  but 
the  angle  of  contact  is  always  bridged  over  by  an  accumulation  of 
material  (varying  in  amount  according  to  its  fluidity)  whose  boundary 
takes  the  form  of  a  circular  arc,  and  which  constitutes  the  "  bourrelet " 
of  Plateau.  In  any  sample  of  the  triradiate  spicules  of  Grantia, 
or  in  any  series  of  careful  drawings,  such  as  Haeckel's,  we  shall  find 
all  these  various  configurations  severally  and  completely  illustrated. 


682  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

The  tetrahedral,  or  rather  tetractineUid,  spicule  needs  no  further 
explanation  in  detail  (Fig.  313,  D,  E).  For  just  as  a  triradiate 
spicule  corresponds  to  the  case  of  three  cells  in  mutual  contact,  so 
does  the  four-rayed  spicule  to  that  of  a  solid  aggregate  of  four  cells : 
these  latter  tending  to  meet  one  another  in  a  tetrahedral  system, 
shewing  four  edges,  at  each  of  which  three  facets  or  partitions  meet, 
their  edges  being  inclined  to  one  another  at  equal  angles  of  about 
109° — the  "Maraldi"  angle.  And  even  in  the  case  of  a  single  layer, 
or  superficial  layer,  of  cells,  if  the  skeleton  originate  in  connection 
with  all  the  edges  of  mutual  contact,  we  shall  (in  complete  and 
typical  cases)  have  a  four-rayed  spicule,  of  which  one  straight  limb 
will  correspond  to  the  hne  of  junction  between  the  three  cells,  and 
the  other  three  limbs  (which  will  then  be  curved  Hmbs)  will  corre- 
spond to  the  three  edges  where  the  three  cells  meet  in  pairs  on  the 
surface  of  the  system. 

But  if  such  a  physical  explanation  of  the  forms  of  our  spicules  is 
to  be  accepted,  we  must  seek  for  some  physical  agency  to  explain 
the  presence  of  the  solid  material  just  at  the  junctions  or  interfaces 
of  the  cells,  and  for  the  forces  by  which  it  is  confined  to,  and  moulded 
to  the  form  of,  these  intercellular  or  interfacial  contacts.  We  owe 
to  Dreyer  the  physical  or  mechanical  theory  of  spicular  conforma- 
tion which  I  have  just  described — a  theory  which  ultimately  rests 
on  the  form  assumed,  under  surface-tension,  by  an  aggregation  of 
cells  or  vesicles.  But  this  fundamental  point  being  granted,  we 
have  still  several  possible  alternatives  by  which  to  explain  the 
details  of  the  phenomenon. 

Dreyer,  if  I  understand  him  aright,  was  content  to  assume  that 
the  soHd  material,  secreted  or  excreted  by  the  organism,  accumu- 
lates in  the  interstices  between  the  cells,  and  is  there  subjected 
to  mechanical  pressure  or  constraint  as  the  cells  get  crowded 
together  by  their  own  growth  and  that  of  the  system  generally. 
As  far  as  the  general  form  of  the  spicules  goes  such  explanation  is 
not  inadequate,  though  under  it  we  might  have  to  renounce  some 
of  our  assumptions  as  to  what  takes  j)lace  at  the  surface  of 
the  system.  But  where  a  few  years  ago  the  concept  of  secretion 
seemed  precise  enough,  we  turn  now-a-days  to  the  phenomenon  of 
adsorption  as  a  further  stage  towards  the  elucidation  of  our  facts, 
and  here  we  have  a  case  in  point.     In  the  tissues  of  our  sponge, 


IX]  OF  THE  SKELETON  OF  SPON.GES  683 

wherever  two  cells  meet,  there  we  have  a  definite  surface  of  contact, 
and  there  accordingly  we  have  a  manifestation  of  surface-energy; 
and  the  concentration  of  surface-energy  will  tend  to  be  a  maximum 
at  the  lines  or  edges  whereby  such  surfaces  are  conjoined.  Of  the 
micro-chemistry  of  the  sponge-cells  our  ignorance  is  great;  but 
(without  venturing  on  any  hypothesis  involving  the  chemical  details 
of  the  process)  we  may  safely  assert  that  there  is  an  inherent  prob- 
abihty  that  certain  substances  will  tend  to  be  concentrated  and 
ultimately  deposited  .just  in  these  hues  of  intercellular  contact  and 
conjunction.  In  other  words,  adsorptive  concentration,  under 
osmotic  pressure,  at  and  in  the  surface-film  which  bounds  contiguous 
cells,  and  especially  in  the  edges  where  these  films  meet  and  intersect, 
emerges  as  an  alternative  (and,  as  it  seems  to  me,  a  highly  preferable 
alternative)  to  Dreyer's  conception  of  an  accumulation  under 
mechanical  pressure  in  the  vacant  spaces  left  between  one  cell  and 
another. 

But  a  purely  chemical,  or  purely  molecular,  adsorption  is  not  the 
only  form  of  the  hypothesis  on  which  we  may  rely.  For  from  the 
purely  physical  point  of  view,  angles  and  edges  of  contact  between 
adjacent  cells  will  be  loci  in  the  field  of  distribution  of  surface- 
energy,  and  any  material  particles  whatsoever  will  tend  to  undergo 
a  diminution  of  freedom  on  entering  one  of  those  boundary  regions. 
Let  us  imagine  a  couple  of  soap  bubbles  in  contact  with  one  another ; 
over  the  surface  of  each  bubble  tiny  bubbles  and  droplets  glide  in 
every  direction ;  but  as  soon  as  these  find  their  way  into  the  groove 
or  re-entrant  angle  between  the  two  bubbles,  there  their  freedom 
of  movement  is  so  far  restrained,  and  out  of  that  groove  they  have 
little  tendency,  or  little  freedom,  to  emerge.  A  cognate  phenomenon 
is  to  be  witnessed  in  microscopic  sections  of  steel  or  other  metals. 
Here,  together  with  its  crystalline  structure,  the  metal  develops  a 
cellular  structure  by  reason  of  its  lack  of  homogeneity;  for  in  the 
molten  state  one  constituent  tends  to  separate  out  into  drops,  while 
the  other  spreads  over  these  and  forms  a  fihny  reticulum  between 
— -the  disperse  phase  and  the  continuous  phase  of  the  colloid  chemists. 
In  a  poHshed  section  we  easily  observe  that  the  little  particles  of 
graphite  and  other  foreign  bodies  common  in  the  matrix  have 
tended  to  aggregate  themselves  in  the  walls  and  at  the  angles  of 
the  polygonal  cells — this  being  a  direct  result  of  the  diminished 


684  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

freedom  which  they  undergo  on  entering  one  of  these  boundary 
regions.  And  the  same  phenomenon  is  turned  to  account  in  the 
various  "separation-processes"  in  which  metalHc  particles  are  caught 
up  in  the  interstices  of  a  froth,  that  is  to  say  in  the  walls  of  the 
foam-cells  or  Schaumkammern* . 

It  is  by  a  combination  of  these  two  principles,  chemical  adsorp- 
tion on  the  one  hand  and  physical  quasi-adsorption  or  concentration 
of  grosser  particles  on  the  other,  that  I  conceive  the  substance  of 
the  sponge-spicule  to  be  concentrated  and  aggregated  at  the  cell 
boundaries;  and  the  forms  of  the  triradiate  and  tetractinellid 
spicules  are  in  precise  conformity  with  this  hypothesis.  A  few 
general  matters,  and  a  few  particular  cases,  remain  to  be  considered. 
It  matters  httle  or  not  at  all  for  the  phenomenon  in  question,  what 
is  the  histological  nature  or  "grade"  of  the  vesicular  structures 
on  which  it  depends.  In  some  cases  (apart  from  sponges),  they 
may  be  no  more  than  httle  alveoh  of  an  intracellular  protoplasmic 
network,  and  this  would  seem  to  be  the  case  at  least  in  the  protozoan 
Entosolenia  aspera,  within  the  vesicular  protoplasm  of  whose  single 
cell  Mobius  has  described  tiny  spicules  in  the  shape  of  little  tetra- 
hedra  with  sunken  or  concave  sides.  It  is  probably  the  case  also 
in  the  small  beginnings  of  Echinoderm  spicules,  which  are  likewise 
intracellular  and  are  of  similar  shape.  Among  the  sponges  we  have 
many  varying  conditions.  In  some  cases  there  is  reason  to  believe 
that  the  spicule  is  formed  at  the  boundaries  of  true  cells  or  histo- 
logical units ;  but  in  the  case  of  the  larger  triradiate  or  tetractineUid 
spicules  they  far  surpass  in  size  the  actual  "cells."  We  find  them 
lying,  regularly  and  symmetrically  arranged,  between  the  "pore- 
canals"  or  "ciliated  chambers,"  and  it  is  in  conformity  with  the 
shape  and  arrangement  of  these  large  rounded  or  spheroidal  struc- 
tures that  their  shape  is  assumed. 

Again,  it  is  not  at  variance  with  our  hypothesis  to  find  that,  in 
the  adult  sponge,  the  larger  spicules  may  greatly  outgrow  the 
bounds  not  only  of  actual  cells  but  also  of  the  ciliated  chambers, 
and  may  even  appear  to  project  freely  from  the  surface  of  the 
sponge.     For  we  have  already  seen  that  the  spicule  is  capable  of 

*  The  crystalline  composition  of  iron  was  recognised  by  Hooke  in  the  Micro- 
graphia  (1665);  and  the  cellular  or  polyhedral  structure  of  the  raetal  was  clearly 
recognised  by  Reaumur,  in  his  Art  de  convertir  le  fer  forge  en  acier,  1722. 


IX]  OF  THE  SKELETON  OF  SPONGES  685 

growing,  without  marked  change  of  form,  by  further  deposition,  or 
crystalhsation,  of  layer  upon  layer  of  calcareous  molecules,  even  in 
an  artificial  solution;  and  we  are  entitled  to  believe  that  the  same 
process  may  be  carried  on  in  the  tissues  of  the  sponge,  without 
greatly  altering  the  symmetry  of  the  spicule,  long  after  it  has 
estabhshed  its  characteristic  but  non-crystalline  form  of  a  system 
of  slender  trihedral  or  tetrahedral  rays. 

Neither  is  it  of  great  importance  to  our  hypothesis  whether  the 
rayed  spicule  ^necessarily  arises  as  a  single  structure,  or  does  so  from 
separate  minute  centres  of  aggregation.  Minchin  has  shewn  that, 
in  some  cases  at  least,  the  latter  is  the  case;  the  spicule  begins, 
he  tells  us,  as  three  tiny  rods,  separate  from  one  another,  each 
developed  in  the  interspace  between  two  sister-cells,  which  are 
themselves  the  results  of  the  division  of  one  of  a  little  trio  of  cells; 
and  the  little  rods  meet  and  fuse  together  while  still  very  minute, 
when  the  whole  spicule  is  only  about  200  ^^  ^  milhmetre  long. 
At  this  stage,  it  is  interesting  to  learn  that  the  spicule  is  non- 
crystaUine;  but  the  new  accretions  of  calcareous  matter  are  soon 
deposited  in  crystaUine  form. 

This  observation  threw  difficulties  in  the  way  of  former  mechanical 
theories  of  the  conformation  of  the  spicule,  and  was  quite  at  variance 
with  Dreyer's  theory,  according  to  which  the  spicule  was  bound  to 
begin  from  a  central  nucleus  coinciding  with  the  meeting-place  of 
three  contiguous  cells,  or  rather  the  interspace  between  them.  But 
the  difficulty  is  removed  when  we  import  the  concept  of  adsorption; 
for  by  this  agency  it  is  natural  enough,  or  conceivable  enough,  that 
deposition  should  go  on  at  separate  parts  of  a  common  system  of 
surfaces;  and  if  the  cells  tend  to  meet  one  another  by  their  interfaces 
before  these  interfaces  extend  to  the  angles  and  so  complete  the 
polygonal  cell,  it  is  again  only  natural  that  the  spicule  should  first 
arise  in  the  form  of  separate  and  detached  Hmbs  or  rays. 

Among  the  "  tetractinelhd "  sponges,  whose  spicules  are  com- 
posed of  amorphous  silica  or  opal,  all  or  most  of  the  above-described 
main  types  of  spicule  occur,  and,  as  the  name  of  the  group  implies, 
the  four-rayed,  tetrahedral  spicules  are  especially  represented. 
A  somewhat  frequent  type  of  spicule  is  one  in  which  one  of  the  four 
rays  is  greatly  developed,  and  the  other  three  constitute  small 
prongs  diverging  at  equal  angles  from  the  main  or  axial  ray.    In 


686 


ON  CONCRETIONS,  SPICULES,  ETC. 


[CH. 


all  probability,  as  Dreyer  suggests,  we  have  here  had  to  do  with  a 

group  of  four  vesicles,  of  which  three  were  large  and  co-equal,  while 

i\  fourth  and  very  much  smaller  one  lay 

above  and  between  the  other  three.     In 

certain  cases  where  we  have  hkewise  one 

large  and  three  much  smaller  rays,  the 

latter  are  recurved,  as  in  Fig.  314,  a-c. 

This  type,  save  for  the  constancy  of  the 

number  of  rays  and  the  limitation  of  the 

terminal  ones  to  three,  and  save  also  for 

the  more  important  difference  that  they 

occur  only  at  one  and  not  at  both  ends  of 

the  long  axis,  is  similar  to  the  type  of 

V  y  1 1  spicule  illustrated  in  Fig.  312,  which  we 

\^  1  ^f      fl  have   explained    as    being  probably   de- 

^fflr  J  veloped  within  an  oval   cell,   by  whose 

■  ^  11  walk  its  branches  have  been  cabined  and 

I  I  confined.     But  it  is  more  probable  that 

'  we  have  here  to  do  with  a  spicule  de- 

Fig.  314.     Spicules  of  tetracti-  i  •      ^i  •  i  ^     r  r  j.\. 

nellid  sponges  (after  SoUas).  veloped  m  the  midst  of  a  group  of  three 

a-e,  anatriaenes;    d-f,  pro-  co-equal  and  more  or  less  elongated  or 

tnaenes.  cylindrical  cells  or  vesicles,  the  long  axial 

ray  corresponding  to  their  common  edge  or  line  of  contact,  and  the 

three  short  rays  having  each  lain  in  the  surface  furrow  between  two 

out  of  the  three  adjacent  cells. 

Just  as  in  the  case  of  the  little  S-shaped  spicules  formed  within 

the  bounds  of  a  single  cell,  so  also  in  the  case  of  the  larger  tetrac- 

tinellid  types  do  we  find  the  same  configurations  reproduced  among 

the  holothuroids  as  we  have  dealt  with  in  the  sponges.     The  holo- 

thurian  spicules  are  a  little  less  neatly  formed,  a  little  rougher,  than 

the  sponge-spicules,  and   certain  forms  occur  among   the   former 

group  which  do  not  present  themselves  among  the  latter;    but  for 

the  most  part  a  community  of  type  is  obvious  and  striking  (Fig.  315). 

The  very  peculiar  spicules  of  the  holothurian  Synapta,  where  a 

tiny  anchor  is  pivoted  or  hinged  on  a  perforated  plate,  are  a  puzzle 

indeed;    but  we  may  at  least  solve  part  of  the  riddle.     How  the 

hinge  is  formed,  I  do  not  know;   the  anchor  gets  its  shape,  perhaps, 

in  some  such  way  as  we  have  supposed  the  "  amphidiscs  "of -ff^^/^Zo- 


IX] 


OF  HOLOTHURIAN  SPICULES 


687 


nema  to  acquire  their  reflexed  spokes,  but  the  perforated  plate  is 
more  comprehensible.  Each  plate  starts  in  a  httle  clump  of  cells 
in  whose  boundary-walls  calcareous  matter  is  deposited,  doubtless 
by  adsorption,  the  holes  in  the  finished  plate  thus  corresponding  to 
the  cells  which  formed  it.  Close-packing  leads  to  an  arrangement 
of  six  cells  round  a  central  one,  and  the  normal  pattern  of  the  plate 
displays  this  hexagonal  configuration.  The  calcareous  plate  begins 
as  a  little  rod  whose  ends  fork,  and  then  fork  again:  in  the  same 
inevitable  trinodal  pattern  which  includes  the  "polar  furrow"  of 
the  embryologists.     The  anchor  had  been  first  formed,  and  the 


Rh 


r^^O 


Fig,  315.     Various  holothurian  spicules.     After  Theel. 

little  plate  is  added  on  beneath  it.  The  first  spicular  rudiment  of 
the  plate  may  lie  parallel  to  the  stock  of  the  anchor  or  it  may  lie 
athwart*  it.  From  the  physical  point  of  view  it  would  seem  to  be 
a  mere  matter  of  chance  'which  way  the  cluster  of  cells  happens  to 
he;  but  this  difference  of  direction  will  cause  a  certain  difference 
in  the  symmetry  of  the  resulting  plate.  It  is  this  very  difference 
which  systematic  zoologists  at  one  time  seized  upon  to  distinguish 
S.    Buskii  from  our  two   commoner   "species."     The  two   latter 

*  Cf.  S.  Becher,  Nicht-funktionelle  Korrelation  in  der  Bildung  selbstandiger 
Skeletelemente,  Zool.  Jahrbucher  (Physiol.),  xxxi,  pp.  1-189,  1912;  Hedwiga 
Wilhelmi,  Skeletbildung  der  fiisslosen  Holothurien,  ibid,  xxxvii,  pp.  493-547, 
1920;  Arch.  f.  Entw.  Mech.  xlvi,  pp.  210-258,  1920.  See  also  W.  Woodland, 
Studies  in  spicule-formation,  Q.J. M.S.  xlix,  pp.  535-559,  1906;  li,  pp.  483-509, 
1907  and  R.  Semon,  Naturgeschichte  der  Synaptiden,  Mitth.  Zool.  St.  Neapel,  vii, 
pp.  272-299,  1886.  On  the  common  species  of  Synapta,  see  Koehler,  Faune  de 
France,  Echinodermes,  1921,  pp.  188-9. 


688 


ON  CONCRETIONS,  SPICULES,  ETC. 


[CH. 


{S.  inhaerens  and  S.  digitata)  are  mainly  distinguislied  from  one 
another  by  the  number  of  holes  in  the  plate,  that  is  to  say,  by  the 
average  number  of  cells  in  the  little  cluster  of  which  the  plate  or 
spicule  was  formed.     In  many  or  perhaps  most  other  holothurians 


Fig.  316.     Development  of  anchor-plate  in  Synapta.     After  Semon. 

the  spicules  consist  of  little  perforated  plates  or  baskets,  developed 
in  the  same  way,  about  cells  or  vesicles  more  or  less  close-packed, 
and  therefore  more  or  less  symmetrically  arranged  (Fig.  316). 


Fig.  317.     Spicules  of  hexactinellid  sponges.     After  F.  E.  Schultze. 

The  six-rayed  siliceous  spicules  of  the  hexactinellid  sponges,  while 
they  are  perhaps  the  most  regular  and  beautifully  formed  spicules 
to  be  found  within  the  entire  group,  have  beert  found  very  difficult 
to  explain,  and  Dreyer  has  confessed  his  complete  inabihty  to 
account  for  their  conformation*.     But,   though  it  may  only  be 

*  Cf.  Albr.  Schwan,  Ueber  die  Funktion  des  Hexactinellidenskelets,  u.  seine 
Vergleichbarkeit  mit  dera  Radiolarienskelet,  Zool.  Jb.,  Ahth.  allg.  Zool.  u.  Physiol. 
XXXIII,  pp.  603-616,  1913;  cf.  V.  Hacker,  Bericht  uber  d.  Tripyleenausbeute 
d.  d.  Tiefsee-Exped,  Verh.  d.  zool.  Ges.  1904. 


IX]  OF  HEXACTINELLID  SPICULES  689 

throwing  the  difficulty  a  Uttle  further  back,  we  may  so  far  account 
for  them  by  considering  that  the  cells  or  vesicles  by  which  they 
are  conformed  are  not  arranged  in  what  is  known  as  '/closest 
packing,"  but  in  hnear  series;  so  that  in  their  arrangement,  and 
by  their  mutual  compression,  we  tend  to  get  a  pattern  not  of 
hexagons  but  of  squares:  or,  looking  to  the  solid,  not  of  dodeca- 
hedra  but  of  cubes  or  parallelepipeda.  This  indeed  appears  to  be 
the  case,  not  with  the  individual  cells  (in  the  histological  sense), 
but  with  the  larger  units  or  vesicles,  which  make  up  the  body  of  the 
hexactinelHd.  And  this  being  so,  the  spicules  formed  between  the 
linear,  or  cubical  series  of  vesicles,  will  have  the  same  tendency 
towards  a  "  hexactinelhd "  shape,  corresponding  to  the  angles  and 
adjacent  edges  of  a  system  of  cubes,  as  in  our  former  case  they  had 
to  a  triradiate  or  a  te tract inellid  form,  when  developed  in  connection 
with  the  angles  and  edges  of  a  system  of  hexagons,  or  a  system  of 
rhombic  dodecahedra. 

However  the  hexactinelhd  spicules  be  arranged  (and  this  is  none 
too  easy  to  determine)  in  relation  to  the  tissues  and  chambers  of 
the  sponge,  it  is  at  least  clear  that,  whether  they  lie  separate  or 
be  fused  together  in  a  composite  skeleton,  th-ey  effect  a  symmetrical 
partitioning  of  space  according  to  the  cubical  system,  in  contrast 
to  that  closer  packing  which  is  represented  and  effected  by  the 
tetrahedral  system^*. 

Histologically,  the  case  is  illustrated  by  a.  well-known  pheno- 
menon in  embryology.  In  the  segmenting  ovum,  there  is  a  tendency 
for  the  cells  to  be  budded  off  in  linear  series;  and  so  they  often 
remain,  in  rows  side  by  side,  at  least  for  a  considerable  time  and 
during  the  course  of  several  consecutive  cell  divisions.  Such  an 
arrangement  constitutes  what  the  embryologists  call  the  "radial 
type"  of  segmentationf.  But  in  what  is  described  as  the  "spiral 
type"  of  segmentation,  it  is  stated  that,  as  soon  as  the  first  hori- 
zontal furrow  has  divided  the  cells  into  an  upper  and  a  lower  layer, 
those  of  "the  upper  layer  are  shifted  in  respect  to  the  lower  layer, 

*  Chall.  Hep.,  Hexactinellida,  pis.  xvi.  liii,  Ixxvi,  Ixxxviii. 

t  See,  for  instance,  the  figures  of  Ihe  segmenting  egg  of  Synapta  (after  Selenka), 
in  Korschelt  and  Heider's  Vergleichende  Entvnckinngsgeschichte.  On  the  spiral 
type  of  segmentation  as  a  secondary  derivative,  due  to  mechanical  causes,  of  the 
"radial"  type  of  segmentation,  see  E.  B.  Wilson,  Cell-lineage  of  Nereis,  Journ, 
Morph.  VI,  p.  450,  1892. 


690  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

by  means  of  a  rotation  about  the  vertical  axis*."  It  is,  of  course, 
evident  that  the  whole  process  is  merely  that  which  is  familiar  to 
physicists  as  "close  packing."  It  is  a  very  simple  case  of  what 
Lord  Kelvin  used  to  call  "a  problem  in  tactics."  It  is  a  mere 
question  of  the  rigidity  of  the  system,  of  the  freedom  of  movement 
on  the  part  of  its  constituent  cells,  whether  or  at  what  stage  this 
tendency  to  shp  into  the  closest  propinquity,  or  position  of  minimmn 
potential,  will  be  found  to  manifest  itself. 

Lastly,  a  curious  case  is  presented  by  the  so-called  "chessman" 
spicules  of  Latrunculia  and  of  a  few  other  sponges,  where  the  spicular 
shaft  is  thickened  at  regular  intervals,  and  the  thickenings  grow 
into  whorled  and  flattened  lobes.  Dendy  suggested  that  the 
developing  spicule  is  in  a  state  of  vibration  (due  perhaps  to  the 
water-currents  of  the  sponge),  and  that  the  whorls  correspond  to 
nodes,  or  loci  of  comparative  rest,  where  the  formative  cells  tend 
to  settle  down  and  do  their  work  undisturbed.  The  position  of 
the  nodes  and  internodes  will  depend  on  many  circumstances,  on 
whether  the  spicule  be  a  fixed  rod  or  a  free  one,  straight  or  curved, 
uniform  in  section  or  tapering  towards  either  end.  In  the  free  bar 
there  should  tend,  in  any  case,  to  be  a  node  in  the  middle,  and 
two  more  at  definite  distances  from  either  end.  It  so  happens  that 
in  the  forms  investigated  there  are  only  two  whorls,  the  median 
and  one  other;  but  J.  W.  Nicholson  has  calculated  the  positions 
of  these  according  to  the  vibration  theory,  and  the  theoretical 
results  are  found  to  agree  with  those  of  observation  very  closely 
indeed.  That  one  of  the  whorls  should  be  lacking  might  seem  to 
imperil  the  proof;  but  on  the  other  hand  among  large  numbers  of 
spicules  no  one  was  found  to  have  its  whorls  in  a  position  incon- 
sistent with  the  theory,  and  there  was  the  required  agreement 
between  the  shape  of  the  spicule  and  the  position  of  the  whorls. 
The  absence  of  a  third  whorl  is  explained  as  due  to  a  lack  of  the 
necessary  formative  cells  at  that  part  of  the  spiculef.  The  theory 
is  in  a  way  supported  by  recent  work  (by  R.  W.  Wood  of  Baltimore 
and  others)  on  "supersonic  vibrations,"  showing  excessively  rapid 

*  Korschelt  and  Heider,  p.  16. 

t  A.  Dendy  and  J.  W.  Nicholson,  On  the  influence  of  vibration  upon  the  form 
of  certain  sponge-spicules,  Proc.  R.S.  (B),  lxxxix,  pp.  .'^73-587,  1917;  A.  Dendy, 
The  chessman  spicules  of  the  genus  Latrunculia,  etc.,  Journ.  Quekett  Microsc.  Club, 
xm,  pp.  1-16,  1917. 


IX]  OF  HEXACTINELLID  SPICULES  691 

vibrations  in  quartz  rods,  more  rapid  even  than  Dendy's  hypothesis 
would  seem  to  require.  But  on  the  other  hand,  it  is  only  to  few 
and  even  exceptional  spicules  that  the  theory  would  seem  to  apply. 

This  question  of  the  origin  and  causation  of  the  forms  of  sponge- 
spicules,  with  which  ^we  have  now  sought  to  deal,  is  all  the  more 
important  and  all  the  more  interesting  because  it  has  been  discussed 
time  and  again,  from  points  of  view  which  are  characteristic  of 
very  different  schools  of  thought  in  biology.  Haeckel  found  in  the 
form  of  the  sponge-spicule  a  typical  illustration  of  his  theory  of 
"bio-crystallisation";  he  considered  that  these  " biocrystals "  re- 
presented something  midway — ein  Mittelding — between  an  inorganic 
crystal  and  an  organic  secretion;  that  there  was  a  ''compromise 
between  the  crystallising  efforts  of  the  calcium  carbonate  and  the 
formative  activity  of  the  fused  cells  of  the  syncytium";  and  that 
the  semi-crystalhne  secretions  of  calcium  carbonate  "were  utihsed 
by  natural  selection  as  'spicules'  for  building  up  a  skeleton,  and 
afterwards,  by  the  interaction  of  adaptation  and  heredity,  became 
modified  in  form,  and  differentiated  in  a  vast  variety  of  ways,  in 
the  struggle  for  existence*."  What  Haeckel  precisely  meant  by 
these  words  is  not  clear  to  me. 

F.  E.  Schultze,  perceiving  that  identical  forms  of  spicule  were 
developed  whether  the  material  were  crystaUine  or  non-crystaUine, 
abandoned  all  theories  based  upon  crystaUisation ;  he  simply  saw 
in  the  form  and  arrangement  of  the  spicules  something  which  was 
"best  fitted"  for  its  purpose,  that  is  to  say  for  the  support  and 
strengthening  of  the  porous  w^alls  of  the  sponge,  and  finding  clear 
evidence  of  "utility"  in  the  specific  characters  of  these  skeletal 
elements,  had  no  difficulty  in  ascribing  them  to  natural  selection. 

SoUas  and  Dreyer,  as  we  have  seen,  introduced  in  various  ways 
the  conception  of  physical  causation — as  indeed  Haeckel  himself 
had  done  in  regard  to  one  particular,  when  he  supposed  the  position 
of  the  spicules  to  be  due  to  the  constant  passage  of  the  water- 

*  "Hierbei  nahm  der  kohlensaure  Kalk  eine  halb-krystallinische  Beschaffen- 
heit  an,  und  gestaltete  sich  unter  Aufnahme  von  Krystallwasser  und  in  Verbindung 
mit  einer  geringen  Quantitat  von  organischer  Substanz  zu  jenen  individuellen, 
festen  Korpern,  welche  durch  die  naturliche  Ziichtung  als  Spicula  zur  Skeletbildung 
benutzt,  und  spaterhin  durch  die  Wechselwirkung  von  Anpassung  und  Vererbung 
ira  Kampfe  urns  Dasein  auf  das  Vielfaltigste  umgebildet  und  differenziert  wurden." 
Die  Kalkschwdmme,  i,  p.  377,  1872;  cf.  also  pp.  482,  483. 


692  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

currents;  though  even  here,  by  the  way,  if  I  understand  Haeckel 
aright,  he  was  thinking  not  of  a  direct  or  immediate  physical 
causation,  but  rather  of  one  manifesting  itself  through  the  agency 
of  natural  selection*.  SoUas  laid  stress  upon  the  "path  of  least 
resistance"  as  determining  the  direction  of  growth;  while  Dreyer 
dealt  in  greater  detail  with  the  tensions  and  pressures  to  which  the 
growing  spicule  was  exposed,  amid  the  alveolar  or  vesicular  structure 
which  was  represented  alike  by  the  chambers  of  the  sponge,  by  the 
constituent  cells,  or  by  the  minute  structure  of  the  intracellular 
protoplasm.  But  neither  of  these  writers,  so  far  as  I  can  discover, 
was  inchned  to  doubt  for  a  moment  the  received  canon  of  biology, 
which  sees  in  such  strucHires  as  these  the  characteristics  of  true 
organic  species,  the  indications  of  blood-relationship  and  family  hke- 
ness,  and  the  evidence  by  which  evolutionary  descent  throughout 
geologic  time  may  be  deduced  or  deciphered. 

Minchin,  in  a  well-known  paperf,  took  sides  with  F.  E.  Schultze, 
and  gave  his  reasons  for  dissenting  from  such  mechanical  theories  as 
those  of  SoUas  and  of  Dreyer.  For  example,  after  pointing  out 
that  all  protoplasm  contains  a  number  of  "granules"  or  micro- 
somes, contained  in  an  alveolar  framework  and  lodged  at  the  nodes 
of  a  reticulum,  he  argued  that  these  also  ought  to  acquire  a  form 
such  as  the  spicules  possess,  if  it  were  the  case  that  these  latter 
owed  their  form  to  their  similar  or  identical  position.  "If  vesicular 
tension  cannot  in  any  other  instance  cause  the  granules  at  the 
nodes  to  assume  a  tetraxon  form,  why  should  it  do  so  for  the 
sclerites?"  The  answer  is  not  far  to  seek.  If  the  force  which  the 
"mechanical"  hypothesis  has  in  view  were  simply  that  of  mechanical 
pressure,  as  between  solid  bodies,  then  indeed  we  should  expect 
that  any  substances  lying  between  the  impinging  spheres  would 
tend  to  assume  the  quadriradiate  or  "tetraxon"  form;  but  this 
conclusion  does  not  follow  at  all,  in  so  far  as  it  is  to  surface-energy 
that  we  ascribe  the  phenomenon.  Here  the  specific  nature  of  the 
substance  makes  all  the  difference.     We  cannot  argue  from  one 

♦  Op.  cit.  p.  483.  "Die  geordnete,  oft  so  sehr  regelmassige  und  zierliche  Zusam- 
mensetzung  des  Skeletsy stems  ist  zum  grossten  Theile  unmittelbares  Product 
der  Wasserstromung;  die  characteristische  Lagerung  der  Spioula  ist  von  der 
constanten  Richtung  des  Wasserstroms  hervorgebracht;  zum  kleinsten  Theile  ist 
Bie  die  Folge  von  Anpassungen  an  untergeordnete  ^ussere  Existenzbedingungen." 

t  Materials  for  a  Monograph  of  the  Ascones,  Q.J. M.S.  XL,  pp.  469-587,  1898. 


IX]  OF  THE  SKELETON  OF  SPONGES  693 

substance  to  another;  adsorptive  attraction  shews  its  effect  on  one 
and  not  on  another;  and  we  have  no  reason  to  be  surprised  if  we 
find  that  the  Httle  granules  of  protoplasmic  material,  which  as  they 
lie  bathed  in  the  more  fluid  protoplasm  have  (presumably,  and  as 
their  shape  indicates)  a  strong  surface-tension  of  their  own,  behave 
towards  the  adjacent  vesicles  in  a  very  different  fashion  to  the 
incipient  aggregations  of  calcareous  or  siliceous  matter  in  a  colloid 
medium.  "The  ontogeny  of  the  spicules,"  says  Professor  Minchin, 
"points  clearly  to  their  regular  form  being  a  phylogenetic  adaptation, 
which  has  becmne  fixed  and  handed  on  by  heredity,  appearing  in  the 
ontogeny  ai  a  prophetic  adaptation.''  And  again,  "The  forms  of  the 
spicules  are  the  result  of  adaptation  to  the  requirements  of  the 
sponge  as  a  whole,  produced  by  the  action  of  natural  selection  upon 
variation  in  every  direction."  It  would  scarcely  be  possible  to 
illustrate  more  briefly  and  more  cogently  than  by  these  few  words 
(or  the  similar  words  of  Haeckel  quoted  on  p.  691),  the  fundamental 
difference  between  the  Darwinian  conception  of  the  causation  and 
determination  of  Form,  and  that  which  is  based  on,  and  characteristic 
of,  the  physical  sciences. 

Last  of  all,  Dendy  took  a  middle  course.  While  admitting  that 
the  majority  of  sponge-spicules  are  "the  outcome  of  conditions 
which  are  in  large  part  purely  physical,"  he  still  saw  in  them  "a  very 
high  taxonomic  value,"  as  "indications  of  phylogenetic  history" 
all  on  the  ground  that  "it  seei^is  impossible  to  account  in  any  other 
way  for  the"^  fact  that  we  can  actually  arrange  the  different  forms 
in  such  well-graduated  series."  At  the  same  time  he  believed  that 
"the  vast  majority  of  spicule-characters  appear  to  be  non-adaptive," 
"that  no  one  form  of  spicule  has,  as  a  rule,  any  greater  survival- 
value  than  another,"  and  that  "the  natural  selection  of  favourable 
varieties  can  have  had  very  Uttle  to  do  with  the  matter*." 

The  quest  after  lines  and  evidences  of  descent  dominated  morpho- 
logy for  many  years,  and  preoccupied  the  minds  of  two  or  three 
generations  of  naturahsts.  We  find  it  easier  to  see  than  they  did 
that  a  graduated  or  consecutive  series  of  forms  may  be  based  on 
physical  causes,  that  forms  mathematically  akin  may  belong  to 

*  Cf.  A.  Dendy,  The  Tetraxonid  sponge-spicule :  a  study  in  evolution,  Acta 
Zoologica,  1921,  pp.  136,  146,  etc.  Cf.  also  Bye-products  of  organic  evolution, 
jQurn.  Quekett  Microscop.  Club,  xii,  pp.  65-82,  1913. 


694  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

organisms  biologically  remote,  and  that,  in  general,  mere  formal 
likeness  may  be  a  fallacious  guide  to  evolution  in  time  and  to 
relationship  by  descent  and  heredity. 

If  I  have  dealt  comparatively  briefly  with  the  inorganic  skeletons 
of  sponges,  in  spite  of  the  interest  of  the  subject  from  the  physical 
point  of  view,  it  has  been  owing  to  several  reasons.  In  the  first 
place,  though  the  general  trend  of  the  phenomena  is  clear,  it  must 
be  admijited  that  many  points  are  obscure,  and  could  only  be 
discussed  at  the  cost  of  a  long  argument.  In  the  second  place,  the 
physical  theory  is  too  often  (as  I  have  shewn)  in  conflict  with  the 
accounts  given  by  embryologists  of  the  development  of  the  spicules, 
and  with  the  current  biological  theories  which  their  descriptions 
embody;  it  is  beyond  our  scope  to  deal  with  such  descriptions 
in  detail.  Lastly,  we  find  ourselves  able  to  illustrate  the  same 
physical  principles  with  greater  clearness  and  greater  certitude  in 
another  group  of  animals,  namely  the  Radiolaria. 

The  group  of  microscopic  organisms  known  as  the  Radiolaria  is 
extraordinarily  rich  in  diverse  forms  or  "species."  I  do  not  know 
how  many  of  such  species  have  been  described  and  defined  by 
naturaUsts,  but  some  fifty  years  ago  the  number  was  said  to  be 
over  four  thousand,  arranged  in  more  than  seven  hundred  genera*; 
of  late  years  there  has  been  a  tendency  to  reduce  the  number. 
But  apart  from  the  extraordinary  multiplicity  of  forms  among  the 
Radiolaria,  there  are  certain  features  in  this  multiphcity  which 
arrest  our  attention.  Their  distribution  in  space  is  curious  and 
vague ;  many  species  are  found  all  over  the  world,  or  at  least  every 
here  and  there,  with  no  evidence  of  specific  limitations  of  geo- 
graphical habitat;  some  occur  in  the  neighbourhood  of  the  two 
poles,  some  are  confined  to  warm  and  others  to  cold  currents  of 
the  ocean.  In  time  their  distribution  is  not  less  vague:  so  much 
so  that  it  has  been  asserted  of  them  that  "from  the  Cambrian  age 
downwards,  the  families  and  even  genera  appear  identical  with 
those  now  Hving."  Lastly,  except  perhaps  in  the  case  of  a  few 
large  "colonial  forms,"  we  seldom  if  ever  find,  as  is  usual  in  most 
animals,  a  local  predominance  of  one  particular  species.     On  the 

*  Haeckel,  in  his  Challenger  Monograph,  p.  clxxxviii  (1887),  estimated  the 
number  of  known  forms  at  4314  species,  included  in  739  genera.  Of  these,  3508 
species  were  described  for  the  first  time  in  that  \i?ork. 


IX]  OF  SNOW  CRYSTALS  695 

contrary,  in  a  little  pinch  of  deep-sea  mud  or  of  some  fossil  "radio- 
larian  earth,"  we  shall  probably  find  scores,  and  it  may  be  even 
hundreds,  of  different  forms.  Moreover,  the  radiolarian  skeletons 
are  of  quite  extraordinary  delicacy  and  complexity,  in  spite  of  their 
minuteness  and  the  comparative  simplicity  of  the  "unicellular" 
organisms  within  which  they  grow;  and  these  complex  conforma- 
tions have  a  wonderful  and  unusual  appearance  of  geometric 
regularity.  All  these  general  considerations  seem  such  as  to  prepare 
us  for  some  physical  hypothesis  of  causation.  The  Httle  skeletons 
remind  us  of  such  things  as  snow-crystals  (themselves  almost  endless 
in  their  diversity),  rather  than  of  a  collection  of  animals,  constructed 
in  accordance  with  functional  needs  and  distributed  in  accordance 
with  their  fitness  for  particular  situations.  Nevertheless,  great 
efforts  have  been  made  to  attach  "a  biological  meaning"  to  these 
elaborate  structures,  and  "to  justify  the  hope  that  in  time  their 
utilitarian  character  will  be  more  completely  recognised*." 

As  Ernst  Haeckel  described  and  figured  many  hundred  "species" 
of  radiolarian  skeletons,  so  have  the  physicists  depicted  snow- 
crystals  in  several  thousand  different  forms  |.  These  owe  their 
multitudinous  variety  to  symmetrical  repetitions  of  qne  simple 
crystalline  form — a  beautiful  illustration  of  Plato's  One  among  the 
Many,  to  ev  napa  ra  iroXXd.  On  the  other  hand,  the  radiolarian 
skeleton  rings  its  endless  changes  on  combinations  of  certain  facets, 
corners  and  edges  within  a  filmy  and  bubbly  mass.  The  broad 
difference  between  the  two  is  very  plain  and  instructive. 

Kepler  studied  the  snowflake  with  care  and  insight,  though  he 
said  that  to  care  for  such  a  trifle  was  like  Socrates  measuring  the 
hop  of  a  flea.  The  first  drawings  I  know  are  by  Dominic  Cassini; 
and  if  that  great  astronomer  was  content  with  them  they  shew  how 
the  physical  sciences  lagged  behind  astronomy.  They  date  from  the 
time  when  Maraldi,  Cassini's  nephew,  was  studying  the  bee's  cell; 

*  Cf.  Gamble,  Radiolaria  (Lankester's  Treatise  on  Zoology),  i,  p.  131,  1909. 
Cf.  also  papers  by  V.  Hacker,  in  Jen.  Zeitschr.  xxxix,  p.  581,  1905;  Z.  f.  vnss. 
Zool.  Lxxxiii,  p.  336,  1905;   Arch.  f.  Protistenkunde,  ix,  p.  139,  1907;   etc. 

I  See  above,  p.  411;  and  see  (besides  the  works  quoted  there)  Kepler,  De  nive 
sexangula  (1611),  Opera,  ed.  Fritsch,  vii,  pp.  715-730;  Erasmus  Bartholin,  De 
figura  nivis,  Diss.,  Hafniae,  1661;  Dom.  Cassini,  Obs.  de  la  figure  de  la  neige 
(Abstr.),  Mem.  Acad.  R.  des  Sciences  (1666-1699),  x,  1730;  J.  C.  Wilcke,  Om  de 
naturliga  sno-figurers,  K.  V.  Akad.  Handl.  xxii,  1761. 


696 


ON  CONCRETIONS,  SPICULES,  ETC. 


[CH. 


and  they  shew  once  more  how  very  rough  his  measurements  of  the 
honeycoipb  are  bound  to  have  been. 

Crystals  He  outside  the  province  of  this  book ;  yet  snow-crystals, 
and  all  the  rest  besides,  have  much  to  teach  us  about  the  variety, 
the  beauty  and  the  very  nature  of  form.  To  begin  with,  the 
snow-crystal  is  a  regular  hexagonal  plate  or  thin  prism;   that  is  to 


Fig.  318  a.     Snow-crystals,  or  "snow -flowers."     From  Dominic  Cassini  (c.  1600). 

say,  it  shews  hexagonal  faces  above  and  below,  with  edges  set  at 
co-equal  angles  of  120°.  Ringing  her  changes  on  this  fundamental 
form,  Nature  superadds  to  the  primary  hexagon  endless  combina- 
tions of  sii^iilar  plates  or  prisms,  all  with  identical  angles  but  varying 
lengths  of  side;    and  she  repeats,   with  an  exquisite  symmetry. 


Fig.  318  6.     Snow-crystals.     From  Bentley  and  Humphreys,  1931. 

about  all  three  axes  of  the  hexagon,  whatsoever  she  may  have 
done  for  the  adornment  and  elaboration  of  one.  These  snow-crystals 
seem  (as  Tutton  says)  to  give  visible  proof  of  the  space-lattice  on 
which  their  structure  is  framed. 

The   beauty   of  a   snow-crystal   depends   on   its   mathematical 
regularity  and  symmetry;    but  somehow  the  association  of  many 


IX]  OF  THE  RADIOLARIAN  SKELETON  697 

variants  of  a  single  type,  all  related  but  no  two  the  same,  vastly 
increases  our  pleasure  and  admiration.  Such  is  the  pecuHar  beauty 
which  a  Japanese  artist  sees  in  a  bed  of  rushes  or  a  clump  of 
bamboos,  especially  when  the  wind's  ablowing;  and  such  (as  we 
saw  before)  is  the  phase-beauty  of  a  flowering  spray  when  it  shews 
every  gradation  from  opening  bud  to  fading  flower. 

The  snow-crystal  is  further  complicated,  and  its  beauty  is  notably 
enhanced,  by  minute  occluded  bubbles  of  air  or  drops  of  water,  whose 
symmetrical  form  and  arrangement  are  very  curious  and  not  always 
easy  to  explain*.  Lastly,  we  are  apt  to  see  our  snow^-crystals  after 
a  slight  thaw  has  rounded  their  sharp  edges,  and  has  heightened 
their  beauty  by  softening  their  contours. 

In  the  majority  of  cases,  the  skeleton  of  the  Radiolaria  is  com- 
posed, like  that  of  so  many  sponges,  of  silica;  in  one  large  family, 
the  Acantharia,  and  perhaps  in  some  others,  it  is  made  of  a  very 
unusual  constituent,  namely  strontium  sulphate  f.  There  is  no 
important  morphological  character  in  which  the  shells  made  of  these 
two  constituents  differ  from  one  another;  and  in  no  case  can  the 
chemical  properties  of  these  inorganic  materials  be  said  to  influence 
the  form  of  the  complex  skeleton  or  shell,  save  only  in  this  general 
way  that,  by  their  hardness,  toughness  and  rigidity,  they  give  rise 
to  a  fabric  more  slender  and  delicate  than  we  find  among  calcareous 
organisms. 

A  shght  exception  to  this  rule  is  found  in  the  presence  of  true 
crystals,  which  occur  within  the  central  capsules  of  certain  Radio- 
laria, for  instance  the  genus  Collos'phaera%.  Johannes  Miiller 
(whose  knowledge  and  insight  never  fail  to  astonish  us§)  remarked 

*  We  may  find  some  suggestive  analogies  to  these  occlusions  in  Emil  Hatschek's 
paper,  Gestalt  und  Orientirung  von  Gasblasen  in  Gelen,  Kolloid.  Ztschr.  xx,  pp. 
226-234,  1914. 

t  Biitschli,  Ueber  die  chemische  Natur  der  Skeletsubstanz  der  Acantharia, 
Zool.  Anz.  XXX,  p.  784,  1906. 

I  For  figures  of  these  crystals  see  Brandt,  F.  n.  Fl.  d.  Golfes  von  Neapel,  xiii, 
Radiolaria,  1885,  pi.  v.  Cf.  Johannes  Miiller,  Ueber  die  Thalassicollen,  etc.,  Abh.  K. 
Akad.  Wiss.  Berlin,  1858. 

§  It  is  interesting  to  think  of  the  lesser  discoveries  or  inventions,  due  to  men 
famous  for  greater  things.  Johannes  Miiller  first  used  th^e  tow-net,  and  Edward 
Forbes  first  borrowed  the  oyster-man's  dredge.  When  we  watch  a  living  polyp 
under  the  microscope  in  its  tiny  aquarium  of  a  glass-cell,  we  are  doing  what  John 
Goodsir  was  the  first  to  do;  and  the  microtome  itself  was  the  invention  of  that 
best  of  laboratory-servants,  "old  Stirling,"  Goodsir's  right-hand  man. 


698  ON  CONCRETIONS,  SPICULES.  [ch. 

that  these  were  identical  in  form  with  crystals  of  celestine,  a 
sulphate  of  strontium  and  barium;  and  Biitschli's  discovery  of 
sulphates  of  strontium  and  of  barium  in  kindred  forms  renders  it 
all  but  certain  that  they  are  actually  true  crystals  of  celestine*. 

In  its  typical  form,  the  radiolarian  body  consists  of  a  spherical 
mass  of  protoplasm,  around  which,  and  separated  from  it  by  some 
sort  of  porous  "capsule,"  lies  a  frothy  protoplasm,  bubbled  up  into 
a  multitude  of  alveoU  or  vacuoles,  filled  with  a  fluid  which  can 
scarcely  differ  much  from  sea-waterf.  According  to  their  surface- 
tension  conditions,  these  vacuoles  may  appear  more  or  less  isolated 
and  spherical,  or  joined  together  in  a  "froth"  of  polyhedral  cells; 
and  in  the  latter,  which  is  the  commoner  condition,  the  cells  tend 
to  be  of  equal  size,  and  the  resulting  polygonal  meshwork  beautifully 
regular.  In  some  cases  a  large  number  of  such  simple  individual 
organisms  are  associated  together,  forming  a  floating  colony;  and 
it  is  probable  that  many  others,  with  whose  scattered  skeletons  we 
are  alone  acquainted,  had  likewise  formed  part  of  a  colonial 
organism. 

In  contradistinction  to  the  sponges,  in  which  the  skeleton  always 
begins  as  a  loose  mass  of  isolated  spicules,  which  only  in  a  few 
exceptional  cases  (such  as  Euplectella  and  Farrea)  fuse  into  a 
continuous  network,  the  characteristic  feature  of  the  radiolarians 
lies  in  the  production  of  a  continuous  skeleton,  of  netted  mesh  or 
perforated  lacework,  sometimes  replaced  by  and  oftener  associated 
with  minute  independent  spicules.  Before  we  proceed  to  treat  of 
the  more  complex  skeletons,  we  may  begin  by  dealing  with  those 
comparatively  few  simple  cases  where  the  skeleton  is  represented 
by  loose,  separate  spicules  or  aciculae,  which  seem,  like  the  spicules 
of  Alcyonium,  to  be  isolated  formations  or  deposits,  precipitated  in 
the  colloid  matrix,  with  no  relation  to  cellular  or  vesicular  boun- 
daries. These  simple  acicular  spicules  occupy  a  definite  position 
in  the  organism.  Sometimes,  as  for  instance  among  the  fresh- water 
Heliozoa  (e.g.  Raphidiophrys),  they  lie  on  the  outer  surface  of  the 
organism,  and  not  infrequently  (when  few  in  number)  they  tend  to 

*  Celestine,  or  celestite,  is  SrS04  with  some  BaO  replacing  SrO. 

t  With  the  colloid  chemists,  we  may  adopt  (as  Rhumbler  has  done)  the  terms 
sputnoid  or  emulsoid  to  denote  an  agglomeration  of  fluid-filled  vesicles,  restricting 
the  na.me  froth  to  such  vesicles  when  filled  with  air  or  some  other  gas. 


IX] 


AND  SPICULAR  SKELETONS 


699 


collect  rouiul  the  bases  of  the  [)seiidopo(lia,  or  around  the  larger 
radiating  spicules  or  axial  rays  in  cases  where  these  latter  are 
present.  When  the  spicules  are  thus  localised  around  some  pro- 
minent centre,  they  tend  to  take  up  a  position  of  symmetry  in 
regard  to  it ;  instead  of  forming  a  tangled  or  felted  layer,  they  come 
to  lie  side  by  side,  in  a  radiating  cluster  round  the  focus.  In  other 
cases  (as  for  instance  in  the  well-known  radiolarian  Aulacantha 
scolymantha)  the  felted  layer  of  aciculae  lies  at  some  depth  below 
the  surface,  forming  a  sphere  concentric  with  the  entire  spherical 
organism.  In  either  case,  whether  the  layer  of  spicules  be  deep  or 
be  superficial,  it  tends  to  mark  a  "surface  of  discontinuity,"  a 
meeting  place  either  between  two  distinct  layers  of  protoplasm  or 
between  the  protoplasm  and  the  water  around;  and  it  is  evident 
that,  in  either  case,  there  are  manifestations  of  surface-energy  at 
the  boundary,  which  cause  the  spicules  to  be  retained  there  and  to 
take  up  their  position  in  its  plane.  The  case  is  analogous  to  that 
of  a  cirrus  cloud,  which  marks  a  surface  of  discontinuity  in  a 
stratified  atmosphere. 

We  have,  then,  to  enquire  w^hat  are  the  conditions,  apart  from 
gravity,  which  confine  an  extraneous  body  to  a  surface-film ;  and 
we  may  do  this  very  simply,  by  con- 
sidering the  surface-energy  of  the  entire 
system.  In  Fig.  319  we  have  two  fluids 
in  contact  with  one  another  (let  us  call  W 
them  water  and  protoplasm),  and  a 
body  (6)  which  may  be  immersed  in 
either,  or  may  be  restricted  to  the 
boundary  between.  We  have  here 
three  possible  "interfacial  contacts," 
each  with  its  own  specific  surface- 
energy  per  unit  of  surface  area: 
namely,  that  between  our  particle  and 
the  water  (let  us  call  it  a),  that  between 
the  particle  and  the  protoplasm  (^),  and 
that  between  water  and  protoplasm  (y).  When  the  body  lies  in 
the  boundary  of  the  two  fluids,  let  us  say  half  in  one  and  half 
in  the  other,  the  surface-energies  concerned  are  equivalent  to 
(>S'/2)  a  4-  (^/2)  ^;  but  we  nmst  also  remember  that,  by  the  presence 


Fig.  319. 


700  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

of  the  particle,  a  small  portion  (equal  to  its  sectional  area  s)  of  the 
original  contact-surface  between  water  and  protoplasm  has  been 
obliterated,  and  with  it  a  proportionate  quantity  of  energy,  equi- 
valent to  sy,  has  been  set  free.  When,  on  the  other  hand,  the  body 
lies  entirely  within,  one  or  other  fluid,  the  surface-energies  of  the 
system  (so  far  as  we  are  concerned)  are  equivalent  to  Sol  +  sy,  or 
S^  +  sy,  as  the  case  may  be.  Accordingly  as  a  be  less  or  greater 
than  ^,  the  particle  will  have  a  tendency  to  remain  immersed  in 
the  water  or  in  the  protoplasm ;  but  if  (aS/2)  (a  +  ^)  -  sy  be  less 
than  either  Sol  or  8/3,  then  the  condition  of  minimal  potential  will 
be  found  when  the  particle  lies,  as  we  have  said,  in  the  boundary 
zone,  half  in  one  fluid  and  half  in  the  other;  and,  if  we  were  to 
attempt  a  more  general  solution  of  the  problem,  we  should  have 
to  deal  with  possible  conditions  of  equilibrium  under  which  the 
necessary  balance  of  energies  would  be  attained  by  the  particle 
rising  or  sinking  in  the  boundary  zone,  so  as  to  adjust  the  relative 
magnitudes  of  the  surface-areas  concerned.  This  principle  may,  in 
certain  cases,  help  us  to  explain  the  position  even  of  a  radial  spicule, 
which  is  just  a  case  where  the  surface  of  the  sohd  spicule  is  dis- 
tributed between  the  fluids  with  a  minimal  disturbance,  or  minimal 
replacement,  of  the  original  surface  of  contact  between  the  one 
fluid  and  the  other. 

In  like  manner  we  may  provide  for  the  case  (a  common  and 
an  important  one)  where  the  protoplasm  "creeps  up"  the  spicule, 
covering  it  with  a  dehcate  film,  and  forming  catenary  curves  or 
festoons  between  one  spicule  and  another;  and  a  less  fluid  or  more 
tenacious  thread  of  protoplasm  may  serve,  like  a  solid  spicule,  to 
extend  the  more  fluid  film,  as  we  see,  for  instance,  in  Chlamydomyxa 
or  in  Gromia.  When  the  spicules  are  numerous  and  close-set,  the 
surface-film  of  protoplasm  stretching  between  them  will  tend  to  look 
like  a  layer  of  concave  or  inverted  bubbles ;  and  this  honeycombed 
bubbly  surface  is  sometimes  beautifully  regular,  to  judge  from  a 
well-known  figure  of  the  living  Globigerina* .  In  Acanthocystis  we 
have  yet  another  special  case,  where  the  radial  spicules  plunge  only 
a  certain  distance  into  the  protoplasm  of  the  cell,  being  arrested 
at  a  boundary-surface   between  an  inner  and  an  outer  layer  of 

*  See  H.  B.  Brady's   Challenger  Monograph,  pi.  Ixxvii;   and  see  the  figure  of 
Chlamydomyxa  in  Doflein's  Protozoenkunde,  p.  374. 


IX]  OF  SO-CALLED  MYONEMES  701 

cytoplasm;  here  we  have  only  to  assume  that  there  is  a 
tension  at  this  surface,  between  the  two  layers  of  protoplasm, 
sufficient  to  balance  the  tensions  which  act  directly  on  the 
spicule*. 

In  various  Acanthometridae,  besides  such  typical  characteristics 
as  the  radial  symmetry,  the  concentric  layers  of  protoplasm,  and 
the  capillary  surfaces  in  which  the  outer  vacuolated  protoplasm 
is  festooned  upon  the  projecting  radii,  we  have  another  curious 
feature.  On  the  surface  of  the  protoplasm  where  it  creeps  up  the 
sides  of  the  long  radial  spicules,  we  find  a  number  of  elongated 
bodies,  forming  in  each  case  one  or  several  little  groups,  and 
lying  neatly  arranged  in  parallel  bundles.  A  Russian  naturalist, 
Schewiakoff,  whose  views  have  been  accepted  in  the  text-books, 
tells  us  that  these  are  muscular  structures,  serving  to  raise  or 
lower  the  conical  masses  of  protoplasm  about  the  radial  spicules, 
which  latter  serve  as  so  many  "tent-poles"  or  masts,  on  which  the 
protoplasmic  membranes  are  hoisted  up;  and  the  little  elongated 
bodies  are  dignified  with  various  names,  such  as  "myonemes"  or 
"myophriscs,"  ,in  allusion  to  their  supposed  muscular  nature  f. 
This  explanation  is  by  no  means  convincing.  To  begin  with,  we 
have  precisely  similar  festoons  of  protoplasm  in  a  multitude  of  other 
cases  where  the  "myonemes"  are  lacking;  from  their  minute  size 
(0-006-0-012  mm.)  and  the  amount  of  contraction  they  are  said  to 
be  capable  of,  the  myonemes  can  hardly  be  very  efficient  instruments 
of  traction;  and  further,  for  them  to  act  (as  is  alleged)  for  a  specific 
purpose,  namely  the  "hydrostatic  regulation"  of  the  organism 
giving  it  power  to  sink  or  to  swim,  would  seem  to  imply  a  mechanism 
of  action  and  of  coordination  not  easy  to  conceive  in  these  minute 
and  simple  organisms.  The  fact  is  that  the  whole  explanation  is 
unnecessary.  Just  as  the  supposed  "hauling  up"  of  the  proto- 
plasmic festoons  may  be  at  once  explained  by  capillary  phenomena, 
so  also  (in  all  probability)  may  the  position  and  arrangement  of 
the  little  elongated  bodies.  Whatever  the  actual  nature  of  these 
bodies  may  be,  whether  they  be  truly  portions  of  differentiated 
protoplasm,  or  whether  they  be  foreign  bodies  or  spicular  structures 
(as  bodies  occupying  a  similar  position  in  other  cases  undoubtedly 

*  Cf.  Koltzoff,  Zur  Frage  der  Zellgestalt,  Anat.  Anzeiger,  xli,  p.  190,  1912. 
t  Mem.  de  VAcad.  des  Sci.,  St  Petersbourg,  xii,  Nr.  10,  1902. 


702  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

are),  we  can  explain  their  situation  on  the  surface  of  the  protoplasm, 
and  their  arrangement  around  the  radial  spicules,  all  by  the  prin- 
ciples of  capillarity. 

This  last  case  is  not  of  the  simplest;  and  I  do  not  forget  that 
my  explanation  of  it,  which  is  wholly  theoretical,  impHes  a  doubt 
of  Schewiakoif 's  statements,  founded  on  his  personal  observation. 
This  I  am  none  too  wiUing  to  do;  but  whether  it  be  justly  done 
in  this  case  or  not,  I  hold  that  it  is  in  principle  justifiable  to  look 
with  suspicion  upon  all  such  statements  where  the  observer  has 
obviously  left  out  of  account  the  physical  aspect  of  the  pheno- 
menon, and  all  the  opportunities  of  simple  explanation  which  the 
consideration  of  that  aspect  might  aiford. 

Whether  it  be  applicable  to  this  particular  and  complex  case  or 
no,  our  general  theorem  of  the  localisation  and  arrestment  of  sohd 
particles  in  a  surface-film  is  of  great  biological  significance ;  for  on 
it  depends  the  power  displayed  by  many  httle  naked  protoplasmic 
organisms  of  covering  themselves  with  an  "agglutinated"  shell. 
Sometimes,  as  in  Difflugia,  Astrorhiza  (Fig.  320)  and  others,  this 
covering  consists  of  sand-grains  picked  up  from  the  surrounding 
medium,  and  sometimes,  on  the  other  hand,  as  in  Quadrula,  it 
consists  of  sohd  particles  said  to  arise  as  inorganic  deposits  or 
concretions  within  the  protoplasm  itself,  and  to  find  their  way 
outwards  to  a  position  of  equihbrium  in  the  surface-layer;  and  in 
both  cases,  the  mutual  capillary  attractions  between  the  particles, 
confined  to  the  boundary-layer  but  enjoying  a  certain  measure  of 
freedom  therein,  tends  to  the  orderly  arrangement  of  the  particles 
one  with  another,  and  even  to  the  appearance  of  a  regular  "pattern" 
as  the  result  of  this  arrangement. 

The  "picking  up"  by  the  protoplasmic  organism  of  a  solid 
particle  with  which  "to  build  its  house"  (for  it  is  hard  to  avoid 
this  customary  use  of  figures  of  speech,  misleading  though  it  be) 
is  a  physical  phenomenon  akin  to  that  by  which  an  amoeba 
"swallows"  a  particle  of  food.  This  latter  process  has  been  repro- 
duced or  imitated  in  various  pretty  experimental  ways.  For 
instance,  Rhumbler  hsts  shewn  that  if  a  splinter  of  glass  be  covered 
with  shellac  and  brought  near  a  drop  of  chloroform  suspended  in 
water,  the  drop  takes  in  the  spicule,  robs  it  of  its  shellac  covering, 


IX]      OF  AGGLUTINATED  SKELETONS      703 


Fig.  320.     Arenaceoi;s  Foraminifera;   Astrorhiza  liinirola  and  nrc)iari(f. 
From  Brady's  Challenger  MonrxjrajJt. 


704  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

and  then  passes  it  out  again*.  In  another  case  a  thread  of  shellac, 
laid  on  a  drop  of  chloroform,  is  drawn  in  and  coiled  within  it: 
precisely  as  we  may  see  a  filament  of  Oscillatoria  ingested  by  an 
Amoeba,  and  twisted  and  coiled  within  its  cell.  It  is  all  a  question 
of  relative  surface-energies,  leading  to  different  degrees  of  "  adhesion  " 
between  the  chloroform  and  the  splinter  of  glass  or  its  shellac 
covering.  Thus  it  is  that  the  Amoeba  takes  in  the  diatom,  dissolves 
off  its  proteid  covering,  and  casts  out  the  shell. 

Furthermore,  as  the  whole  phenomenon  depends  on  a  distribu- 
tion of  surface-energy,  the  amount  of  which  is  specific  to  certain 
particular  substances  in  contact  with  one  another,  we  have  no 
difficulty  in  understanding  the  selective  action  which  is  very  often 
a  conspicuous  feature  in  the  phenomenon")*.  Just  as  some  caddis- 
worms  make  their  houses  of  twigs,  and  others  of  shells  and  again 
others  of  stones,  so  some  Rhizopods  construct  their  agglutinated 
"test"  out  of  stray  sponge-spicules,  or  frustules  of  diatoms,  or 
again  of  tiny  mud  particles  or  of  larger  grains  of  sand.  In  all  these 
cases,  we  have  to  deal  with  specific  surface-energies,  and  also 
doubtless  with  differences  in  the  total  available  amount  of  surface- 

*  Rhumbler,  Physikalische  Analyse  von  Lebenserscheinungen  der  Zelle,  Arch, 
f.  Entw.  Mech.  vii,  p.  250,  1898. 

t  The  whole  phenomenon  has  been  described  as  a  "surprising  exhibition  of 
constructive  and  selective  activity,"  and  ascribed,  in  varying  phraseology,  to 
inteUigence,  skill,  purpose,  psychical  activity,  or  "microscopic  mentality":  that  is 
to  say,  to  Galen's  t(x^lk7]  (pvais,  or  "artistic  creativeness "  (cf.  Brock's  Galen,  1916, 
p.  xxix);  cf.  Carpenter,  Mental  Physiology,  1874,  p.  41;  Norman,  Architectural 
achievements  of  Little  Masons,  etc.,  Ann.  Mag.  Nat.  Hist.  (5),  i,  p.  284,  1878;  Heron- 
Allen,  Contributions ...  to  the  study  of  the  Foraminifera,  Phil.  Trans.  (B),  ccvi, 
pp.  227-279,  1915;  Theory  and  phenomena  of  purpose  and  intelligence  exhibited  by 
the  Protozoa,  as  illustrated  by  selection  and  behaviour  in  the  Foraminifera,  Journ. 
R.  Microsc.  Soc.  1915,  pp.  .547-.557;  ibid.,  1916,  pp.  1.37-140.  Sir  J.  A.  Thomson 
{New  Statesman,  Oct.  23,  1915)  describes  a  certain  little  foraminifer,  whose  proto- 
plasmic body  is  overlaid  by  a  crust  of  sponge-spicules,  as  "a  psycho-physical 
individuality,  whose  experiments  in  self-expression  include  a  masterly  treatment  of 
sponge-spicules,  and  illustrate  that  organic  skill  which  came  before  the  dawn  of  Art." 
tSir  Kay  Lankester  finds  it  "not  difficult  to  conceive  of  the  existence  of  a  mechanism 
in  the  protoplasm  of  the  Protozoa  which  selects  and  rejects  building-material, 
and  determines  the  shapes  of  the  structures  built,  comparable  to  that  mechanism 
which  is  assumed  to  exist  in  the  nervous  system  of  insects  and  other  animals  which 
'automatically'  go  through  wonderfully  elaborate  series  of  complicated  actions." 
And  he  agrees  with  "Darwin  and  others  [who]  have  attributed  the  building  up  of 
these  inherited  mechanisms  to  the  age-long  action  of  Natural  Selection,  and  the 
survival  of  those  individuals  possessing  qualities  or  'tricks'  of  life-saving  value," 
■Iniirn.  R.  Microsc.  Soc.  April,  1916,  p.  136. 


IX]      OF  AGGLUTINATED  SKELETONS      705 

energy  in  relation  to  gravity  or  other  extraneous  forces.  In  my 
early  student  days,  Wyville  Thomson  used  to  tell  us  that  certain 
deep-sea  "Difflugias,"  after  constructing  a  shell  out  of  particles  of 
the  black  volcanic  sand  common  in  parts  of  the  North  Atlantic, 
finished  it  off  with  "a  clean  white  collar"  of  httle  grains  of  quartz. 
Even  this  phenomenon  may  be  accounted  for  on  surface-tension 
principles,  if  we  may  assume  that  the  surface-energy  ratios  have 
tended  to  change,  either  with  the  growth  of  the  protoplasm  or  by 
reason  of  external  variation  of  temperature  or  the  like;  we  are  by 
no  means  obhged  to  attribute  even  this  phenomenon  to  a  mani- 
festation of  volition,  or  taste,  or  aesthetic  skiU,  on  the  part  of  the 
microscopic  organism.  Nor,  when  certain  Radiolaria  tend  more 
than  others  to  attract  into  their  own  substance  diatoms  and  such- 
hke  foreign  bodies,  is  it  scientifically  correct  to  speak,  as  some 
text-books  do,  of  species  "in  which  diatom-selection  has  become 
a  regular  habits  To  do  so  is  an  exaggerated  misuse  of  anthropo- 
morphic phraseology. 

The  formation  of  an  "agglutinated"  shell  is  thus  seen  to  be  a 
purely  physical  phenomenon,  and  indeed  a  special  case  of  a  more 
general  physical  phenomenon  which  has  important  consequences  in 
biology.  For  the  shell  to  assume  the  sohd  and  permanent  character 
which  it  acquires  in  Difflugia,  we  have  only  to  make  the  further 
assumption  that  small  quantities  of  a  cementing  substance  are 
secreted  by  the  animal,  and  that  this  substance  flows  or  creeps  by 
capillary  attraction  through  all  the  interstices .  of  the  little  quartz 
grains,  and  ends  by  binding  them  together.  Rhumbler*  has  shewn 
us  how  these  agglutinated  tests  of  spicules  or  of  sand-grains  can 
be  precisely  imitated,  and  how  they  are  formed  with  greater  or  less 
ease  and  greater  or  less  rapidity  according  to  the  nature  of  the 
materials  employed,  that  is  to  say  according  to  the  specific  surface- 
tensions  which  are  involved.  If  we  mix  up  a  little  powdered  glass 
with  chloroform,  and  set  a  drop  of  the  mixture  in  water,  the  glass 
particles  gather  neatly  round  the  surface  of  the  drop  so  quickly  that 
the  eye  cannot  follow  the  operation.  If  we  do  the  same  with  oil 
and  fine  sand,   dropped  into   70  per  cent,   alcohol,   a  still  more 

*  Rhumbler,  Beitrage  z.  Kenntniss  d.  Rhizopoden,  i-v,  Z.  f.  w.  Z.  1891-o;  Das 
Protoplasma  als  physikalisches  System,  Jena,  p.  591,  1914;  also  in  Arch.  J.  Ent- 
wickelungsmech.  vii,  pp.  279-335,  1898;   Biol.  Centralbl.  xvm,  1898;   etc. 


706  ON  CONCRETIONS,  SPICULES,  [ch. 

beautiful  artificial  Rhizopod-shell  is  formed,  but  it  takes  some  three 
hours  to  do.  Where  the  action  is  quick  the  Uttle  test  forms  as  the 
droplet  exudes  from  the  pipette :  precisely  as  in  the  living  Difflugia 
when  new  protoplasm,  laden  with  soHd  particles,  is  being  extruded 
from  the  mouth  of  the  parent-cell.  The  experiment  can  be  varied, 
simply  and  easily.  Instead  of  a  spherical  drop  a  pear-shaped  one 
may  easily  be  formed,  so  exactly  like  the  common  Difflugia  pyriformis 
that  -Rhumbler  himself  was  unable,  sometimes,  to  tell  under  the 
microscope  the  real  from  the  artefact.  Again  he  found  that,  when 
the  alcohol  dissolved  the  oily  substance  of  the  (^rop  and  shrinkage 
took  place  accordingly,  the  surface-layer  with  its  solid  particles  got 
kinked  or  folded  in — and  reproduced  in  doing  so,  with  startUng 
accuracy,  a  httle  shell  of  common  occurrence,  known  by  the  generic 
name  of  Lesqueureusia,  or  Diffiugia  spiralis.  The  pecuhar  shape 
of  this  little  twisted  and  bulging  shell  has  been  taken  to  shew  that 
"it  had  enlarged  after  its  first  formation,  a  very  rare  occurrence  in  this 
group*";  the  very  opposite  is  the  case.  Neither  here  nor  in  any. 
allied  form  does  the  agglutinated  test,  once  set  in  order  by  capillary 
forces,  yield  scope  for  intercalation  and  enlargement. 

At  the  very  tinje  when  Rhumbler  was  thus  demonstrating  the 
physical  nature  of  the  Difflugian  shell,  Verworn,  a  very  notable 
person,  was  studying  the  same  and  kindred  organisms  from  the 
older  standpoint  of  an  incipient  psychology!.  But  as  Rhumbler 
himself  admits,  Verworn  (unlike  many  another)  was  doing  his  best 
not  to  over-estimate  the  appearance  of  vohtion,  or  selective  choice, 
in  the  little  organism's  use  of  materials  to  construct  its  dweUing. 

This  long  parenthesis  has  led  us  away,  for  the  time  being,  from 
the  subject  of  the  radiolarian  skeleton,  and  to  that  subject  we  must 
now  return.  Leaving  aside,  then,  the  loose  and  scattered  spicules, 
which  we  Have  sufficiently  discussed,  the  more  perfect  radiolarian 
skeletons  consist  of  a  continuous  and  regular  structure;  and  the 
sihceous  (or  other  inorganic)  material  of  which  this  framework  is 
composed  tends  to  be  deposited  in  one  or  other  of  two  ways  or  in 
both  combined:  (1)  in  long  radial  spicules,  emanating  symmetrically 
from,  and  usually  conjoined  at,  the  centre  of  the  protoplasmic  body; 

*  Cf.  Cambridge  Natural  History,  Protozoa,  p.  55. 

t  Max  Verworn,  Psycho -physiologische  Protisten-Studien,  Jena,  1889  (219  pp.); 
Biologische  Protisten-Studien,  Z.  f.  wiss.  Z.  L,  pp.  445-467,  1890. 


IX]  AND  SPIOULAR  SKELETONS  707 

(2)  in  the  form  of  a  crust,  developed  either  on  the  outer  surface  of 
the  organism  or  in  relation  to  one  or  more  of  the  internal  surfaces 
which  separate  its  concentric  layers  or  its  component  vesicles. 
Not  infrequently,  this  superficial  skeleton  comes  to  constitute  a 
spherical  shell,  or  a  system  of  concentric  spheres. 

We  have  already  seen  that  a  great  part  of  the  body  of  the 
Radiolarian,  and  especially  that  outer  portion  to  which  Haeckel 
has  given  the  name  of  the  "calymma,"  is  built  up  of  a  mass  of 
"vesicles,"  forming  a  sort  of  stiff  froth,  and  equivalent  in  the 
physical  though  not  necessarily  in  the  biological  sense  to  "cells," 
inasmuch  as  the  little  vesicles  have  their  own  well-defined  boun- 
daries, and  their  own  surface  phenomena.  In  short,  all  that  we 
have  said  of  cell-surfaces  and  cell-conformations  in  our  discussion 
of  cells  and  of  tissues  will  apply  in  like  manner,  and  under  appro- 
priate conditions,  to  these.  In  certain  cases,  even  in  so  common 
and  so  simple  a  one  as  the  vacuolated  substance  of  an  Actino- 
sphaerium,  we  may  see  a  close  resemblance,  or  formal  analogy,  to 
a  cellular  or  parenchymatous  tissue  in  the  close-packed  arrangement 
and  consequent  configuration  of  these  vesicles,  and  even  at  times 
in  a  shght  membranous  hardening  of  their  walls.  Leidy  has  figured  * 
some  curious  httle  bodies  hke  small  masses  of  consoUdated  froth, 
which  seem  to  be  nothing  else  than  the  dead  and  empty  husks,  or 
filmy  skeletons,  of  Actinosphaerium ;  and  Carnoyf  has  demon- 
strated in  certain  cell-nuclei  an  all  but  precisely  similar  framework, 
of  extreme  minuteness  and  tenuity,  formed  by  adsorption  or  partial 
sohdification  of  interstitial  matter  in  a  close-packed  system  of 
alveoh  (Fig.  321).  In  short,  we  are  again  dealing  or  about  to  deal 
with  a  network  or  basketwork,  whose  meshes  correspond  to  the 
boundary  fines  between  associated  cells  or  vesicles.  It  is  just  in 
those  boimdary  walls  or  films,  still  more  in  their  edges  or  at  their 
corners,  that  surface-energy  will  be  concentrated  and  adsorption 
will  be  hard  at  work;  and  the  whole  arrangement  will  follow,  or 
tend  to  follow,  the  rules  of  areas  minimae — the  partition-walls 
meeting  at  co-equal  angles,  three  by  three  in  an  edge,  and  their 
edges  meeting  four  by  four  in  a  corner. 

Let  us  suppose  the  outer  surface  of  our  Radiolarian  to  be  covered 

*  J.  Leidy,  Fresh-water  Rhizopods  of  North  America,  1879,  p.  262,  pi.  xli,  figs.  11,12. 
t  Carnoy,  Biologie  Cellulaire,  p.  244,  fig.  108;   cf.  Dreyer,  op.  cit.  1892,  fig.  185. 


708  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

by  a  layer  of  froth-like  vesicles,  uniform  in  size  or  nearly  so.  We 
know  that  their  mutual  tensions  will  tend  to  conform  them  into  the 
fashion  of  a  honeycomb,  or  regular  meshwork  of  hexagons,  and 
that  the  free  end  of  each  hexagonal  prism  will  be  a  little  spherical 
cap.  Suppose  now  that  it  be  at  the  outer  surface  of  the  protoplasm 
(in  contact  with  the.  surrounding  sea- water)  that  the  sihceous 
particles  have  a  tendency  to  be  secreted  or  adsorbed;  the  distribu- 
tion of  surface-energy  will  lead  them  to  accumulate  in  the  grooves 
which  separate  the  vesicles,  and  the  result  will  be  the  development 


Fig.  321.     ''''  Reticulum  plasmatique J" 
After  Carnoy. 


Fig.  322.     Aulonia  hexagona  Hkl. 


of  a  dehcate  sphere  composed  of  tiny  rods  arranged,  or  apparently 
arranged,  in  a  hexagonal  network  after  the  fashion  of  Carnoy's 
reticulum  jplasmatique,  only  more  solid,  and  still  more  neat  and 
regular.  Just  such  a  spherical  basket,  looking  like  the  finest 
imaginable  Chinese  ivory  ball,  is  found  in  the  siliceous  skeleton  of 
Aulonia,  another  of  Haeckel's  Radiolaria  from  the  Challenger. 

But  here  a  strange  thing  comes  to  light.  No  system  of  hexagons 
can  enclose  space;  whether  the  hexagons  be  equal  or  unequal, 
regular  or  irregular,  it  is  still  under  all  circumstances  mathematically 
impossible.  So  we  learn  from  Euler:  the  array  of  hexagons  may 
be  extended  as  far  as  you  please,  and  over  a  surface  either  plane  or 
curved,  but  it  never  closes  in.     Neither  our  reticulum  plasmatique 


IX] 


OF  THE  RADIOLARIAN  SKELETON 


709 


nor  what  seems  the  very  perfection  of  hexagonal  symmetry  in 
Aulonia  are  as  we  are  wont  to  conceive  them;  hexagons  indeed 
predominate  in  both,  but  a  certain  number  of  facets  are  and  must 
be  other  than  hexagonal.  If  we  look  carefully  at  Carnoy's  careful 
drawing  we  see  that  both  pentagons  and  heptagons  are  shewn  in 
his  reticulum,  and  Haeckel  actually  states,  in  his  brief  description 
of  his  Aulonia  hexagona,  that  a  few  square  or  pentagonal  facets  are 
to  be  found  among  the  hexagons. 

Such  skeletal  conformations  are  common:   and  Nature,  as  in  all 
her  handiwork,  is  quick  to  ring  the  changes  on  the  theme.     Among 


Fig,  323.    Actinomma  arcadophorum  Hkl. 

its  many  variants  may  be  found  cases  (e.g.  Actinonima)  where  the 
vesicles  have  been  less  regular  in  size;  and  others  in  which  the 
meshwork  has  been  developed  not  on  an  outer  surface  only  but  at 
successive  levels,  producing  a  system  of  concentric  spheres.  If  the 
sihceous  material  be  not  limited  to  the  linear  junctions  of  the  cells 
but  spread  over  a  portion  of  the  outer  spherical  surfaces  or  caps, 
then  we  shall  have  the  condition  represented  in  Fig.  324  (Eihmo- 
sphaera),  where  the  shell  appears  perforated  by  circular  instead 
of  hexagonal  apertures  and  the  circular  pores  are  set  on  shght 
spheroidal  eminences;   and,  interconnected  with  such  types  as  this, 


710 


ON  CONCRETIONS,  SPICULES,  ETC. 


[CH. 


we  have  others  in  which  the  accumulating  peUicles  of  skeletal  matter 
have  extended  from  the  edges  into  the  substances  of  the  boundary 
walls  and  have  so  produced  a  system  of  films,  normal  to  the  surface 
of  the  sphere,  constituting  a  very  perfect  honeycomb,  as  in  Ceno- 
sphaera  favosa  and  vesparia*. 

In  one  or  two  simple  forms,  such  as  the  fresh-water  Clathrulina, 
just  such  a  spherical  perforated  shell  is  produced  out  of  some 


Fig.  324.     Ethmosphaera  conoslphonia  Fig.  325.  Portions  of  shells 

Hkl.  of    two    "species"     of 

Cenosphaera :         upper 

figure,  C.  favosa ;  lower, 

C.  vesparia  Hkl. 

organic,  acanthin-like  substance;  and  in  some  examples  of  Clath- 
rulina the  chitinous  lattice-work  of  the  shell  is  just  as  regular  and 
delicate,  with  the  meshes  for  the  most  part  as  beautifully  hexagonal 
as  in  the  siHceous  shells  of  the  oceanic  Radiolaria.  This  is  only 
another  proof  (if  proof  be  needed)  that  the  peculiar  form  and 
character  of  these  little  skeletons  are  due  not  to  the  material  of 
which  they  are  composed,  but  to  the  moulding  of  that  material 
upon  an  underlying  vesicular  structure. 

Let  us  next  suppose  that  another  and  outer  layer  of  cells  or 
vesicles  develops   upon  some  such  lattice- work  as  has  just  been 

*  In  all  these  latter  cases  we  recognise  a  relation  to,  or  extension  of,  the  principle 
of  Plateau's  bourrelet,  or  van  der  Mensbrugghe's  masse  annulaire,  or  Gibbs's  ring, 
of  which  we  have  had  much  to  say. 


IX]  OF  THE  EADIOLARIAN  SKELETON  711 

described;  and  that  instead  of  forming  a  second  hexagonal  lattice- 
work, the  skeletal  matter  tends  to  be  developed  normally  to  the 
surface  of  the  sphere,  that  is  to  say  along  the  radial  edges  where 
the  external  vesicles  (now  compressed  into  hexagonal  prisms)  meet 
one  another  three  by  three.  The  result  will  be  that,  if  the  vesicles 
be  removed,  a  series  of  radiating  spicules  will  be  left,  directed 
outwards  from  the  angles  of  the  original  polyhedron  mesh  work,  all 
as  is  seen  in  Fig.  326.  And  it  may  further  happen  that  thes6 
radiating  skeletal  rods  branch  at  their  outer  ends  into  divergent 
rays,  forming  a  triple  fork,  and  corresponding  (after  the  fashion 


Fig.  326.     Aulaatrum  triceros  Hkl. 


which  we  have  already  described  as  occurring  in  certain  sponge- 
spicules)  to  the  superficial  furrows  between  the  three  adjacent  cells; 
this  is,  as  it  were,  a  halfway  stage  between  simple  rods  or  radial 
spicules  and  the  full  completion  of  another  sphere  of  latticed 
hexagons.  Another  possible  case,  among  many,  is  when  the  large, 
uniform  vesicles  of  the  outer  protoplasm  are  replaced  by  smaller 
vesicles,  piled  on  one  another  in  concentric  layers.  In  this  case 
the  radial  rods  will  no  longer  be  straight,  but  will  be  bent  zig-zag, 
with  their  angles  in  three  vertical  planes  corresponding  to  the 
alternate  contacts  of  the  successive  layers  of  cells  (Fig.  327). 

The  solid  skeleton  is  confined,  in  all  these  cases,  to  the  boundary- 
lines,  or  edges,  or  grooves  between  adjacent  cells  or  vesicles,  but 


712  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

adsorptive  energy  may  extend  throughout  the  intervening  walls. 
This  happens  in  not  a  few  Radiolaria,  and  in  a  certain  group  called 
the  Nassellaria  it  produces  geometrical  forms  of  peculiar  elegance 
and  mathematical  beauty. 

When  Plateau  made  the  wire  framework  of  a  regular  tetrahedron 
and  dipped  it  in  soap-solution,  he  obtained  in  an  instant  (as  we 
well  know)  a  beautifully  symmetrical  system  of  six  films,  meeting 


/^JvVvi 


Fig.  327. 


Fig.  328. .  A  Nassellarian  skeleton,  Callimitra  agnesae  Hkl. 
(0-15  mm.  diameter). 


three  by  three  in  four  edges,  and  these  four  edges  running  from  the 
corners  of  the  figure  to  its  centre  of  symmetry.  Here  they  meet, 
two  by  two,  at  the  Maraldi  angle;  and  the  films  meet  three  by 
three,  to  form  the  re-entrant  soHd  angle  which  we  have  called  a 
"Maraldi  pyramid"  in  our  account  of  the  architecture  of  the  honey- 
comb. The  very  same  configuration  is  easily  recognised  in  the 
minute  siliceous  skeleton  of  Callimitra.  There  are  two  discrepancies, 
neither  of  which  need  raise  any  difiiculty.  The  figure  is  not  a 
rectilinear  but  a  spherical  tetrahedron,  such   as   might   be  formed 


IX]  OF  THE  NASSELLARIAN  SKELETON  713 

by  the  boundary-edges  of  a  tetrahedral  cluster  of  four  co-equal 
bubbles;  and  just  as  Plateau  extended  his  experiment  by  blowing 
a  small  bubble  in  the  centre  of  his  tetrahedral  system,  so  we  have 
a  central  bubble  also  here. 

This  bubble  may  be  of  any  size*;  but  its  situation  (if  it  be 
present  at  all)  is  always  the  same,  and  its  shape  is  always  such 
as  to  give  the  Maraldi  angles  at  its  own  four  corners.     The  tensions 


A  B 

Fig.  329.    Diagrammatic  construction  of  Callimitra.    A,  a  bubble  suspended  within 
a  tetrahedral  cage.     B,  another  bubble  within  a  skeleton  of  the  former  bubble. 

of  its  own  walls,  and  those  of  the  films  by  which  it  is  supported  Or 
slung,  all  balance  one  another.  Hence  the  bubble  appears  in  plane 
projection  as  a  curvihnear  equilateral  triangle;  and  we  have  only 
got  to  convert  this  plane  diagram 
into  the  corresponding  sohd  to  obtain 
the  spherical  tetrahedron  we  have 
been  seeking  to  explain  (Fig.  329). 

We  may  make  a  simpHfied  model 
(omitting  the  central  bubble)  of  the 
tetrahedral  skeleton  of  Callimitra,  after 
the  fashion  of  that  of  the  bee's  cell 
(p.  535).  Take  OC  =^  CD  =  DB,  and 
draw  a  circle  with  radius  OB  and 
diameter  AB.    Erect  a  perpendicular  Fig.  330.  Geometrical  construction 

to  AB  at  C,  cutting  the  circle  at  E,  F.  °^  Callimitra-sl.eleton. 

AOE,  ^Oi^  will  be  (as  before)  Maraldi  angles  of  109°;  the  arcs  AE, 

*  Plateau  introduced  the  central  bubble  into  his  cube  or  tetrahedron  by  dipping 
the  cage  a  second  time,  and  so  adding  an  extra  face-film;  under  these  circum- 
stances the  bubble  has  a  definite  magnitude. 


714  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

AF  will  be  edges  of  the  spherical  tetrahedron,  and  0  will  be  the 
centre  of  s)niimetry.  Make  the  angle  FOG  =  FOA  =  EOA.  Cut  out 
the  circle  EAFG,  and  cut  through  the  radius  E0\  fold  at  AO,  FO, 
GO,  and  fasten  together,  using  EOG  for  a  flap.  Make  four  such 
sheets,  and  fasten  together  back  to  back.  The  model  will  be  much 
improved  if  little  cusps  be  left  at  the  corners  in  the  cuttmg  out. 

The  geometry  of  the  little  inner  tetrahedron  is  not  less  simple 
and  elegant.  Its  six  edges  and  four  faces  are  all  equal.  The  films 
attaching  it  to  the  outer  skeleton  are  all  planes.  Its  faces  are 
spherical,  and  each  has  its  centre  in  the  opposite  corner.  The 
edges  are  circular  arcs,  with  cosine  ^ ;  each''is  in  a  plane  perpendicular 
to  the  chord  of  the  arc  opposite,  and  each  has  its  centre  in  the  middle 
of  that  chord.  Along  each  edge  the  two  intersecting  spheres  meet 
each  other  at  an  angle  of  120°*. 

This  completes  the  elelmentary  geometry  of  the  figure;  but  one 
or  two  points  Temain  to  be  considered. 

We  may  notice  that  the  outer  edges  of  the  little  skeleton  are 
thickened  or  intensified,  and  these  thickened  edges  often  remain 
whole  or  strong  while  the  rest  of  the  surfaces  shew  signs  of  imper- 
fection or  of  breaking  away;  moreover,  the  four  corners  of  the 
tetrahedron  are  not  re-entrant  (as  in  a  group  of  bubbles)  but  a 
surplus  of  material  forms  a  little  point  or  vcusp  at  each  corner. 
In  all  this  there  is  nothing  anomalous,  and  nothing  new.  For  we 
have  already  seen  that  it  is  at  the  margins  or  edges,  and  a  fortiori 
at  the  corners,  that  the  surface-energy  reaches  its  maximum — with 
the  double  effect  of  accumulating  protoplasmic  material  in  the  form 
of  a  Gibbs's  ring  or  bourrelet,  and  of  intensifying  along  the  same 
Hues  the  adsorptive  secretion  of  skeletal  matter.  In  some  other 
tetrahedral  systems  analogous  to  Callimitra,  the  whole  of  the 
skeletal  matter  is  concentrated  along  the  boundary-edges,  and  none 
left  to  spread  over  the  boundary-planes  or  interfaces:  just  as  among 
our  spherical  Radiolaria  it  was  at  the  boundary-edges  of  their  many 
cells  or  vesicles,  and  often  there  alone,  that  skeletal  formation 
occurred,  and  gave  rise  to  the  spherical  skeleton  and  its  mesh  work 

*  For  proof,  see  Lamarle,  op.  cit.  pp.  6-8.  Lamarle  shewed  that  the  sphere 
can  be  so  divided  in  seven  ways,  but  of  these  seven  figures  the  tetrahedron  alone 
is  stable.  The  other  six  are  the  cube  and  the  regular  dodecahedron;  prisms, 
triangular  and  pentagonal,  with  equilateral  base  and  a  certain  ratio  of  base  to 
height;   and  two  polyhedra  constructed  of  pentagons  and  quadrilaterals. 


IX].  OF  THE  NASSELLARIAN  SKELETON  715 

of  hexagons.  In  the  beautiful  form  which  Haeckel  calls  Archi- 
scenium  the  boundary  edges  disappear,  the,  four  edges  converging 
on  the  median  point  are^  intensified,  and  only  three  of  the  six 
convergent  facets  are  retained;  but,  much  as  the  two  differ  in 
appearance,  the  geometry  of  this  and  of  Callimitra  remain  essentially 
the  same. 

We  learned  also  from  Plateau  that,  just  as  a  tetrahedral  bubble 
can  be  inserted  within  the  tetrahedral  skeleton  or  cage,  so  may  a 
cubical  bubble  be  introduced  within  a  cubical  cage;  and  the  edges 
of  the  inn^er  cube  will  be  just  so  curved  as  to  give  the  Maraldi  angles 
at  the  corners.  We  find  among  Haeckel's  Radiolaria  one  (he  calls 
it  Lithocubus  geometricus)  which  precisely  corresponds  to  the  skeleton 
A  B 


Fig.  331.   A,  bubble  suspended  within  i  cubical  cage. 
B,  Lithocubus  geometricus  Hkl. 


of  this  inner  cubical  bubble;  and  the  httle  spokes  or  spikes  which 
project  from  the  corners  are  parts  of  the  edges  which  once  joined 
the  corners  of  the  enclosing  figure  to  those  of  the  bubble  within 
(Fig.  331). 

Again,  if  we  construct  a  cage  in  the  form  of  an  equilateral 
triangular  prism,  and  proceed  as  before,  we  shall  probably  see  a 
vertical  edge  in  the  centre  of  the  prism  connecting  two  nodes  near 
either  end,  in  each  of  which  the  Maraldi  figure  is  displayed.  But 
if  we  gradually  shorten  our  prism  there  comes  a  point  where  the 
two  nodes  disappear,  a  plane  curvilinear  triangle  appears  horizon- 
tally in  the  middle  of  the  figure,  and  at  each  of  its  three  corners  four 
curved  edges  meet  at  the  famihar  angle.  Here  again  we  may  insert 
a  central  bubble,  which  will  now  take  the  form  of  a  curvihnear 


716  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

equilateral  triangular  prism;    and  Haeckel's  Prismatium  tripodium 
repeats  this  configuration  (Fig.  332). 

In  a  framework  of  two  crossed  rectangles,  we  may  insert  one 
bubble  after  another,  producing  a  chain  of  superposed  vesicles  whose 
shapes  vary  as  we  alter  the  relative  positions  of  the  rectangular 
frames.  Various  species  of  Triolampas,  Theocyrtis,  etc.  are  more 
or  less  akin  to  these  complicated  figures  of  equilibrium.  A  very 
beautiful  series  of  forms  may  be  made  by  introducing  successive 
bubbles   within   the   film-system  formed   by  a  tetrahedron  or  a 


Fig.  332.    Prismatium  tripodium  Hkl. 

parallelepipedon.  The  shape  and  the  curvature  of  the  bubbles 
and  of  their  suspensory  films  become  extremely  beautiful,  and  we 
have  certain  of  them  reproduced  unmistakably  in  various  Nassel- 
larian  genera,  such  as  Podocyrtis  and  its  allies. 

In  Fig.  333  we  see  a  curious  little  skeletal  structure  or  complex 
spicule,  whose  conformation  is  easily  accounted  for.  Isolated  spicules 
such  as  this  form  the  skeleton  in  the  genus  Dictyocha,  and  occur 
scattered  over  the  spherical  surface  of  the  organism  (Fig.  334).  The 
basket-shaped  spicule  has  evidently  been  developed  about  a  cluster 
of  four  cells  or  vesicles,  lying  in  or  on  the  surface  of  the  organism, 
and  therefore  arranged,  not  in  the  three-dimensional,  tetrahedral 
form  of  Callimitra,  but  in  the  manner  in  which  four  contiguous  cells 
lying  side  by  side  in  one  plane  normally  set  themselves,  like  the 


IX]  OF  THE  NASSELLARIAN  SKELETON  717 

four  cells  of  a  segmenting  egg:  that  is  to  say  with  an  intervening 
"polar  furrow,"  whose  ends  mark  the  meeting  place,  at  equal  angles, 
of  the  four  cells  in  groups  of  three.  The  Uttle  projecting  spokes,  or 
spikes,  which  are  set  normally  to  the  main  basket-work,  seem  to  be 
uncompleted  portions  of  a  larger  basket,  corresponding  to  a  more 
numerous  aggregation  of  cells.     Similar  but  more  complex  forma- 


Fig.  333.     An  isolated  portion  of  the  skeleton  of  Dityocha. 

tions,  all  explicable  as  basket-like  frameworks  developed  around 
a  cluster  of  cells,  and  adsorbed  or  secreted  in  the  grooves  common 
to  adjacent  cells  or  bubbles,  are  found  in  great  variety. 


Kg.  334.    Dictyocha  stapedia  Hkl. 

The  Dicfyocha-si^icule,  laid  down  as  a  siliceous  framework  in 
the  grooves  between  a  few  clustered  cells,  is  too  simple  and  natural 
to  be  confined  to  one  group  of  animals.  We  have  already  seen  it, 
as  a  calcareous  spicule,  in  the  holothurian  genus  Thyone,  and  we 
may  find  it  again,  in  many  various  forms,  in  the  protozoan  group 
known  as  the  Silicoflagellata*.  Nothing  can  better  illustrate  the 
physico-mathematical  character  of  these  configurations  than  their 

*  See  {iyit.  al.)  G.  Deflandre,  Les  Silicoflagelles,  etc.,  Bull.  Soc.  Fr.  de  Microscopic, 
I,  p.  1,  1932:  the  figures  in  which  article  are  mostly  drawn  from  Ehrenberg's 
Mikrogeologie,  1854. 


718  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

common  occurrence  in  diverse  groups  of  organisms.  And  the  simple 
fact  is,  that  we  seem  to  know  less  and  less  of  these  things  on  the 
biological  side,  the  more  we  come  to  understand  their  physical  and 


Fig.  335.     Various  species  of  Distephanus  (SilicojEIagellata). 
From  Deflandre,  after  Ehrenberg. 

mathematical   characters.     I    have   lost   faith    in   Haeckel's   four 
thousand  "species"  of  Radiolaria. 

In  Callimitra  itself,  and  elsewhere  where  the  boundary- walls  (and 
not  merely  their  edges)  are  sihcified,  the  skeletal  matter  is  not 


Fig.  336.     Holothurian  spicules.     A,  of  Thyone  (Mortensen) ;   B,  Holothuria  ladea 
(Perrier);   C,  D,  Holothuria  and  Phyllophorus  (Deichman). 

deposited  in  an  even  layer,  like  the  waxen  walls  of  a  bee's  cell,  but 
in  a  close  mesh  work  of  fine  curvilinear  threads;  and  the  curves 
seem  to  form  three  main  series  more  or  less  closely  related  to  the 
three  edges  of  the  partition.  Sometimes  (as  may  also  be  seen  in 
our  figure)  the  system  is  further  comphcated  by  a  radial  series 
running  from  the  centre  towards  the  free  edge  of  each  partition. 


IX]  OF  THE  NASSELLARIAN  SKELETON  719 

As  to  the  former,  their  arrangement  is  such  as  would  result  if 
deposition  or  soKdification  had  proceeded  in  waves,  starting  inde- 
pendently from  each  of  the  three  boundary-edges  of  the  Httle 
partition- wall,  and  something  of  this  kind  is  doubtless  what  actually 
happened.  We  are  reminded  of  the  wave-hke  periodicity  of  the 
Liesegang  phenomenon,  and  especially,  perhaps,  of  the  criss-cross 
rings  which  Liesegang  observed  in  frozen  gelatine  {supra,  p.  663). 
But  there  may  be  other  explanations.  For  instance  the  film,  hquid 
or  other,  which  originally  constituted  the  partition,  might  conceivably 
be  thrown  into  vibrations,  and  then  (hke  the  dust  upon  a  Chladni 
plate)  minute  particles  in  or  on  the  film  would  tend  to  take  up 
position  in  an  orderly  way,  in  relation  to  the  nodal  points  or  fines 
of  the  vibrating  surface*.  Some  such  hypothetical  vibration  may 
(to  my  thinking)  account  for  the  minute  and  varied  and  very 
beautiful  patterns  upon  many  diatoms,  the  resemblance  of  which 
patterns  to  the  Chladni  figures  (in  certain  of  their  simpler  cases) 
seems  here  and  there  striking  and  obvious.  But  I  have  not  attempted 
to  investigate  the  many  special  problems  suggested  by  the  diatom- 
skeleton. 

The  cusps  at  the  four  corners  of  the  tetrahedral  skeleton  are  a 
marked  peculiarity  of  our  Nassellarian  shell,  and  we  should  by  no 
means  expect  to  see  them  in  a  skeleton  formed  at  the  boundary- 
edges  of  ai  simple  tetrahedral  pyramid  of  four  bubbles  or  cells.  But 
when  we  introduce  another  bubble  into  the  centre  of  a  system  of 
four,  then,  as  Plateau  shewed,  the  tensions  of  its  walls  and  of  the 
surrounding  partitions  so  balance  one  another  that  it  becomes  a 
regular  curvifinear  tetrahedron,  or,  as  seen  in  plane  projection 
(Fig.  337),  a  curvilinear,  equilateral  triangle,  with  prominent,  not 
re-entrant  angles.  A  drop  of  fluid  tends  to  accumulate  at  each 
corner  where  four  edges  meet,  and  forms  a  bourreletf ;  it  is  drawn 
out  in  the  directions  of  the  four  films  which  impinge  upon  it,  and 

*  Cf.  Faraday's  beautiful  experiments,  On  the  moving  groups  of  particles  found 
on  vibrating  elastic  surfaces,  etc.,  Phil.  Trans.  1831,  p.  299;  Researches,  1859, 
pp.  314-358. 

t  The  bourrelet  is  not  on4y,  as  Plateau  expresses  it,  a  "surface  of  continuity," 
but  we  also  recognise  that  it  tends  (so  far  as  material  is  available  for  its  production) 
to  further  lessen  the  free  surface-area.  On  its  relation  to  vapour-pressure  and  to 
the  stability  of  foam,  see  FitzGerald's  interesting  note  in  Nature,  Feb.  1,  1894 
{Works,  p.  309);  and  on  its  effect  in  thinning  the  soap-bubble  to  bursting-point, 
see  Willard  Gibbs,  Coll.  Papers,  i,  p.  307  seq. 


720  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

so  tends  to  assume  in  miniature  the  very  same  shape  as  the  tetra- 
hedron to  whose  corner  it  is  attached.  Out  of  these  bourrelets,  then, 
the  cusps  at  the  four  corners  of  our  httle  skeleton  are  formed. 

A  large  and  curiously  beautiful  family  of  radiolarian  (or  "poly- 
cystine")  skeletons  look,  in  a  general  way,  Uke  tiny  hehnets  or 
Pickelhauben,  with  spike  above,  and  three  (or  sometimes  six)  curved 
lobes,  Hke  helmet-straps,  below.  We  recognise  a  family  likeness, 
even  a  mathematical  identity,  between  this  figure  and  the  last,  for 
both  alike  are  based  on  a  tetrahedral  symmetry:  the  body  of  the 
helmet  corresponding  to  the  inner  vesicle  of  Callimitra,  and  the 
spike  and  the  three  straps  to  the  four  edges  which  ran  out  from  the 
inner  to  the  outer  tetrahedron.     In  the  one  case  an  inner  vesicle 


is  surrounded  by  a  tetrahedral  figure  whose  outer  walls,  indeed,  are 
absent,  but  its  edges  remain,  and  so  do  the  walls  connecting  the 
outer  and  inner  vesicles.  In  the  other  case  the  outer  edges  are 
gone,  and  so  are  the  filmy  partition- walls,  save  parts  which  corre- 
spond to  the  four  internal  edges  between  them*.  There  are  apt 
to  be  two  slight  discrepancies.  The  helmet  is  often  of  somewhat 
complicated  form,  easily  explained  as  due  to  the  presence  of  two 
superposed  bubbles  instead  of  one.  The  other  apparent  anomaly 
is  that  the  three  helmet-straps  are  curved,  while  the  corresponding 
edges  are  straight  in  Plateau's  figure  of  the  regular  tetrahedron. 
But  it  is  a  paramount  necessity  (as  we  well  know)  for  each  set  of 
four  edges  in  a  system  of  fluid  films  to  meet  in  a  point  two  and  two 
at  the  Maraldi  angle  \    just  as  it  is  necessary  for  the  faces  to  meet 

*  Looking  through  Haeckel's  very  numerous  figures,  we  see  that  now  and  then 
something  more  is  left  than  the  mere  edges  of  the  partition-walls. 


IX] 


OF  THE  RADIOLARIAN  SKELETON 


721 


CO 


4^ 


•2  a 


00 
CO 


722  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

at  120°;  and  faces  and  edges  become  curved,  whenever  necessary, 
in  order  that  these  conditions  may  be  fulfilled.  The  need  may  arise 
in  various  ways.     Suppose  (as  in  Fig.  339)  that  our  httle  central 

bubble  be  no  longer  in  the  centre  of 
symmetry,  but  near  one  corner  of  the 
enclosing  tetrahedron;  the  short  edge 
running  out  from  that  corner  will  tend 
to  remain  straight  (and  so  form  the 
spike  of  the  helmet);  while  the  other 
three  will  each  form  an  S-shaped  curve, 
as  a  condition  of  making  co-equal  angles 
at  their  two  extremities.  An  analogous 
case  is  figured  in  one  of  Sir  David 
Brewster's  papers  where  he  repeats  and 
amplifies  some  experiments  of  Plateau's*. 
Fig.  339.  Diagram  of  one  of  the  He  made  a  tetrahedral  cage,  and  fitted 
helmet-shaped  radiolaria,  e.g.  it  with  three  more  wires,  leading  from 

Podocyrtis:  to  shew  its  tetra-   ,^  ,     ,-,  •  t  t,  •    ,     n        i    i         i 

hedral  symmetry.  ^^^  ^P®^  ^^  the  middle  pomt  of  each  basal 

edge.  On  dipping  this  into  soap-solution, 
various  comphcations  were  seen.  At  the  apex,  six  films  must  not 
meet  together,  for  no  more  than  three  surfaces  may  meet  in  an  edge ; 
intermediate  or  interstitial  films  make  their  appearance,  with  which 
we  are  not  greatly  concerned.  But  six  films  now  ascend  towards  the 
apex  from  the  base  instead  of  three,  and  the  three  which  come  from 
the  corners  have  a  longer  path  than  the  other  three  which  come  from 
the  mid-points  of  the  basal  edges ;  they  must  be  curved  in  different 
degrees  in  order  that  all  t&ree  may  make  at  either  end  their  co-equal 
angles.  And  if  we  now  introduce  a  bubble  (or  two  bubbles)  into  the 
interior  of  the  system,  we  obtain  the  characteristic  form  of  our  helmet- 
shaped  Radiolarian — the  spike  above,  the  single  or  double  vesicle 
of  the  body,  and  the  straps  or  lappets  with  their  peculiar  and 
characteristic  curvatures. 

The  Httle  shell  is  perforated  with  many  rounded  holes,  and  it 
remains  to  account  for  these.  They  are  required,  so  we  are  told, 
for  the  passage  of  pseudopodia;  well  and  good.  We  have  referred 
the  Dictyocha-si^icule  and  the  hexagonal  meshwork  of  Aulonia  to 
froth-like  associations  of  vesicles  and  to  adsorption  taking  place 
*  On  the  figures  of  equihbrium  in  liquid  films,  Trans.  R.S.E.  xxiv,  1866. 


IX]  OF  THE  RADIOLARIAN  SKELETON  723 

between;  so  far  so  good,  again.  But  the  irregular  lacework  of  the 
Uttle  helmets  does  not  suit  this  explanation  very  well,  and  there 
may  be  yet  other  possibilities.  We  have  already  mentioned 
Tomhnson's  "cohesion  figures*."  Experimenting  with  a  great 
variety  of  substances,  Tomhnson  studied  the  innumerable  ways  in 
which  drops,  jets  or  floating  films  "cohere,"  disrupt  or  otherwise 
behave,  under  the  resultant  influences  of  surface-tension,  cohesion, 
viscosity  and  friction;  in  one  case  a  film  runs  out  into  a  wavy  or 
broken  edge,  in  another  it  gives  way  here  and  there,  and  makes 
rents  or  holes  in  its  surface.  I  take  the  Httle  holes  in  our  poly  cystine 
skeleton  to  be  cohesion-figures,  in  Tomhnson's  sense  of  the  word — 
spots  where  the  delicate  film  has  given  way  and  run  into  holes,  and 
where  surface-tension  has  rounded  off  the  broken  edges,  and  made 
the  rents  into  rounded  apertures. 

In  the  foregoing  examples  of  Radiolaria,  the  symmetry  which 
the  organism  displays  seems  identical  with  that  symmetry  of  forces 
which  results  from  the  play  and  interplay  of  surface-tensions  in  the 
whole  system :  this  symmetry  being  displayed,  in  one  class  of  cases, 
in  a  more  or  less  spherical  mass  of  froth,  and  in  another  class  in 
a  simpler  aggregation  of  a  few,  otherwise  isolated,  vesicles.  In 
either  case  skeletons  are  formed,  in  great  variety,  by  one  and  the 
same  kind  of  surface-action,  namely  by  the  adsorptive  deposition 
of  sihca  in  walls  and  edges,  corresponding  to  the  manifold  surfaces 
and  interfaces  of  the  system.  But  among  the  vast  number  of 
known  Radiolaria,  there  are  certain  forms  (especially  among  the 
Phaeodaria  and  Acantharia)  which  display  a  no  less  remarkable 
symmetry  the  origin  of  which  is  by  no  means  clear,  though  surface- 
tension  may  play  a  part  in  its  causation.  Even  this  is  doubtful; 
for  the  fact  that  three-way  nodes  are  no  longer  to  be  seen  at  the 
junctions  of  the  cells  suggests  that  another  law  than  that  of  minimal 
areas  had  been  in  action  here.  They  are  cases  in  which  (as  in  some 
of  those  already  described)  the  skeleton  consists  (1)  of  radiating 
spicular  rods,  definite  in  number  and  position,  and  (2)  of  inter- 
connecting rods  or  plates,  tangential  to  the  more  or  less  spherical 
body  of  the  organism,  whose  form  becomes,  accordingly,  that  of  a 
geometric,  polyhedral  sohd.     The  great  regularity,  the  numerical 

♦  Cf.  supra,  p.  418 


724  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

symmetries  and  the  apparent  simplicity  of  these  latter  forms  makes 
of  them  a  class  apart,  and  suggests  problems  which  have  not  been 
solved  or  even  investigated. 

The  matter  is  partially  illustrated  by  the  accompanying  figures 
(Fig.  340)  from  Haeckel's  Monograph  of  the  Challenger  Radiolaria*. 
In  one  of  these  we  see  a  regular  octahedron,  in  another  a  regular, 
or  pentagonal,  dodecahedron,  in  a  third  a  regular  icosahedron.  In 
all  cases  the  figure  appears  to  be  perfectly  symmetrical,  though 
neither  the  triangular  facets  of  the  octahedron  and  icosahedron, 
nor  the  pentagonal  facets  of  the  dodecahedron,  are  necessarily  plane 
surfaces.  In  all  of  these  cases,  the  radial  spicules  correspond  to  the 
comers  of  the  figure;  and  they  are,  accordingly,  six  in  number 
in  the  octahedron,  twenty  in  the  dodecahedron,  and  twelve  in  the 
icosahedron.  If  we  add  to  these  three  figures  the  regular  tetra- 
hedron which  we  have  just  been  studying,  and  the  cube  (which  is 
represented,  at  least  in  outline,  in  the  skeleton  of  the  hexactinelhd 
sponges),  we  have  completed  the  series  of  the  five  regular  poly hedra 
known  to  geometers,  the  Platonic  bodies  "f  of  the  older  mathema- 
ticians. It  is  at  first  sight  all  the  more  remarkable  that  we  should 
here  meet  with  the  whole  five  regular  polyhedra,  when  we  remember 
that,  among  the  vast  variety  of  crystalline  forms  known  among 
minerals,  the  regular  dodecahedron  and  icosahedron,  simple  as  they 
are  from  the  mathematical  point  of  view  never  occur.  Not  only 
do  these  latter  never  occur  in  crystallography  but  (as  is  explained 
in  textbooks  of  that  science)  it  has  been  shewn  that  they  cannot 
occur,  owing  to  the  fact  that  their  indices  (or  numbers  expressing 
the  relation  of  the  faces  to  the  three  primary  axes)  involve  an 
irrational  quantity:  whereas  it  is  a  fundamental  law  of  crystallo- 
graphy, involved  in  the  whole  theory  of  space-partitioning,  that 
"the  indices  of  any  and  every  face  of  a  crystal  are  small  whole 
numbers  J."     At  the  same  time,  an  imperfect  pentagonal  dodeca- 

*  Of  the  many  thousand  figures  in  the  hundred  and  forty  plates  of  this  beautifully 
illustrated  book,  there  is  scarcely  one  which  does  not  depict  some  subtle  and 
elegant  yeornetrical  configuration. 

t  They  were  known  long  before  Plato-    llXdnov  5^  kuI  iv  tovtols  irvdayopl^ei. 

X  If  the  equation  of  any  plane  face  of  a  crystal  be  written  in  the  form 
hx  +  ky -\-lz-\,  then  h,  k,  I  are  the  indices  of  which  we  are  speaking.  They  are 
the  reciprocals  of  the  parameters,  or  reciprocals  of  the  distances  from  the  origin 
at  which  the  plane  meets  the  several  axes.  In  the  case  of  the  regular  or  pentagonal 
do«lecahedron  these  indices  are  2,  1  -r  Vo,  0.     Kepler  described  as  follows,  briefly 


IX]  OF  JOHANNES  MtJLLER'S  LAW  725 

hedron,  whose  pentagonal  sides  are  non-equilateral,  is  common 
among  crystals.  If  we  may  safely  judge  from  Haeckel's  figures, 
the  pentagonal  dodecahedron  of  the  Radiolarian  (Circorhegma)  is 
perfectly  regular,  and  we  may  rest  assured,  accordingly,  that  it  is 
not  brought  about  by  principles  of  space-partitioning  similar  to 
those  which  manifest  themselves  in  the  phenomenon  of  crystallisa- 
tion. It  will  be  observed  that  in  all  these  radiolarian  polyhedral 
shells,  the  surface  of  each  external  facet  is  formed  of  a  minute 
hexagonal  network,  whose  probable  origin,  in  relation  to  a  vesicular 
structure,  is  such  as  we  have  already  discussed. 

In  certain  allied  Radiolaria  of  the  family  Acanthometridae  (Fig. 
341),  which  have  twenty  radial  spines,  the  arrangement  of  these 
spines  is  commonly  described  in  a  somewhat  singular  way.  The 
twenty  spines  are  referred  to  five  whorls  of  four  spines  each,  arranged 
as  parallel  circles  on  the  sphere,  and  corresponding  to  the  equator, 
the  tropics  and  the  polar  circles.  This  rule  was  laid  down  by  the 
celebrated  Johannes  Miiller,  and  has  ever  since  been  used  and 
quoted  as  Miiller's  law*.  But  when  we  come  to  examine  the  figure, 
we  find  that  Miiller's  law  hardly  does  justice  to  the  facts,  and  seems 
to  overlook  a  simpler  symmetry.  We  see  in  the  first  place  that 
here,  unlike  our  former  cases,  the  twenty  radial  spines  issue  through 
the  facets  (and  all  the  facets)  of  the  polyhedron,  instead  of  coming 
from  its  corners;  and  that  our  twenty  .spines  correspond,  therefore, 
not  to  the  corners  of  a  dodecahedron,  but  to  the  facets  of  some  sort 
of  an  icosahedron.  We  see,  in  the  next  place,  that  this  icosahedron 
is  composed  of  faces  of  two  kinds,  hexagonal  and  pentagonal ;  and 
that  the  whole  figure  may  be  described  as  a  hexagonal  prism,  whose 
twelve  corners  are  truncated,  and  replaced  by  pentagonal  facets. 
Both  hexagons  and  pentagons  appear  to  be  equilateral,  but  if  we 

but  adequately,  the  common  characteristics  of  the  dodecahedron  and  icosahedron : 
"Duo  sunt  corpora  regularia,  dodecaedron  et  icosaedron,  quorum  illud  quin- 
quangulis  figuratur  expresse,  hoc  triangulis  quidem  sed  in  quinquanguli  formam 
coaptati^.  Utriusque  horum  corporum  Jpsiusque  adeo  quinquanguli  structura 
perfici  non  potest  sine  proportione  ilia,  quam  hodierni  geometrae  divinam  appellant'' 
(De  nive  sexangula  (1611),  Opera,  ed.  Fritsch,  vn,  p.  723).  Here  Kepler  was  dealing, 
somewhat  after  the  manner  of  Sir  Thomas  Browne,  with  the  mysteries  of  the 
quincunx,  and  also  of  the  hexagon;  and  was  seeking  for  an  explanation  of  the 
mysterious  or  even  mystical  beauty  of  the  5-petalled  or  3-petalled  flower — pulchri- 
tudinis  aut  proprietatis  fgurae,  quae  animam  harum  plantarum  characterisavit. 

*  See  Johannes  Miiller,  Ueber  die  Thalassicollen,  Polycistinen  und  Acantho- 
metren  des  Mittelmeeres,  Abh.  d.  Akad.  Wiss.  Berlin,  1858,  pp.  1-62,  11  pi. 


726  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 

try  to  construct  a  plane-sided  polyhedron  of  this  kind,  we  find  it 
to  be  impossible;    for  into  the  angles  between  the  six  equatorial 


Fig.  340.  Skeletons  of  various  Radiolarians,  after  Haeckel.  1,  Circoporus  sexfurcus; 
2,  C.  odahe.drus;  3,  Circogonia  icosahcdra;  4,  Circospathis  novena;  5,  Circorrhegma 
dodecahedra. 

regular  hexagons  six  regular  pentagons  will  not  fit.     The  figure, 
however,  can  be  easily  constructed  if  we  replace  the  straight  edges 


IX] 


OF  JOHANNES  MtJLLER'S  LAW 


727 


(or  some  of  them)  by  curves,  the  plane  facets  by  slightly  curved 
surfaces,  or  the  regular  by  non-equilateral  polygons*. 

In  some  cases,  such  as  Haeckel's  Phatnaspis  cristata  (Fig.  342), 
we  have  an  elHpsoidal  body  from  which  the  spines  emerge  in  the 
order  described,  but  which  is  not  obviously  divided  into  facets. 
In  Fig.  234  I  have  indicated  the  facets  corresponding  to  the  rays, 
and  dividing  the  surface  in  the  usual  symmetrical  way. 


6F, 


eF. 


-6F, 


Fig.  341.  Dorataspis  cristata  Hkl.  A,  viewed  according  to  Muller's  law:  a,  four 
polar  plates;  h,  four  intermediate  or  "tropical"  plates;  c,  four  equatorial 
plates.  B,  an  alternative  description:  Fg,  two  polar  and  six  equatorial 
hexagonal  plates;   F5,  two  rows  of  six  intermediate  pentagonal  plates. 

About  any  polyhedron  (within  or  without)  we  may  describe 
another  whose  corners  correspond  to  the  sides,  and  whose  sides  to 
the  corners,  of  the  original  figure;  or  the  one  configuration  may 
be  developed  from  the  other  by  bevelling  off,  to  a  certain  definite 
extent,  the  corners  of  the  original  polyhedron.  The  two  figures, 
thus  reciprocal  to  one  another,  form  a  "conjugate  pair,"  and  the 
principle  is  known  as  the  "principle  of  duahty"  in  polyhedral . 
Of  the   regular   sohds,   cube   and   octahedron,   dodecahedron   and 

*  Muller's  interpretation  was  emended  by  Brandt,  and  what  is  known  as  Brandt's 
Law,  viz.  that  the  symmetry  consists  of  two  polar  rays  and  three  whorls  of  six 
each,  coincides  so  far  with  the  above  description :  save  only  that  Brandt  says  plainly 
that  the  intermediate  whorls  stand  equidistant  between  the  equator  and  the  poles, 
i.e.  in  latitude  45",  which,  though  not  very  far  wrong,  is  geometrically  inaecurate. 
But  Brandt,  if  I  understand  him  rightly,  did  not  propose  his  "law"  as  a  substitute 
for  Muller's,  but  rather  as  a  second  law,  applicable  to  a  few  special  cases. 

t  First  proved  by  Legendre,  EUm.  de  Geometrie,  vii.  Prop.  25,  1794. 


728  ON  CONCRETIONS,  SPICULES,  ETC.  [ch. 


Kg.  342  A.     Phatnaspis  cristata  Hkl. 


Fig.  342  B.     The  same,  diagrammatic. 


IX]  OF  JOHANNES  MULLER'S  LAW  729 

icosahedron,  are  conjugate  pairs;  but  the  first  and  simplest  of  all 
solid  figures,  the  tetrahedron,  has  no  conjugate  but  itself. 

In  our  httle  shell  of  Dorataspis,  the  twenty  spicules  (as  we  have 
seen)  spring  from  and  correspond  to  the  twenty  facets  of  the  poly- 
hedron, twelve  pentagonal  and  eight  hexagonal,  meeting  at  thirty-six 
corners,  in  all  cases  three  by  three;  we  may  write  the  formula  of 
the  polyhedron,  accordingly,  as 

12i^5  +  8i?'6  +  36C3. 

If  we  now  connect  up  the  twenty  spicules,  three  by  three,  we 
shall  obtain  thirty-six  triangles,  completely  covering  the  figure; 
but  we  shall  find  that  of  the  twenty  corners  twelve  are  surrounded 
by  five,  and  eight  by  six  triangles.     The  formula  is  now 

362^3 +I2C5+8C6, 

and  the  two  figures  are  fully  reciprocal  or  conjugate. 

I  do  not  know  of  any  radiolarian  in  which  this  configuration  is 
to  be  found;  nor  does  it  seem  a  likely  one,  owing  to  the  large  and 
variable  number  of  edges  which  meet  in  its  corners.  But  we  may 
have  polyhedra  related  to,  or  derived  from,  one  another  in  a  less 
full  and  perfect  degree.  For  instance,  letting  the  twenty  spicules 
of  Dorataspis  again  serve  as  corners  for  the  new  figure,  let  four 
facets  meet  in  each  corner;  or  (which  comes  to  the  same  thing)  let 
each  spicule  give  off  four  branches  or  offshoots,  which  shall  meet 
their  corresponding  neighbours,  and  form  the  boundary-edges  of  a 
new  network  of  facets.  The  result  (Fig.  34'3)  is  a  symmetrical 
figure,  not  geometrically  perfect  but  elegant  in  its  own  way,  which 
we  recognise  in  a  number  of  described  forms*.  It  shews  eight 
triangular  and  fourteen  rhomboidal  facets;    and  its  formula  is 

8^3+142^4  +  2004. 

Many  subsidiary  varieties  may  arise  in  turn:  when,  for  instance, 
certain  of  the  little  branches  fail  to  meet,  or  others  grow  large  and 
widely  confluent,  always  in  symmetrical  fashion. 

We  now  see  how  in  all  such  cases  as  these  there  is  a  double 
symmetry  involved,  that  of  two  superimposed,  and  conjugate  or 
semi-conjugate,  figures.  And  the  ambiguity  which  attends  such 
descriptions  as  that  which  Johannes  Miiller  embodied  in  his  "law" 

*  Cf.  W.  Mielck,  Acanthometren  aus  Neu-Pommern,  Diss.,  Kiel,  1907. 


730 


ON  CONCRETIONS,  SPICULES,  ETC. 


[CH. 


seems  due  to  a  failure  to  recognise  this  twofold  or  alternative 
symmetry. 

In  all  these  latter  cases  it  is  the  arrangement  of  the  axial  rods — 
the  "polar  symmetry"  of  the  entire  organism — which  hes  at  "the 
root  of  the  matter;  and  which,  if  only  we  could  account  for  it, 
would  make  it  comparatively  easy  to  explain  the  superficial  con- 
figuration. But  there  are  no  obvious  mechanical  forces  by  which 
we  can  so  explain  this  peculiar  polarity.  This  at  least  is  evident, 
that  it  arises  in  the  central  mass  of  protoplasm,  which  is  the  essential 
living  portion  of  the  organism  as  distinguished  from  that  frothy 


Fig.  343.  Acanthometra  sp.  A 
derivative  of  the  Dorataspis 
figure.    After  Mielck. 


Fig,  344.    Phractaapis  prototypus  Hkl. 


peripheral  mass  whose  structure  has  helped  us  to  explain  so  many 
phenomena  of  the  superficial  or  external  skeleton.  To  say  that  the 
arrangement  depends  upon  a  specific  polarisation  of  the  cell  is 
merely  to  refer  the  problem  to  other  terms,  and  to  set  it  aside  for 
future  solution.  But  it  is  possible  that  we  may  learn  something 
about  the  fines  in  which  to  seek  for  such  a  solution  by  considering 
the  cage  of  Lehmann's  "fluid  crystals,"  and  the  light  which  they 
throw  upon  the  phenomena  of  molecular  aggregation. 

The  phenomenon  of  "fluid  crystallisation"  is  found  in  a  number 
of  chemical  bodies;  it  is  exhibited  at  a  specific  temperature  for  each 
substance;  and  it  would  seem  to  be  limited  to  bodies  in  which 
there  is  an  elongated,  or  "long-chain"  arrangement  of  the  atoms 
in  the  molecule.  Such  bodies,  at  the  appropriate  temperature,  tend 
to  aggregate  themselves  into  masses,  which  are  sometimes  spherical 


IX]  OF  LIQUID  OR  FLUID  CRYSTALS  731 

drops  or  globules  (the  so-called  "spherulites"),  and  sometimes  have 
the  definite  form  of  needle-like  or  prismatic  crystals.  In  either  case 
they  remain  Hquid,  and  are  also  doubly  refractive,  polarising  Hght 
in  brilhant  colours.  Together  with  them  are  formed  ordinary  sohd 
crystals,  also  with  characteristic  polarisation,  and  into  such  soHd 
crystals  all  the  fluid  material  ultimately  turns.  It  seems  that  in 
these  hquid  crystals,  though  the  molecules  are  freely  mobile,  just 
as  are  those  of  water,  they  are  yet  subject  to,  or  endowed  w4th, 
a  ''directive  force,"  a  force  which  confers  upon  them  a  definite 
configuration  or  "polarity,"  the  " Gestaltungskraft "  of  Lehmann. 

Such  an  hypothesis  as  this  has  been  gradually  extruded  from  the 
theories  of  mathematical  crystallography*;  and  it  has  come  to  be 
understood  that  the  s^onmetrical  conformation  of  a  homogeneous 
crystalhne  structure  is  sufficiently  explained  by  the  mere  mechanical 
fitting  together  of  appropriate  structural  units  along  the  easiest  and 
simplest  lines  of  "close  packing":  just  as  a  pile  of  oranges  becomes 
definite,  both  in  outward  form  and  inward  structural  arrangement, 
without  the  play  of  any  specific  directive  force.  But  w^hile  our 
conceptions  of  the  tactical  arrangement  of  crystalline  molecules 
remain  the  same  as  before,  and  our  hypotheses  of  "modes  of 
packing"  or  of  "space-lattices"  remain  as  useful  and  as  adequate 
as  ever  for  the  definition  and  explanation  of  the  miolecular  arrange- 
ments, a  new  conception  is  introduced  when  we  find  something  hke 
such  space-lattices  maintained  in  what  has  hitherto  been  considered 
the  molecular  freedom  of  a  liquid  field;  and  Lehmann  would  per- 
suade us,  accordingly,  to  postulate  a  specific  molecular  force,  or 
"  Gestaltungskraft "  (not  unlike  Kepler's  "facultas  formatrix"),  to 
account  for  the  phenomenon^. 

Now  just  as  some  sort  of  specific  "  Gestaltungskraft "  had  been 
of  old  the  deus  ex  machina  accounting  for  all  crystalline  phenomena 
(gnara  totius  geometriae,  et  in  ea  exercita,  as  Kepler  said),  and  as 

*  Cf.  Tutton,  Crystallography,  1911,  p.  932. 

t  Kepler,  if  I  understand  him  aright,  saw  his  way  to  account  for  the  shape  of  the 
bee's  cell  or  the  pomegranate-seed;  and  it  was  for  want  of  any  such  mechanical 
explanation,  and  as  little  more  than  a  confession  of  ignorance,  that  he  fell  back 
on  a  facultas  formatrix  to  account  for  the  six  rays  of  the  snow-crystal  or  the  five 
petals  of  the  flower.  He  was  equally  ready,  unfortunately,  to  explain,  by  the 
same  facultas  formatrix  in  acre,  the  appearance  of  a  plague  of  locusts  or  a  swarm 
of  flies. 


732  A  NOTE  ON  POLYHEDRA  [ch. 

such  an  hypothesis,  after  being  dethroned  and  repudiated,  has  now 
fought  its  way  back  and  claims  a  right  to  be  heard,  so  it  may  be 
also  in  biology.  We  begin  by  an  easy  and  general  assumption  of 
specific  properties,  by  which  each  organism  assumes  its  own  specific 
form ;  we  learn  later  (as  it  is  the  purpose  of  this  book  to  shew)  that 
throughout  the  whole  range  of  organic  morphology  there  are  innu- 
merable phenomena  of  form  which  are  not  pecuhar  to  living  things, 
but  which  are  more  or  less  simple  manifestations  of  ordinary  physical 
law.  But  every  now  and  then  we  come  to  deep-seated  signs  of 
protoplasmic  symmetry  or  polarisation,  which  seem  to  lie  beyond 
the  reach  of  the  ordinary  physical  forces.  It  by  no  means  follows 
that  the  forces  in  question  are  not  essentially  physical  forces,  more 
obscure  and  less  familiar  to  us  than  the  rest;  and  this  would  seem 
to  be  a  great  part  of  the  lesson  for  us  to  draw  from  Lehmann's 
beautiful  discovery.  For  Lehmann  claims  to  have  demonstrated, 
in  non-living,  chemical  bodies,  the  existence  of  just  such  a  deter- 
minant, just  such  a  "Gestaltungskraft,"  as  would  be  of  infinite  help 
to  us  if  we  might  postulate  it  for  the  explanation  (for  instance)  of 
our  Radiolarian's  axial  symmetry.  Further  than  this  we  cannot 
go;  such  analogy  as  we  seem  to  see  in  the  Lehmann  phenomenon 
soon  evades  us,  and  refuses  to  be  pressed  home.  The  symmetry 
of  crystallisation,  which  Haeckel  tried  hard  to  discover  and  to  reveal 
in  these  and  other  organisms,  resolves  itself  into  remote  analogies 
from  which  no  conclusions  can  be  drawn.  Many  a  beautiful 
protozoan  form  has  lent  itself  to  easy  physico-mathematical  ex- 
planation ;  others,  no  less  simple  and  no  more  beautiful  prove  harder 
to  explain.  That  Nature  keeps  some  of  her  secrets  longer  than 
others — that  she  tells  the  secret  of  the  rainbow  and  hides  that  of 
the  northern  hghts — is  a  lesson  taught  me  when  I  was  a  boy. 

A  note  on  Polyhedra. 

The  theory  of  Polyhedra,  Euler's  doctrina  solidorum,  is  a  branch 
of  geometry  which  deals  with  the  more  or  less  regular  solids;  and 
the  rudiments  of  the  theory  may  help  us  to  study  certain  more  or 
less  symmetrical  organic  forms.  Euler,  a  contemporary  of  Lin- 
naeus, is  the  most  celebrated  of  the  many  mathematicians  who 
have  carried  this  subject  beyond  where  Pythagoras,  Plato,  Euclid 
and  Archimedes  had  left  it.     He  drew  up  a  classification  of  poly- 


IX]  OF  EULER'S  LAW  733 

hedral  solids,  using  a  binomial  nomenclature  based  on  the  number 
of  their  corners  or  vertices,  and  sides  or  faces.  Thus,  for  example, 
he  called  a  figure  with  eight  corners  and  seven  faces  Octogonum 
hsptaedrum;  and  the  analogy  between  this  and  Linnaeus's  botanical 
classification  and  nomenclature — e.g.  Hexandria  trigynia  and  the 
rest — is  very  close  and  curious. 

A  simple  theorem,  of  which  Euler  was  vastly  proud  and  which 
we  still  speak  of  as  Euler 's  Law*,  is  fundamental  to  the  theory  of 
polyhedra.  It  tells  us  that  in  every  polyhedron  whatsoever,  the 
faces  and  corners  together  outnumber  the  edges  by  two  t  : 

C-E+F=2  (1). 

Another  fundamental  theorem  follows.  We  know  from  Euchd 
that  the  three  angles  of  a  triangle  are  equal  to  two  right  angles; 
consequently,  that  in  a  polygon  of  C  angles,  the  sum  of  the  angles 
-=  2  (C  —  2)  right  angles.  And  there  follows  from  this — but  by  no 
means   expectedly — the   analogous  and  extremely  simple  relation 

*  Euler,  Elementa  doctrinae  solidorum,  Novi  Comment.  Acad.  Sci.  Imp. 
Petropol.  IV,  p.  109  seq.  (ad  annos  17.52  et  1753),  1758:  "In  omni  solido  hedris 
planis  inclusum,  aggregatura  ex  numero  angulorum  solidorum  et  ex  numero 
hedraruiu  binario  excedit  numerum  acierum."  For  a  proof,  see  {int.  al.)  De 
Morgan,  article  Polyhedron  in  the  Penny  Cyclopaedia.  There  is  reason  to  believe 
that  Descartes  was  acquainted  with  this  theorem  between  1672  and  1676;  cf. 
Foucher  de  Careil,  (Euvres  inedites  de  Descartes,  Paris,  ii,  p.  214.  Cf.  Baltzer, 
Monatsber.  Berlin.  Akad.  1861,  p.  1043;  and  de  Jonquieres,  C.R.  1890,  p.  261. 
(The  student  will  be  struck  by  the  resemblance  between  this  formula  and  the  phase 
rule  of  Willard  Gibbs.) 

t  If  we  include,  besides  the  corners,  edges  and  faces  (i.e.  points,  lines  and 
surfaces)  the  solid  figure  itself,  Euler's  Law  becomes 

C-E  +  F-S  =  \. 

And  in  this  form  the  theorem  extends  to  n  dimensions,  as  follows: 

With  equal  beauty  and  simplicity,  the  simplest  figure  in  each  ^-dimensional 
space  is  given  as  follows: 

Kq  rC%  A^2  "'S  "^4  ^^^» 

w  =  0         1  "                  =1  (point) 

12-1  =1  (line) 

2  3-3  +1                                      =1  (triangle) 

3  4          -6  +4            -1                      —1  (tetrahedron) 

4  5          -10  +10          -5          +1        =1  (pentahedroid) 
etc. 

And,  in  a  figure  of  /i- dimensions,  the  sum  of  the  plane  angles  =2*"-^'  (C-2) 
right  angles. 


734  A  NOTE  ON  POLYHEDRA  [ch. 

that    in   a    polyhedron    of    C    corners,    the    sum    of    the    plane 
angles 

=  4  (C  -  2)  right  angles  (2)*. 

Hence,  if  the  polyhedron  be  isogonal,  the  sum  of  the  plane  angles 
at  each  corner 

4  (C  -  2) 


C 


90°,  or  U  -  ^i  right  angles       (3). 


The  five  regular  sohds,  or  Platonic  bodies — there  can  be  no  more 
— ^have  been  known  from  remote  antiquity ;  they  have  their  corners 
all  ahke  and  their  face's  all  alike,  they  are  isogonal  and  isohedral. 
Three  of  them,  the  tetrahedron,  octahedron  and  icosahedron,  have 
triangular  faces;  three  of  them,  the  tetrahedron,  cube  and  dodeca- 
hedron, have  trihedral  or  three-way  corners.  One  or  other  of  these, 
triangles  or  three-way  corners,  must  (as  we  shall  soon  see)  be  present 
in  every  polyhedron  whatsoever. 

The  semi-regular  solids  are  -regular  in  one  respect  or  other,  but 
not  in  both;  they  are  either  isogonal  or  isohedral — isohedral,  when 
every  face  is  an  identical  polygon  and  isogonal  when  at  every  corner 
the  same  set  of  faces  is  combined.  The  semi-regular  isogonal  gohds, 
with  all  their  corners  ahke  but  with  two  or  more  kinds  of  regular 
polygons  for  their  faces,  are  thirteen  in  number — there  can  be  no 
more;  they  were  all  described  by  Archimedes,  and  we  call  them 
by  his  name.  One  of  them,  with  six  square  and  eight  hexagonal 
facets,  derived  by  truncating  the  octahedron  or  the  cube,  we  have 
found  to  be  of  pecuhar  interest,  and  it  has  become  famihar  to  us 
as,  of  all  homogeneous  space-fillers,  the  one  which  encloses  a  given 
volume  within  ,a  minimal  area  of  surface.  It  is  the  cuho-octahedron 
of  Kepler  or  of  Fedorow,  the  tetrakaidekahedron  of  Kelvin,  which 
latter  name  we  commonly  use. 

Of  semi-regular  isohedral  bodies,  with  all  their  sides  alike  (though 
no  longer  regular  polygons)  and  their  corners  of  two  kinds  or  more, 
only  one  was  known  to  antiquity;  it  is  the  rhombic  dodecahedron, 
which  is  the  crystalline  form  of  the  garnet,  and  appears  in  part 

*  On  this  remarkable  parallel  see  Jacob  Steiner,  Gesammelte  Werke,  i,  p.  97. 
It  follows  that  the  sum  of  the  plane  angles  in  a  polyhedron,  as  in  a  plane  polygon, 
is  at  once  determined  by  the  number  of  its  comers:  a  result  which  dehghted 
Euler,  and  led  him  to  base  his  primary  or  generic  classification  of  polyhedra  on 
their  comers  rather  than  their  sides. 


IX]  OF  THE  SEMIREGULAR  SOLIDS  735 

again  as  a  "space-filler"  at  the  base  of  the  bee's  cell.  A  closely 
related  rhombic  icosahedron  was  known  to  Kepler;  but  it  was 
left  to  Catalan*  to  discover,  only  some  seventy-five  years  ago, 
that  the  isohedral  bodies  were  thirteen  in  number,  and  were 
precisely  comparable  with  and  reciprocal  to  the  Archimedean 
solids. 

The  semi-regular  solids,  both  of  Archimedes  and  of  Catalan,  are 
all,  like  the  Platonic  bodies,  related  to  the  sphere  |,  for  a  circum- 
scribing sphere  meets  all  the  corners  of  an  Archimedean  solid,  and 
an  inscribed  sphere  touches  all  the  faces  of  a  solid  of  Catalan;  and 
while  the  isogonal  bodies  can  be  constructed  by  various  simple 
geometrical  means,  the  general  method  of  constructing  the  thirteen 
isohedral  bodies  is  by  dividing  the  sphere  into  so  many  similar  and 
equal  areas  J.  It  is  a  matter  of  spherical  trigonometry  rather  than  of 
simple  geometry,  and  the  problem,  for  that  very  reason,  remained 
long  unsolved. 

The  thirteen  Archimedean  bodies  are  derivable  from  the  five 
Platonic  bodies,  in  most  cases  easily,  by  so  truncating  their  corners 
and  their  edges  as  to  produce  new  and  rpgularly  polygonal  faces  in 
place  of  the  old  faces,  corners  and  edges,  and  the  possible  number 
of  faces  in  the  new  figure  will  be  easily  derived  from  the  edges, 
corners  and  faces  of  the  old.  Part  of  the  old  faces  will  remain; 
each  truncated  corner  will  yield  one  new  face;  but  each  edge  may 
be  truncated,  or  bevelled,  more  than  once,  so  as  to  yield  one,  two, 
or  possibly  three  new  faces.  In  short,  if  the  faces,  corners  and 
edges  of  a  regular  solid  be  F,  C,  E,  those  of  the  Archimedean  solids 
derivable  from  it  {Fj)  will  be 

F^  =  F  +  mC  +  nE, 

where  m  =  0  or  1,  and  n  =  0,  1,  or  2. 

From  the  cube  six  Archimedean  bodies  may  be  derived,  from  the 
dodecahedron  six,  and  from  the  tetrahedron  one. 

*  Journal  de  Vecole  imper.  polytechnique,  xli,  pp.  1-71,  1865. 

t  It  follows  that  the  Chinese  carved  and  perforated  ivory  balls,  which  are 
based  on  regular  and  symmetrical  division  of  the  sphere,  can  all  be  referred  to  one 
or  another  of  the  Platonic  or  Archimedean  bodies. 

J  As  a  matter  of  fact,  the  Catalan  bodies  can  be  formed  by  adding  to  the  Platonic 
bodies,  just  as  (but  not  so  easily  as)  the  Archimedean  bodies  can  be  formed  by 
truncating:  them. 


736  A  NOTE  ON  POLYHEDRA  [ch. 

The  derivatives  of  the  cube  (with  its  six  sides  and  eight  corners) 
have  the  following  numbers  of  sides : 

F^  =  F+C  =6  +  8  =14 

F+C  +  E   =6  +  8+12  =  26 

i^+C+2j6;=6  +  8  +  24  =  38. 

The  derivatives  of  the  dodecahedron  have,  in  hke  manner,  32,  62 
or  92  sides;  while  the  tetrahedron  yields,  by  truncation  of  its  four 
corners,  a  solid  with  eight  sides. 

The  growth  and  form  of  crystals  is  a  subject  alien  to  our  own, 
yet  near  enough  to  attract  and  tempt  us.  It  is  a  curious  thing 
(probably  traceable  to  the  Index  Law  of  the  crystallographer)  that 
the  Archimedean  or  isogonal  bodies  seldom  occur  and  certainly  play 
no  conspicuous  part  in  crystallography,  while  several  of  Catalan's 
isohedral  figures  are  the  characteristic  forms  of  well-known  minerals  *. 

Just  as  we  pass  from  the  Platonic  to  the  Archimedean  bodies  by 
truncating  the  corners  or  edges  of  the  former,  so  conversely,  by 
producing  their  faces  to  a  limit  we  obtain  another  family  of  figures 
— in  all  cases  save  the  tetrahedron,  which  admits  of  no  such 
extension;  and  the  figures  of  this  family  are  remarkable  for  the 
"twinned,"  or  duphcate  or  multiple  appearance  which  they  present. 
If  we  extend  the  faces  of  an  octahedron  we  get  what  looks  hke  two 
tetrahedra,  "twinned  with"  or  interpenetrating  one  another;  but 
there  has  been  no  interpenetration  in  the  construction  of  this  twin- 
like figure,  only  further  accretion  upon,  and  extension  of,  the  facets 
of  the  octahedron.  Among  the  higher  polyhedra  there  are  many 
figures  which  look,  in  a  far  more  complicated  way,  like  the  twinning 
of  simpler  but  still  comphcated  forms;  and  these  also  have  been 
constructed,  not  by  interpenetration,  but  by  the  mere  superposition 
of  new  parts  on  old. 

An  elementary,  even  a  very  elementary,  knowledge  of  the  theory 
of  polyhedra  becomes  useful  to  the  naturahst  in  various  ways. 
Among  organic  structures  we  often  find  many-sided  boxes  (or  what 
may  be  regarded  as  such),  like  the  capsular  seed-vessels  of  plants, 
the  skeletons  of  certain  Radiolaria,  the  shells  of  the  Peridinia,  the 
carapace  of  a  tortoise,  and  a  great  many  more.     Or  we  may  go 

*  E.g.  the  triakis,  tetrakis  and  hexakis  octahedra  of  fluor-spar.  However  the 
Archimedean  tetrakaidekahedron  [QF^  SFq)  occurs  in  ahmi. 


IX]  OF  EULER'S  LAW  737 

further  and  treat  any  cluster  of  cells,  such  as  a  segmenting  ovum, 
as  a  species  of  polyhedron  and  study  it  from  the  point  of  view 
of  Euler's  Law  and  its  associated  theorems.  We  should  have  to 
include,  as  the  geometer  seldom  does,  the  case  of  two-sided  facets — 
facets  with  two  corners  and  two  curved  sides  or  edges — hke  the 
"  hths  "  *  of  a  peeled  orange ;  but  the  general  formula  would  include 
these  as  a  matter  of  course.  On  the  other  hand,  we  need  very 
seldom  consider  any  other  than  trihedral  or  three-way  corners. 

When  we  limit  ourselves  to  polyhedra  with  trihedral  corners  the 
following  formula  applies : 

4A+  3/3+  2/4+/6± 0./,-/, -  2/3-  ...  =  12  (4). 

That  this  formula  apphes  to  the  tetrahedron  with  its  four  triangles, 
the  cube  with  its  six  squares  and  the  dodecahedron  with  its  twelve 
pentagons,  is  at  once  obvious.  The  now  famihar  case  of  our  four- 
celled  egg  with  its  polar  furrows  (Fig.  486,  B,  etc.)  appears  in  two 
forms,  according  as  the  polar  furrows  run  criss-cross  or  parallel. 
In  the  one  case  we  have  a  curvilinear  tetrahedron,  in  the  other  a 
figure  with  two  two-sided  and  two  four-sided  facets;  in  either  case 
the  formula  is  obviously  satisfied. 

But  the  main  lesson  for  us  to  learn  is  the  broad,  general  principle 
that  we  cannot  group  as  we  please  any  number  and  sort  of  polygons 
into  a  polyhedron,  but  that  the  number  and  kind  of  facets  in  the 
latter  is  strictly  Umited  to  a  narrow  range  of  possibiHties.  For 
example,  the  case  of  Aulonia  has  already  taught  us  that  a  poly- 
hedron composed  entirely  of  hexagons  is  a  mathematical  impossi- 
bihtyt;  and  the  zero-coefficient  which  defines  the  number  of 
Jiexagons  in  the  above  formula  (4)  is  the  mathematical  statement 
of  the  fact  J.  We  can  state  it  still  more  simply  by  the  following 
corollary,  hkewise  hmited  to  the  case  of  three-way  corners : 

*  Lith,  a  useful  Scottish  word  for  a  joint  X)t  segment.  Cromwell,  according  to 
Carlyle,  "gar'd  kings  ken  they  had  a  lith  in  their  necks." 

t  That  hexagons  cannot  enclose  space,  or  form  a  "three-way  graph,"  has  been 
recognised  as  a  significant  fact  in  organic  chemistry :  where,  for  instance,  it  limits, 
somewhat  unexpectedly,  the  ways  in  which  a  closed  cyclol,  or  space-enclosing 
protein  molecule,  can  be  imagined  to  be  built  up.  Cf.  Dorothy  Wrinch,  in 
Proc.  R.  S.  (A),  No.  907,  p.  510,  1937. 

J  Euler  shewed  at  the  same  time  the  singular  fact  that  no  polyhedron  can  exist 
with  seven  edges. 


738  A  NOTE  ON  POLYHEDRA  [ch. 

This  applies  at  once  to  the  tetrahedron,  the  cube  and  the  regular 
dodecahedron,  and  at  once  excludes  the  possibihty  of  the  closed 
hexagonal  network. 

We  found  in  Dorataspis  a  closed  shell  consisting  only  of  hexagons 
and  pentagons;  without  counting  these  latter  we  know,  by  our 
formula,  that  they  must  be  twelve  in  number,  neither  more  nor 
less.  Lord  Kelvin's  tetrakaidekahedron  consists  only  of  squares 
and  hex:agons;   the  squares  are,  and  must  be,  six  in  number. 

In  a  typical  Peridinian,  such  as  Goniodoma,  there  are  twelve 
plates,  all  meeting  by  three-way  nodes  or  corners;  we  know^  and 
we  have  no  difficulty  in  verifying  the  fact,  that  the  twelve  plates 
are  all  pentagonal. 


T.  345.     Goniodoma,  from  above  and  below. 

Without  going  beyond  the  elements  of  our  subject  we  may  want 
to  extend  our  last  formula,  and  remove  the  restriction  to  three-way 
corners  under  which  it  lay.  We  know  that  a  tetrahedron  has  four 
triangles  and  four  trihedral  corners ;  that  a  cube  has  six  squares  and 
eight  trihedral  corners ;  an  octahedron  eight  triangles  and  six  four- way 
corners ;  an  icosahedron  twenty  triangles  and  twelve  five-way  corners. 
By  inspection  of  these  numbers  we  are  led  to  the  following  rule,  and 
may  estabhsh  it  as  a  deduction  from  Euler's  Law : 

ifs  +  c,)  =  8  +  0if,  +  c,)  +  {f,  +  c,)  +  2if,  +  c,)+etc (5). 

This  important  formula  further  illustrates  the  limitations  to  which 
all  polyhedra  are  subject;  for  it  shews  us,  among  other  things,  that 
(if  we  neglect  the  exceptional  case  of  dihedral  facets  or  'Uiths") 
every  polyhedron  must  possess  either  triangular  faces  or  trihedral 
corners,  and  that  these  taken  together  are  never  less  than  eight  in 
number. 


IX]  OF  EULER'S  LAW  739 

We  may  add  yet  two  more  formulae,  both  related  to  the  last,  and 
all  derivable,  ultimately,  from  Euler's  Law  * : 

3/3+2/4+/5-^12  +  0.C3+2c4+4c5+...  +  0./e+/7  +  2/8  (6) 

and 

3C3+  2c4  +  C5  =  12  +  0,/3+  2/4+  4/5  +  ...  +  O.Ce  +  C7  +  2c,    ...(7). 

These  imply  that  in  every  polyhedron  the  triangular,  quadrangular 
and  pentagonal  faces  (or  corners)  must,  taken  together  and  multi- 
plied as  above,  be  at  least  twelve  in  number.  Therefore  no  poly- 
hedron can  exist  which  has  not  a  certain  number  of  triangles, 
squares  or  pentagons  in  its  composition;  and  the  impossibihty  of 
a  polyhedron  consisting  only  of  hexagons  is  demonstrated  once 
again. 

Formulae  (5),  (6)  and  (7)  further  shew  us  that  not  only  is  a  three- 
way  polyhedron  of  hexagons  impossible,  but  also  a  four- way 
polyhedron  of  quadrangles,  or  one  of  six- way  corners  and  triangular 
facets ;  all  of  which  become  the  more  obvious  when  we  reflect  that 
the  plane  angles  meeting  in  each  point  or  node  must  be,  on  the 
average,  in  the  first  case  3  x  120°,  in  the  second  4  x  90°,  and  in  the 
third  6  X  60°. 

Lastly,  having  now  considered  the  case  of  other  than  trihedral 
corners,  we  may  learn  a  simple  but  very  curious  relation  between 
the  number  of  faces  and  corners,  arising  (like  so  much  else)  out  of 
Euler's  Law.  In  a  polyhedron  whose  corners  are  all  n-hedral, 
nC=2E;  therefore  (by  Euler)  7iC/2  +  2  =  F  +  C;  therefore 
2F  ={n-2)C  +  4. 

Therefore,  if 

n=3,    2F=4:+    C] 

=  4,  =4  +  2cl  (8). 

=  5,  =  4  +  3Cj 

Let  us  look  again  at  the  microscopic  skeleton  of  Dorataspis 
(Fig.  341).  We  have  seen  that  some  of  its  facets  are  hexagonal, 
the  rest  pentagonal;  there  happen  to  be  eight  of  the  former,  and 
therefore  (as  we  now  know)  there  must  be  twelve  of  the  latter. 

*  Derivable  from  Euler  together  with  the  formulae  for  the  "edge-counts," 
viz.  llnF^  =  2E,  and  SnC„  =  2^;  which  merely  mean  that  each  edge  separates 
two  faces,  and  joins  two  corners. 


740  A  NOTE  ON  POLYHEDRA  [ch.  ix 

We  know  also  that,  having  no  triangular  facets,  the  polyhedral 
skeleton  must  possess  trihedral  corners;  and  these,  moreover,  must 
(by  equation  5)  be  eight  in  number,  plus  the  number  of  pentagons, 
plus  twice  the  number  of  hexagons.     The  total, 

8  +/5+  2/6=  8  +  12  +  (2  X  8)  =  36, 

is  precisely  the  number  of  corners  in  the  figure,  all  of  them  trihedral. 
We  also  know  (from  equation  2)  that  the  sum  of  its  plane  angles 

=  4  (36  -  2)  X  90°  =12,240°, 

which  agrees  with  the  sum  of  the  angles  of  twelve  pentagons  and 
eight  hexagons.     The  configuration,  then,  is  a  possible  one. 

So  here  and  elsewhere  an  apparently  infinite  variety  of  form  is 
defined  by  mathematical  laws  and  theorems,  and  hmited  by  the 
properties  of  space  and  number.  And  the  whole  matter  is  a  running 
commentary  on  the  cardinal  fact  that,  under  such  foedera  Natural 
"as  Lucretius  recognised  of  old,  there  are  things  which  are  possible, 
and  things  which  are  impossible,  even  to  Nature  herself. 


CHAPTER  X 


A  PARENTHETIC  NOTE  ON  GEODESICS 


We  have  made  use  in  the  last  chapter  of  the  mathematical  principle 
of  Geodesies  (or  Geodetics)  in  order  to  explain  the  conformation 
of  a  certain  class  of  sponge-spicules ;  but  the  principle  is  of  much 
wider  appHcation  in  morphology,  and  would  seem  to  deserve  atten- 
tion which  it  has  not  yet  received.  The  subject  is  not  an  easy  one, 
and  if  we  are  to  avoid  mathematical  difficulties  we  must  keep  within 
narrow  bounds. 


Fig.  346.     Annular  and  spiral  thickenings  in  the  walls  of  plant-cells. 

Defining,  meanwhile,  our  geodesic  hne  (as  we  have  already  done) 
as  the  shortest  distance  between  two  points  on  the  surface  of  a 
soUd  of  revolution,  we  find  that  the  cyhnder  gives  us  some  of  the 
simplest  of  cases.  Here  it  is  plain  that  the  geodesies  are  of  three 
kinds:  (1)  a  series  of  annuh  around  the  cyhnder,  that  is  to  say,  a 
system  of  circles,  in  planes  parallel  to  one  another  and  at  right 
angles  to  the  axis  of  the  cylinder  (Fig.  346,  A);  (2)  a  series  of 
straight  lines  parallel  to  the  axis;  and  (3)  a  series  of  spiral  curves 
winding  round  the  wall  of  the  cyhnder  (B,  C).     These  three  systems 


742  A  PARENTHETIC  NOTE  [ch. 

are  all  of  frequent  occurrence,  and  are  illustrated  in  the  local 
thickenings  of  the  wall  of  the  cyHndrical  cells  or  vessels  of  plants. 

The  spiral,  or  rather  helical,  geodesic  is  particularly  common  in 
cylindrical  structures,  and  is  beautifully  shewn  for  instance  in  the 
spiral  coil  which  stiffens  the  tracheal  tubes  of  an  insect,  or  the 
so-called  tracheides  of  a  woody  stem.  A  Hke  phenomenon  is 
often  witnessed  in  the  splitting  of  a  glass  tube.  If  a  crack  appear 
in  a  test-tube  it  has  a  tendency  to  be  prolonged  in  its  own  direction, 
and  the  more  isotropic  be  the  glass  the  more  evenly  will  the  split 
tend  to  follow  the  straight  course  in  which  it  began.  As  a  result, 
the  crack  often  continues  till  our  test-tube  is  split  into  a  continuous 
spiral  ribbon. 

One  may  stretch  a  tape  along  a  cylinder,  but  it  no  sooner  swerves 
to  one  side  than  it  begins  to  wind  itself  around;  it  is  tracing  its 
geodesic*. 

In  a  circular  cone,  the  spiral  geodesic  falls  into  closer  and  closer 
coils  as  the  cone  narrows,  till  it  comes  to  the  end,  and  then  it  winds 
back  the  same  way;  and  a  beautiful  geodesic  of  this  kind  is 
exemplified  in  the  sutural  line  of  a  spiral  shell,  such  as  Turritella, 
or  in  the  striations  which  run  parallel  with  the  spiral  suture.  On 
a  prolate  spheroid,  the  coils  of  a  spiral  geodesic  come  closer  together 
as  they  approach  the  ends  of  the  long  axis  of  the  ellipse,  and  wind 
back  and  forward  from  one  pole  to  the  other  "j".  We  have  a  case  of 
this  kind  in  an  Equisetum-sipoTe,  when  the  integument  splits  into  the 
spiral  "elaters,"  though  the  spire  is  not  long  enough  to  shew  all  its 
geodesic  features  in  detail. 

We  begin  to  see  that  our  first  definition  of  a  geodesic  requires  to 
be  modified;  for  it  is  only  subject  to  conditions  that  it  is  ''the 
shortest  distance  between  two  points  on  the  surface  of  the  solid," 
and  one  of  the  commonest  of  these  restricting  conditions  is  that 
our  geodesic  may  be  constrained  to  go  twice,  or  many  times,  round 
the  surface  on  its  way.  In  short,  we  may  re-define  a  geodesic,  as 
a  curve  drawn  upon  a  surface  such  that,  if  we  take  any  two  adjacent 

*  It  is  not  that  the  geodesic  is  rectified  into  a  straight  line;  but  that  a  straight 
line  (the  midline  of  the  tape  or  ribbon)^  is  converted  into  the  geodesic  on  the  given 
surface. 

t  In  all  these  cases,  r  cos  a  =  a  constant,  where  r  is  the  radius  of  the  circular 
section,  and  a  the  angle  at  which  it  is  crossed  by  the  geodesic.  And  the  constant 
is  measured  by  the  smallest  circle  which  the  geodesic  can  reach. 


X]  ON  GEODESICS  743 

points  on  the  curve,  the  curve  gives  the  shortest  distance  between 
them.  It  often  happens,  in  the  geodesic  systems  which  we  meet 
with  in  morphology,  that  two  opposite  spirals  or  rather  helices 
run  separate  and  distinct  from  one  another,  as  in  Fig.  346,  C; 
and  it  is  also  common  to  find  the  two  interfering  with  one  another, 
and  forming  a  criss-cross  or  reticulated  arrangement.  This  indeed 
is  a  common  source  of  reticulated  patterns. 

The  microscopic  and  even  ultramicroscopic  structure  of  the  cell- 
wall  shews  analogous  configurations:  as  in  the  large  cells  of  the 
alga  Valonia,  where  the  wall  consists  of  many  lamellae,  each  com- 
posed of  parallel  fibrillae  running  in  spiral  geodesies,  and  alternating 
in  direction  from  one  lamella  to  another.  Here,  and  not  less  clearly 
in  the  young  parenchyma  of  seedHng  oats,  it  is  the  long-chain 
cellulose  molecules  which  follow  a  spiral  course  around  the  cell- wall, 
right-handed  or  left-handed  as  the  case  may  be,  and  inchned  more 
or  less  steeply  according  to  the  elongation  of  the  cell.  But  these 
highly  interesting  questions  of  molecular,  or  micellar,  structure 
lie  beyond  our  scope*. 

Among  the  ciliated  infusoria,  we  have  a  variety  of  beautiful 
geodesic  curves  in  the  spiral  patterns  in  which  their  ciha  are 
arranged;  though  it  is  probable  enough  that  in  some  comphcated 
cases  these  are  not  simple  geodesies,  but  developments  of  curves  other 
than  a  straight  hne  upon  the  surface  of  the  organism.  In  other  words, 
they  seem  to  be  instances  of  "geodesic  curvature." 

Lastly,  an  instructive  case  is  furnished  by  the  arrangement  of 
the  muscular  fibres  on  the  surface  of  a  hollow  organ,  such  as  the 
heart  or  the  stomach.  Here  we  may  consider  the  phenomenon 
from  the  point  of  view  of  mechanical  efficiency,  as  well  as  from 
that  of  descriptive  anatomy.  In  fact  we  have  a  right  to  expect 
that  the  muscular  fibres  covering  such  hollow  organs  will  coincide 
with  geodesic  lines,  in  the  sense  in  which  we  are  using  the  term. 
For  if  we  imagine  a  contractile  fibre,  .or  an  elastic  band,  to  be  fixed 
by  its  two  ends  upon  a  curved  surface,  it  is  obvious  that  its  first 
effort  of  contraction  will  tend  to  expend  itself  in  accommodating 

*  Cf.  (e.g.)  C.  Correns,  Innere  Struktur  einiger  Algenmembranen,  Beitr.  zur 
Morphol.  u.  Physiol,  d.  Pflanzenzelle,  1893,  p.  260;  W.  T.  Astbury  and  others, 
Proc,  E.S.  (B),  cix,  p.  443,  1932,  and  other  papers;  G.  van  Iterson,  jr..  Nature, 
cxxxviii,  p.  364,  1936;   R.  D.  Preston,  Proc.  U.S.  (B),  cxxv,  p.  772,  1938;   etc. 


744  A  PARENTHETIC  NOTE  [ch. 

the  band  to  the  form  of  the  surface,  in  "stretching  it  tight,"  or  in 
other  words  in  causing  it  to  assume  a.  direction  which  is  the  shortest 
possible  hne  upon  the  surface  between  the  two  extremes:  and  it  is 
only  then  that  further  contraction  will  have  the  effect  of  constricting 
the  tube  and  so  exercising  pressure  on  its  contents.  Thus  the 
muscular  fibres,  as  they  wind  over  the  curved  surface  of  an  organ, 
arrange  themselves  automatically  in  geodesic  curves:  in  precisely 
the  same  manner  as  we  also  automatically  construct  complex 
systems  of  geodesies  whenever  we  wind  a  ball  of  wool  or  a  spindle 
of  tow,  or  when  the  skilful  surgeon  bandages  a  Hmb ;  indeed  the 
surgeon  must  fold  and  crease  his  bandage  if  it  is  not  to  keep  on 
geodesic  lines.  It  is  as  a  simple,  necessary  result  of  geodesic 
principles  that  we  see  those  "  figures-of -eight "  produced,  to  which, 
in  the  case  for  instance  of  the  heart-muscles,  Pettigrew  and  other 
anatomists  have  ascribed  pecuhar  importance.  In  the  case  of 
both  heart  and  stomach  we  must  look  upon  these  organs  as  de- 
veloped from  a  simple  cyhndrical  tube,  after  the  fashion  of  the 
glass-blower,  as  is  further  discussed  on  p.  1049  of  this  book,  the 
modification  of  the  simple  cyhnder  consisting  of  various  degrees  of 
dilatation  and  of  twisting.  In  the  primitive  undistorted  cyhnder, 
as  in  an  artery  or  in  the  intestine,  the  muscles  run  in  simple  geodesic 
fines,  and  constitute  the  circular  and  longitudinal  coats  which  form 
(or  are  said  to  form)  the  normal  musculature  of  all  tubular  organs, 
or  the  cylindrical  body  of  a  worm.  However,  we  can  often  recog- 
nise, in  a  small  artery  for  instance,  that  the  so-called  circular  fibres 
tend  to  take  a  shghtly  obhque  or  spiral  course;  and  that  the 
so-called  annular  muscle-fibres  are  really  spirals  is  an  old  statement 
which  may  very  hkely  be  true*.  If  we  consider  each  muscular 
fibre  as  an  elastic  strand  embedded  in  the  elastic  membrane  which 
constitutes  the  wall  of  the  organ,  it  is  evident  that,  whatever  be 
the  distortion  suffered  by  the  entire  organ,  the  individual  fibre  will 
follow  its  own  course,  which  will  still,  in  a  sense,  be  geodesic. 
But  if  the  distortion  be  considerable,  as  for  instance  if  the  tube 

*  See  A  Discourse  concerning  the  Spiral,  instead  of  the  supposed  Annular, 
structure  of  the  Fibres  of  the  Intestins;  discover'd  and  shewn  by  the  Learn'd 
and  Inquisitive  Dr.  William  Cole  to  the  Royal  Society,  Phil.  Trans,  xi,  pp.  603-609, 
1676.  Cf.  Eben  J.  Carey,  Studies  on  the. .  .small  intestine,  Anat.  Record,  xxi,  pp. 
189-215,  1921 ;  F.  T.  Lewis,  The  spiral  trend  of  intestinal  muscle  fibres.  Science,  LV, 
June  30,  1922. 


X]  ON  GEODESICS  745 

become  bent  upon  itself,  or  if  at  some  point  its  walls  bulge  outwards 
in  a  diverticulum  or  pouch,  then  the  old  system  of  geodesies  will 
only  mark  the  shortest  distance  between  two  points  more  or  less 
approximate  to  one  another,  and  new  systems  of  geodesies, 
peculiar  to  the  new  surface,  will  tend  to  appear,  and  link 
up  points  more  remote  from  one  another.  This  is  evidently  the 
case  in  the  human  stomach.  We  still  have  the  systems,  or  their 
unobhterated  remains,  of  circular  and  longitudinal  muscles;  but 
we  also  see  two  new  systems  of  fibres,  both  obviously  geodesic 
(or  rather,  when  we  look  more  closely,  both  parts  of  one  and  the 
same  geodesic  system),  in  the  form  of  annuh  encircling  the  pouch 
or  diverticulum  at  the  cardiac  end  of  the  stomach,  and  of  obhque 
fibres  taking  a  spiral  course  from  the  neighbourhood  of  the 
oesophagus  over  the  sides  of  the  organ. 

In  the  heart  we  have  a  similar,  but  more  complicated  pheno- 
menon. Its  musculature  consists,  in  great  part,  of  the  original 
simple  system  of  circular  and  longitudinal  muscles  which  enveloped 
the  original  arterial  tubes,  which  tubes,  after  a  process  of  local 
thickening,  expansion,  and  especially  tuisting,  came  together  to 
constitute  the  composite,  or  double,  mammalian  heart;  and  these 
systems  of  muscular  fibres,  geodesic  to  begin  with,  remain  geodesic 
(in  the  sense  in  which  we  are  using  the  word)  after  all  the  twisting 
which  the  primitive  cylindrical  tube  or  tubes  have  undergone. 
That  is  to  say,  these  fibres  still  run  their  shortest  possible  course, 
from  start  to  finish,  over  the  complicated  curved  surface  of  the 
organ;  and,  as  Borelli  well  understood,  it  is  only  because  they  do 
so  that  their  contraction,  or  longitudinal  shortening,  is  able  to 
produce  its  direct  eifect  in  the  contraction  or  systole  of  the  heart*. 

As  a  parenthetic  corollary  to  the  case  o-f  the  spiral  pattern  upon 
the  wall  of  a  cyhndrical  cell,  we  may  consider  for  a  moment  the 
spiral  fine  which  many  small  organisms  tend  to  follow  in  their  path 

*  The  spiral  fibres,  or  a  large  portion  of  them,  constitute  what  Searle  called 
"the  rope  of  the  heart"  (Todd's  Cyclopaedia,  ii,  p.  621,  1836).  The  "twisted 
sinews  of  the  heart"  were  known  to  early  anatomists,  and  have  been  frequently 
and  elaborately  studied:  for  instance,  by  Gerdy  {Bull.  Fac.  Med.  Paris,  1820, 
pp.  40-148),  and  by  Pettigrew  (Phil.  Trans.  1864),  and  again  by  J.  B.  Macallum 
{Johns  Hopkins  Hospital  Report,  ix,  1900)  and  by  Franklin  P.  Mall  {Amer.  Journ. 
Anat.  XI,  1911). 


746  A  PARENTHETIC  NOTE  [ch. 

of  locomotion*.  A  certain  physiologist  observed  that  an  Amoeba, 
crawling  within  a  narrow  tube,  wound  its  slow  way  in  a  spiral  course 
instead  of  going  straight  along  the  tube.  The  creature  was  going 
nowhere  in  particular,  but  merely  following  the  direction  in  which 
it  had  begun :  in  curious  illustration  of  a  familiar  statement  in  the 
"dynamics  of  a  particle,"  that  a  particle  moving  on  a  surface 
without  constraint  will  describe  geodesic  lines. 

But  it  is  after  a  different  fashion,  and  without  any  constraint  to 
a  surface,  that  the  smaller  ciliated  organisms,  such  as  the  ciliate 
and  flagellate  infusoria,  the  rotifers,  the  swarm-spores  of  various 
Protista,  and  so  forth,  shew  a  tendency  to  pursue  a  spiral  path  in 
their  ordinary  locomotion.  The  means  of  locomotion  which  they 
possess  in  their  cilia  are  at  best  somewhat  primitive  and  inefficient; 
they  have  no  apparent  means  of  steering,  or  modifying  their 
direction;  and,  if  their  course  tended  to  swerve  ever  so  little  to 
one  side,  the  result  would  be  to  bring  them  round  and  round  again 
in  an  approximately  circular  path  (such  as  a  man  astray  on  the 
prairie  is  said  to  follow),  with  little  or  no  progress  in  a  definite 
longitudinal  direction.  But  as  a  matter  of  fact,  by  reason  of  a 
more  or  less  unsymmetrical  form  of  the  body,  all  these  creatures 
tend  more  or  less  to  rotate  about  their  long  axis  while  they  swim. 
And  this  axial  rotation,  just  as  in  the  case  of  a  rifle-bullet,  causes 
their  natural  swerve,  which  is  always  in  the  same  direction  as 
regards  their  own  bodies,  to  be  in  a  continually  changing  direction 
as  regards  space:  in  short,  to  make  a  spiral  course  around,  and  more 
or  less  near  to,  a  straight  axial  line|. 

In  this  short  chapter  we  have  touched  on  phenomena  where  form 
repeats  itself,  and  mathematical  analogies  recur,  in  very  different 
things  and  very  different  orders  of  magnitude.  The  spiral  muscles 
of  heart  or  stomach  are  the  mechanical  outcome  of  twists  which 
these  tubular  organs  have  undergone  in  the  course  of  their  develop- 
ment, and  come,  accordingly,  under  the  general  category  of  organic 

*  Cf.  Butschli,  "Protozoa,"  in  Bronn's  Thierreich,  ii,  p.  848,  in,  p.  1785,  etc., 
1883-87;  Jennings,  Amer.  Nat.  xxxv,  p.  369,  1901;  Putter,  Thigmotaxie  bei 
Protisten,  Arch.  f.  Anat.  u.  Phys.  {Phys.  Abth.  Swppl.),  pp.  243-302,  1900. 

t  Cf.  W.  Ludwig,  Ueber  die  Schraubenbahnen  niederer  Organismen,  Arch.  f. 
vergl.  Physiologic,  ix,  1919. 


x]  ON  GEODESICS  747 

or  embryological  growth.  But  the  spiral  thickenings  in  the  woody 
fibres  of  a  plant  are  of  another  order  of  things,  and  lie  in  the  region 
of  molecular  phenomena.  The  delicate  spirals  of  the  cell-wall  of 
a  cotton-hair  are  based  on  a  complicated  cellulose  space-lattice, 
recaUing  Nageh's  micellar  hypothesis  in  a  new  setting;  and  giving 
us  a  glimpse  of  organic  growth  after  the  very  fashion  of  crystaUine 
growth,  that  is  to  say  from  the  starting-point  of  molecular  structure 
and  configuration*. 

*  W.  Lawrence  Balls,  Determiners  of  cellulose  structure  as  seen  in  the  cell-wall 
of  cotton-hairs,  Proc.  R.S.  (B),  xcv,  pp.  72-89,  1923,  and  other  papers.  Cf.  also 
Wilfred  Robinson,  Microscopical  features  of  mechanical  strains  in  timber,  and  the 
bearing  of  these  on  the  structure  of  the  cell-wall  in  plants,  Phil.  Trans.  (B),  ccx, 
pp.  49-82,  1920. 


CHAPTER  XI 

THE  EQUIANGULAR  SPIRAL 

The  very  numerous  examples  of  spiral  conformation  which  we 
meet  with  in  our  studies  of  organic  form  are  pecuUarly  adapted 
to  mathematical  methods  of  investigation.  But  ere  we  begin  to 
study  them  we  must  take  care  to  define  our  terms,  and  we  had 
better  also  attempt  some  rough  preliminary  classification  of  the 
objects  with  which  we  shall  have  to  deal. 

In  general  terms,  a  Spiral  is  a  curve  which,  starting  from 
a  point  of  origin,  continually  diminishes  in  curvature  as  it  recedes 
from  that  point;  or,  in  other  words,  whose  radius  of  curvature 
continually  increases.  This  definition  is  wide  enough  to  include 
a  number  of  different  curves,  but  on  the  other  hand  it  excludes 
at  least  one  which  in  popular  speech  we  are  apt  to  confuse  with 
a  true  spiral.  This  latter  curve  is  the  simple  screw,  or  cyhndrical 
helix,  which  curve  neither  starts  from  a  definite  origin  nor  changes 
its  curvature  as  it  proceeds.  The  "spiral"  thickening  of  a  woody 
plant-cell,  the  "spiral"  thread  within  an  insect's  tracheal  tube,  or 
the  "spiral"  twist  and  twine  of  a  climbing  stem  are  not,  mathe- 
matically speaking,  spirals  at  all,  but  screws  or  helices.  They  belong 
to  a  distinct,  though  not  very  remote,  family  of  curves. 

Of  true  organic  spirals  we  have  no  lack*.  We  think  at  once  of 
horns  of  ruminants,  and  of  still  more  exquisitely  beautiful  molluscan 
shells — in  which  (as  Pliny  says)  tnagna  ludentis  Naturae  varietas. 
Closely  related  spirals  may  be  traced  in  the  florets  of  a  sunflower; 
a  true  spiral,  though  not,  by  the  way,  so  easy  of  investigation,  is  seen 
in  the  outhne  of  a  cordiform  leaf;  and  yet  again,  we  can  recognise 
typical  though  transitory  spirals  in  a  lock  of  hair,  in  a  staple  of 
woolt,  in  the  coil  of  an  elephant's  trunk,  in  the  "circling  spires" 

*  A  great  number  of  spiral  forms,  both  organic  and  artificial,  are  described 
and  beautifully  illustrated  in  Sir  T.  A.  Cook's  Spirals  in  Nature  and  Art,  1903,  and 
Curves  of  Life,  1914. 

t  On  this  interesting  case  see,  e.g.  J.  E.  Duerden,  in  Science,  May  25,  1934. 


CH.  XI]  THE  EQUIANGULAR  SPIRAL  749 

of  a  snake,  in  the  coils  of  a  cuttle-fish's  arm,  or  of  a  monkey's  or 
a  chameleon's  tail. 


Fig.  347.  The  shell  of  Nautilus  pompilius,  from  a  radiograph:  to  shew  the 
equiangular  spiral  of  the  shell,  together  with  the  arrangement  of  the  internal 
septa.     From  Green  and  Gardiner,  in  Proc.  Malacol.  Soc.  ii,  1897. 

Among  such  forms  as  these,  and  the  many  others  which  we 
might  easily  add  to  them,  it  is  obvious  that  we  have  to  do  with 
things    which,    though    mathematically    similar,    are    biologically 


750  THE  EQUIANGULAR  SPIRAL  [ch. 

speaking  fundamentally  different ;  and  not  only  are  they  biologically 
remote,  but  they  are  also  physically  different,  in  regard  to  the  causes 
to  which  they  are  severally  due.  For  in  the  first  place,  the  spiral 
coil  of  the  elephant's  trunk  or  of  the  chameleon's  tail  is,  as  we  have 
said,  but  a  transitory  configuration,  and  is  plainly  the  result  of 
certain  muscular  forces  acting  upon  a  structure  of  a  definite,  and 
normally  an  essentially  different,  form.  It  is  rather  a  position,  or 
an  attitude,  than  a  form.,  in  the  sense  in  which  we  have  been  using 
this  latter  term;  and,  unhke  most  of  the  forms  which  we  have  been 
studying,  it  has  little  or  no  direct  relation  to  the  phenomenon  of 
growth. 


Fig.  348.     A  foraminiferal  shell  {Pulvinnlina). 

Again,  there  is  a  difference  between  such'  a  spiral  conformation 
as  is  built  up  by  the  separate  and  successive  florets  in.  the  sunflower, 
and  that  which,  in  the  snail  or  Nautilus  shell,  is  apparently  a  single 
and  indivisible  unit.  And  a  similar  if  not  identical  difference  is 
apparent  between  the  Nautilus  shell  and  the  minute  shells  of  the 
Foraminifera  which  so  closely  simulate  it:  inasmuch  as  the  spiral 
shells  of  these  latter  are  composite  structures,  combined  out  of 
successive  and  separate  chambers,  while  the  molluscan  shell,  though 
it  may  (as  in  Nautilus)  become  secondarily  subdivided,  has  grown 
as  one  continuous  tube.  It  follows  from  all  this  that  there  cannot 
be  a  physical  or  dynamical,  though  there  may  well  be  a  mathematical 
law  of  growth,  which  is  common  to,  and  which  defines,  the  spiral 
form  in  Nautilus,  in  Globigerina,  in  the  ram's  horn,  and  in  the 
inflorescence  of  the  sunflower.  Nature  at  least  exhibits  in  them  all 
^'un  reflet  des  formes  rigoureuses  qu'etudie  la  geometrie"^ .'' 

*  Haton  de  la  Goupilliere,  in  the  introduction  to  his  important  study  of  the 
Surfaces  Nautiloides,  Annaes  sci.  da  Acad.  Polytechnica  do  Porto,  Coimbra,  ni,  1908. 


XI]  OF  SUNDRY  SPIRALS  751 

Of  the  spiral  forms  which  we  have  now  mentioned,  every  one 
(with  the  single  exception  of  the  cordate  outline  of  the  leaf)  is  an 
example  of  the  remarkable  curve  known  as  the  equiangular  or 
logarithmic  spiral.  But  before  we  enter  upon  the  mathematics  of 
the  equiangular  spiral,  let  us  carefully  observe  that  the  whole  of  the 
organic  forms  in  which  it  is  clearly  and  permanently  exhibited, 
however  different  they  may  be  from  one  another  in  outward  appear- 
ance, in  nature  and  in  origin,  nevertheless  all  belong,  in  a  certain 
sense,  to  one  particular  class  of  conformations.  In  the  great 
majority  of  cases,  when  we  consider  an  organism  in  part  or  whole, 
when  we  look  (for  instance)  at  our  own  hand  or  foot,  or  contemplate 
an  insect  or  a  worm,  we  have  no  reason  (or  very  Httle)  to  consider 
one  part  of  the  existing  structure  as  older  than  another;  through 
and  through,  the  newer  particles  have  been  merged  and  commingled 
among  the  old ;  the  outhne,  such  as  it  is,  is  due  to  forces  which  for 
the  most  part  are  still  at  work  to  shape  it,  and  which  in  shaping  it 
have  shaped  it  as  a  whole.  But  the  horn,  or  the  snail-shell,  is 
curiously  different;  for  in  these  the  presently  existing  structure  is, 
so  to  speak,  partly  old  and  partly  new.  It  has  been  conformed  by 
successive  and  continuous  increments;  and  each  successive  stage  of 
growth,  starting  from  the  origin,  remains  as  an  integral  and  un- 
changing portion  of  the  growing  structure. 

We  may  go  further,  and  see  that  horn  and  shell,  though  they 
belong  to  the  living,  are  in  no  sense  alive*.  They  are  by-products 
of  the  animal;  they  consist  of  "formed  material,"  as  it  is  sometimes 
called ;  their  growth  is  not  of  their  own  doing,  hut  comes  of  Hving 
cells  beneath  them  or  around.  The  many  structures  which  display 
the  logarithmic  spiral  increase,  or  accumulate,  rather  than  grow. 
The  shell  of  nautilus  or  snail,  the  chambered  shell  of  a  foraminifer, 
the  elephant's  tusk,  the  beaver's  tooth,  the  cat's  claws  or  the 
canary-bird's— all  these  shew  the  same  simple  and  very  beautiful 
spiral  curve.  And  all  alike  consist  of  stuff  secreted  or  deposited  by 
living  cells ;  all  grow,  as  an  edifice  grows,  by  accretion  of  accumulated 

*  For  Oken  and  Goodsir  the  logarithmic  spiral  had  a  profound  significance,  for 
they  saw  in  it  a  manifestation  of  life  itself.  For  a  like  reason  Sir  Theodore  Cook 
spoke  of  the  Curves  of  Life;  and  Alfred  Lartigues  says  (in  his  Biodynamique  generale, 
1930,  p.  60):  "Nous  verrons  la  Conchy liologie  apporter  une  magnifique  contribution 
a  la  Stereodynamique  du  tourbillon  vital."  The  fact  that  the  spiral  is  always 
formed  of  non-living  matter  helps  to  contradict  these  mystical  conceptions. 


752  THE  EQUIANGULAR  SPIRAL  [ch. 

material;    and  in  all  alike  the  parts  once  formed  remain  in  being, 
and  are  thenceforward  incapable  of  change. 

In  a  shghtly  different,  but  closely  cognate  way,  the  same  is  true 
of  the  spirally  arranged  florets  of  the  sunflower.  For  here  again 
we  are  regarding  serially  arranged  portions  of  a  composite  structure, 
which  portions,  similar  to  one  another  in  form,  differ  in  age;  and 
differ  also  in  magnitude  in  the  strict  ratio  of  their  age.  Somehow 
or  other,  in  the  equiangular  spiral  the  time-element  always  enters 
in;  and  to  this  important  fact,  full  of  curious  biological  as  well  as 
mathematical  significance,  we  shall  afterwards  return. 

In  the  elementary  mathematics  of  a  spiral,  we  speak  of  the  point 
of  origin  as  the  pole  (0) ;  a  straight  hne  having  its  extremity  in  the 
pole,  and  revolving  about  it,  is  called  the  radius  vector;  and  a 
point  (P),  travelling  along  the  radius  vector  under  definite  conditions 
of  velocity,  will  then  describe  our  spiral  curve. 

Of  several  mathematical  curves  whose  form  and  development 
may  be  so  conceived,  the  two  most  important  (and  the  only  two 
with  which  we  need  deal)  are  those  which  are  known  as  (1)  the 
equable  spiral,  or  spiral  of  Archimedes,  and  (2)  the  equiangular  or 
logarithmic  spiral. 

The  former  may  be  roughly  illustrated  by  the  way  a  sailor 
coils  a  rope  upon  the  deck;  as  the  rope  is  of  uniform  thickness,  so 
in  the  whole  spiral  coil  is  each  whorl  of  the  same  breadth  as  that 
which  precedes  and  as  that  which  follows  it.  Using  its  ancient 
definition,  we  may  define  it  by  saying,  that  ''If  a  straight  fine 
revolve  uniformly  about  its  extremity,  a  point  which  Hkewise  travels 
uniformly  along  it  will  describe  the  equable  spiral*."  Or,  putting 
the  same  thing  into  our  more  modern  words,  "If,  while  the  radius 
vector  revolve  uniformly  about  the  pole,  a  point  (P)  travel  with 
uniform  velocity  along  it,  the  curve  described  will  be  that  called 
the  equable  spiral,  or  spiral  of  Archimedes."  It  is  plain  that  the 
spiral  of  Archimedes  may  be  compared,  but  again  roughly,  to  a 
cylinder  coiled  up.  It  is  plain  also  that  a  radius  {r  =  OP),  made 
up  of  the  successive  and  equal  whorls,  will  increase  in  arithmetical 
progression :  and  will  equal  a  certain  constant  quantity  (a)  multiplied 

*  Leslie's  Geometry  of  Curved  Lines,  1821,  p.  417.  This  is  practically  identical 
with  Archimedes'  own  definition  (ed.  Torelli,  p.  219);  cf.  Cantor,  Geschichte  der 
Mathematik,  i,  p.  262,  1880. 


XI]  AND  THE  SPIRAL  OF  ARCHIMEDES  753 

by  the  whole  number  of  whorls,  (or  more  strictly  speaking)  multiplied 
by  the  whole  angle  (6)  through  which  it  has  revolved:  so  that 
r  =  ad.  And  it  is  also  plain  that  the  radius  meets  the  curve  (or 
its  tangent)  at  an  angle  which  changes  slowly  but  continuously, 
and  which  tends  towards  a  right  angle  as  the  whorls  increase  in 
number  and  become  more  and  more  nearly  circular. 

But,  in  contrast  to  this,  in  the  equiangular  spiral  of  the  Nautilus 
or  the  snail-shell  or  Globigerina,  the  whorls  continually  increase 
in  breadth,  and  do  so  in  a  steady  and  unchanging  ratio.  Our 
definition  is  as  follows:  "If,  instead  of  travelhng  with  a  uniform 
velocity,  our  point  move  along  the  radius  vector  with  a  velocity 
increasing  as  its  distance  from  the  pole,  then  the  path  described  is 


Fig.  349.    The  spiral  of  Archimedes. 

called  an  equiangular  spiral."  Each  whorl  which  the  radius  vector 
intersects  will  be  broader  than  its  predecessor  in  a  definite  ratio; 
the  radius  vector  will  increase  in  length  in  geometrical  progression,  as 
it  sweeps  through  successive  equal  angles ;  and  the  equation  to  the 
spiral  will  be  r  =  a^.  As  the  spiral  of  Archimedes,  in  our  example  of 
the  coiled  rope,  might  be  looked  upon  as  a  coiled  cyfinder,  so  (but 
equally  roughly)  may  the  equiangular  spiral,  in  the  case  of  the  shell, 
be  pictured  as  a  cone  coiled  upon  itself;  and  it  is  the  conical  shape 
of  the  elephant's  trunk  or  the  chameleon's  tail  which  makes  them  coil 
into  a  rough  simulacrum  of  an  equiangular  spiral. 

While   the   one   spiral   was   known  in   ancient   times,   and   was 
investigated  if  not  discovered  by  Archimedes,  the  other  was  first 


754  THE  EQUIANGULAR  SPIRAL  [ch. 

recognised  by  Descartes,  and  discussed  in  the  year  1638  in  his  letters 
to  Mersenne*.  Starting  with  the  conception  of  a  growing  curve 
which  should  cut  each  radius  vector  at  a  constant  angle — just  as 
a  circle  does — Descartes  shewed  how  tt  would  necessarily  follow  that 
radii  at  equal  angles  to  one  another  at  the  pole  would  be  in  con- 
tinued proportion ;  that  the  same  is  therefore  true  of  the  parts  cut  off 
from  a  common  radius  vector  by  successive  whorls  or  convolutions 
of  the  spire;  and  furthermore,  that  distances  measured  along  the 
curve  from  its  origin,  and  intercepted  by  any  radii,  as  at  B,  C,  are 
proportional  to  the  lengths  of  these  radii,  OB,  OC.     It  follows  that 


Fig.  350.     The  equiangular  spiral. 

the  sectors  cut  off  by  successive  radii,  at  equal  vectorial  angles,  are 
similar  to  one  another  in  every  respect;  and  it  further  follows  that 
the  figure  may  be  conceived  as  growing  continuously  without  ever 
changing  its  shape  the  while. 

If  the  whorls  increase  very  slowly,  the  equiangular  spiral  will  come  to  look 
like  a  spiral  of  Archimedes.  The  Nummulite  is  a  case  in  point.  Here  we  have 
a  large  number  of  whorls,  very  narrow,  very  close  together,  and  apparently  of 
equal  breadth,  which  give  rise  to  an  appearance  similar  to  that  of  our  coiled 
rope.  And,  in  a  case  of  this  kind,  we  might  actually  find  that  the  whorls 
were  of  equal  breadth,  being  produced  (as  is  apparently  the  case  in  the 
Nummulite)  not  by  any  very  slow  and  gradual  growth  in  thickness  of  a  con- 
tinuous tube,  but  by  a  succession  of  similar  cells  or  chambers  laid  on,  round 
and  round,  determined  as  to  their  size  by  constant  surface-tension  conditions 
and  therefore  of  unvarying  dimensions.  The  Nummulite  must  always  have 
a  central  core,  or  initial  cell,  around  which  the  coil  is  not  only  wrapped,  but 
out  of  which  it  springs ;  and  this  initial  chamber  corresponds  to  our  a'  in  the 
expression  r  =  a'  +  ad  cot  a. 

*  (Euvres,  ed.  Adam  et  Tannery,  Paris,  1898,  p.  360. 


XI]  OR  LOGARITHMIC  SPIRAL  755 

The  many  specific  properties  of  the  equiangular  spiral  are  so 
interrelated  to  one  another  that  we  may  choose  pretty  well  any 
one  of  them  as  the  basis  of  our  definition,  and  deduce  the  others 
from  it  either  by  analytical  methods  or  by  elementary  geometry. 
In  algebra,  when  m^  =  n,  x  is  called  the  logarithm  of  n  to  the  base 
m.  Hence,  in  this  instance,  the  equation  r  =  a^  may  be  written  in 
the  form  log  r  =  6  log  a,  or  6  =  log  r/log  a,  or  (since  a  is  a  constant) 
6  =  A;  log  /•*.  Which  is  as  much  as  to  say  that  (as  Descartes  dis- 
covered) the  vector  angles  about  the  pole  are  proportional  to  the 
logarithms  of  the  successive  radii;  from  which  circumstance  the 
alternative  name  of  the  "logarithmic  spiral"  is  derivedf. 

Moreover,  for  as  many  properties  as  the  curve  exhibits,  so  many 
names  may  it  more  or  less  appropriately  receive.  ^James  Bernoulli 
called  it  the  logarithmic  spiral,  as  we  still  often  do ;  P.  Nicolas  called 
it  the  geometrical  spiral,  because  radii  at  equal  polar  angles  are  in 
geometrical  progression;  Halley,  the  proportional  spiral,  because 
the  parts  of  a  radius  cut  off  by  successive  whorls  are  in  continued 
proportion;  and  lastly,  Roger  Cotes,  going  back  to  Descartes'  first 
description  or  first  definition  of  all,  called  it  the  equiangular  spiral  J. 
We  may  also  recall  Newton's  remarkable  demonstration  that,  had 
the  force  of  gravity  varied  inversely  as  the  cube  instead  of  the 
square  of  the  distance,  the  planets,  instead  of  being  bound  to  their 

*  Instead  of  r=a^,  we  might  write  r  =  rQa^;  in  which  case  r^  is  the  value  of  r 
for  zero  value  of  6. 

f  Of  the  two  names  for  this  spiral,  equiangular  and  logarithmic,  I  used  the 
latter  in  my  first  edition,  but  equiangular  spiral  seems  to  be  the  better  name; 
for  the  constant  angle  is  its  most  distinguishing  characteristic,  and  that  which 
leads  to  its  remarkable  property  of  continuous  self-similarity.  Equiangular  spiral 
is  its  name  in  geometry;  it  is  the  analyst  who  derives  from  its  geometrical  pro- 
perties its  relation  to  the  logarithm.  The  mechanical  as  well^as  the  mathematical 
properties  of  this  curve  are  very  numerous.  A  Swedish  admiral,  in  the  eighteenth 
century,  shewed  an  equiangular  spiral  (of  a  certain  angle)  to  be  the  best  form  for 
an  anchor-fluke  {Sv.  Vet.  Akad.  Hdl.  xv,  pp.  1-24,  1796),  and  in  a  parrot's 
beak  it  has  the  same  efl&ciency.  Macquorn  Rankine  shewed  its  advantages  in 
the  pitch  of  a  cam  or  non-circular  wheel  {^Manual  of  Mechanics,  1859,  pp.  99-102; 
cf.  R.  C.  Archibald,  Scripta  Mathem.  m  (4),  p.  366,  1935). 

X  James  Bernoulli,  in  Acta  Eruditorum,  1691,  p.  282;  P.  Nicolas,  De  novis 
spiralibus,  Tolosae,  1693,  p.  27;  E.  Halley,  Phil.  Trans,  xix,  p.  58,  1696;  Roger 
Cotes,  ibid.  1714,  and  Harmonia  Mensurarum,  1722,  p.  19.  For  the  further  history 
of  the  curve  see  (e.g.)  Gomes  de  Teixeira,  Traite  des  courbes  remarqicables,  Coimbre, 
1909,  pp.  76-86;  Gino  Loria,  Spezielle  algebrdische  Kurven,  ii,  p.  60  scq.,  1911; 
R.  C.  Archibald  (to  whom  I  am  much  indebted)  in  Amer.  Mathem.  Monthly,  xxv, 
pp.  189-193,  1918,  and  in  Jay  Hambidge's  Dynamic  Symmetry,  1920,  pp.  146-157. 


756 


THE  EQUIANGULAR  SPIRAL 


[CH. 


ellipses,  would  have  been  shot  off  in  spiral  orbits  from  the  sun,  the 
equiangular  spiral  being  one  case  thereof.* 

A  singular  instance  of  the  same  spiral  is  given  by  the  route  which 
certain  insects  follow  towards  a  candle.  Owing  to  the  structure 
of  their  compound  eyes,  these  insects  do  not  look  straight  ahead 
but  make  for  a  light  which  they  see  abeam,  at  a  certain  angle. 
As. they  continually  adjust  their  path  to  this  constant  angle,  a  spiral 
pathway  brings  them  to  their  destination  at  last|. 

In  mechanical  structures,  curvature  is  essentially  a  mechanical 
phenomenon.     It  is  found  in  flexible  structures  as  the  result  of 


Fig.  351.  Spiral  path  of  an  insect, 
as  it  draws  towards  a  light. 
From  Wigglesworth  (after  van 
Buddenbroek). 


Fig.  352.      Dynamical   aspect 
of  the  equiangular  spiral. 


bending,  or  it  may  be  introduced  into  the  construction  for  the 
purpose  of  resisting  such  a  bending-moment.  But  neither  shell  nor 
tooth  nor  claw  are  flexible  structures ;  they  have  not  been  bent  into 
their  pecuhar  curvature,  they  have  grown  into  it. 

We  may  for  a  moment,  however,  regard  the  equiangular  or  logarithmic 
spiral  of  our  shell  from  the  dynamical  point  of  view,  by  looking  on  growth 
itself  as  the  force  concerned.  In  the  growing  structure,  let  growth  at  any 
point  P  be  resolved  into  a  force  F  acting  along  the  line  joining  P  to  a  pole  0, 
and  a  force  T  acting  in  a  direction  perpendicular  to  OP;  and  let  the  magnitude 
of  these  forces  (or  of  these  rates  of  growth)  remain  constant.     It  follows  that 

*  Principia,  I,  9;  ii,  15,    On  these  "Cotes's  spirals"  see  Tait  and  Steele,  p.  147. 
t  Cf.  W.  Buddenbroek,  Sitzungsber.  Heidelb.  Akad.,  1917;   V.  H.  Wigglesworth, 
Insect  Physiology,  1839,  p.  167. 


XI]  IN  ITS  DYNAMICAL  ASPECT  757 

the  resultant  of  the  forces  F  and  T  (as  PQ)  makes  a  constant  angle  with 
the  radius  vector.  But  a  constant  angle  between  tangent  and  radius  vector 
is  a  fundamental  property  of  the  "equiangular"  spiral:  the  very  property 
with  which  Descartes  started  his  investigation,  and  that  which  gives  its 
alternative  name  to  the  curve. 

In  such  a  spiral,  radial  growth  and  growth  in  the  direction  of  the  curve 
bear  a  constant  ratio  to  one  another.  For,  if  we  consider  a  consecutive 
radius  vector,  0P\  whose  increment  as  compared  with  OP  is  dr,  while  ds  is 
the  small  arc  PP',  then  dr/ds  =  cos  a  =  constant. 

In  the  growth  of  a  shell,  we  can  conceive  no  simpler  law  than 
this,  namely,  that  it  shall  widen  and  lengthen  in  the  same  unvarying 
proportions:  and  this  simplest  of  laws  is  that  which  Nature  tends 
to  follow.  The  shell,  hke  the  creature  within  it,  grows  in  size 
but  does  not  change  its  shape;  and  the  existence  of  this  constant 
relativity  of  growth,  or  constant  similarity  of  form,  is  of  the  essence, 
and  may  be  made  the  basis  of  a  definition,  of  the  equiangular  spiral*. 

Such  a.  definition,  though  not  commonly  used  by  mathematicians, 
has  been  occasionally  employed;  and  it  is  one  from  which  the  other 
properties  of  the  curve  can  be  deduced  with  great  ease  and  sim- 
plicity. In  mathematical  language  it  would  run  as  follows:  "Any 
[plane]  curve  proceeding  from  a  fixed  point  (which  is  called  the 
pole),  and  such  that  the  arc  intercepted  between  any  two  radii  at 
a  given  angle  to  one  another  is  always  similar  to  itself,  is  called  an 
equiangular,  or  logarithmic,  spiral." 

In  this  definition,  we  have  the  most  fundamental  and  "intrinsic" 
property  of  the  curve,  namely  the  property  of  continual  similarity, 
and  the  very  property  by  reason  of  which  it  is  associated  with 
organic  growth  in  such  structures  as  the  horn  or  the  shell.  For  it 
is  peculiarly  characteristic  of  the  spiral  shell,  for  instance,  that  it 
does  not  alter  as  it  grows ;  each  increment  is  similar  to  its  predecessor, 
and  the  whole,  after  every  spurt  of  growth,  is  just  Uke  what  it  was 
before.  We  feel  no  surprise  when  the  animal  which  secretes 
the  shell,  or  any  other  animal  whatsoever,  grows  by  such  sym- 
metrical expansion  as  to  preserve- its  form  unchanged;  though  even 
there,  as  we  have  already  seen,  the  unchanging  form  denotes  a  nice 
balance  between  the  rates  of  growth  in  various  directions,  which  is 

*  See  an  interesting  paper  by  W.  A.  Whitworth,  The  equiangular  spiral,  its  chief 
properties  proved  geometrically,  Messenger  of  Mathematics  (1),  i,  p.  5,  1862.  The 
celebrated  Christian  Wiener  gave  an  explanation  on  these  lines  of  the  logarithmic 
spiral  of  the  shell,  in  his  highly  original  Grundzuge  der  Weltordnung,  1863. 


758  THE  EQUIANGULAR  SPIRAL  [ch. 

but  seldom  accurately  maintained  for  long.  But  the  shell  retains 
its  unchanging  form  in  spite  of  its  asymmetrical  growth;  it  grows 
at  one  end  only,  and  so  does  the  horn.  And  this  remarkable 
property  of  increasing  by  terminal  growth,  but  nevertheless  retaining 
.unchanged  the  form  of  the  entire  figure,  is  cnaracteristic  of  the 
equiangular  spiral,  and  of  no  other  mathematical  curve.  It  well 
deserves  the  name,  by  which  James  Bernoulli  was  wont  to  call  it, 
of  spira  mirahilis. 

We  may  at  once  illustrate  this  curious  phenomenon  by  drawing 
the  outline  of  a  Uttle  Nautilus  shell  within  a  big  one.  We  know, 
or  we  may  see  at  once,  that  they  are  of  precisely  the  same  shape ; 
so  that,  if  we  look  at  the  little  shell  through  a  magnifying  glass, 
it  becomes  identical  with  the  big  one.  But  we  know,  on  'the  other 
hand,  that  the  little  Nautilus  shell  grows  into  the  big  one,  not  by 
growth  or  magnification  in  all  parts  and  directions,  as  when  the  boy 
grows  into  the  man,  but  by  growing  at  one  end  only. 

If  we  should  want  further  proof  or  illustration  of  the  fact  that  the  spiral 
shell  remains  of  the  same  shape  while  increasing  in  magnitude  by  its  terminal 
growth,  we  may  find  it  by  help  of  our  ratio  W  :  L^,  which  remains  constant 
so  long  as  the  shape  remains  unchanged.  Here  are  weights  and  measurements 
of  a  series  of  small  land-shells  {Clausilia):* 

W  (mgm.)  L  (mm.)  </lf/L 


50 

14-4 

2-56 

53 

151 

2-49 

56 

15-2 

2-52 

56 

15-2 

2-52 

56 

15-4 

2-44 

58 

15-5 

2-50 

61 

16-4 

2-40 

63 

160 

2-49 

67 

160 

2-54 

69 

161 

2-56 
Mean     2-50 

Though  of  all  plane  curves,  this  property  of  continued  similarity 
is  found  only  in  the  equiangular  spiral,  there  are  many  rectilinear 
figures  in  which  it  may  be  shewn.     For  instance,  it  holds  good  of 

*  In  100  specimens  of  Clausilia  the  mean  value  of  '^'WjL  was  found  to  be 
2-517,  the  coefficient  of  variation  0092,  and  the  standard  deviation  3-6.  That 
is  to  say,  over  90  per  cent,  grouped  themselves  about  a  mean  value  of  2-5  with 
a  deviation  of  less  than  4  per  /;ent.  Cf.  C.  Petersen,  Das  Quotientengesetz,  1921, 
p.  55. 


XI]  CONCERNING  GNOMONS  759 

any  cone;  for  evidently,  in  Fig.  353,  the  little  inner  cone  (repre- 
sented in  its  triangular  section)  may  become  identical  with  the  larger 
one  either  by  magnification  all  round  (as  in  a),  or  by  an  increment 
at  one  end  (as  in  b) ;  or  for  that  matter  on  the  rest  of  its  surface, 
represented  by  the  other  two  sides,  as  in  c.  All  this  is  associated 
with  the  fact,  which  we  have  already  noted,  that  the  Nautilus  shell 
is  but  a  cone  rolled  up;  that,  in  other  words,  the  cone  is  but  a 
particular  variety,  or  "hmiting  case,"  of  the  spiral  shell. 

This  singular  property  of  continued  similarity,  which  we  see  in 
the  cone,  and  recognise  as  characteristic  of  the  logarithmic  spiral, 
would  seem,  under  a  more  general  aspect,  to  have  engaged  the 
particular  attention  of  ancient  mathematicians  even  from  the  days 
of  Pythagoras,  and  so,  with  little  doubt,  from  the  still  more  ancient 
days  of  that  Egyptian  school  whence  he  derived  the  foundations  of 


his  learning*;  and  its  bearing  on  our  biological  problem  of  the 
shell,  however  indirect,  is  close  enough  to  deserve  our  very  careful 
consideration. 

There  are  certain  things,  says  Aristotle,  which  suffer  no  alteration 
(save  of  magnitude)  when  they  growf .  Thus  if  we  add  to  a  square 
an  L-shaped  portion,  shaped  Hke  a  carpenter's  square,  the  resulting 
figure  is  still  a  square ;  and  the  portion  which  we  have  so  added,  with 
this  singular  result,  is  called  in  Greek  a  "gnomon." 

EucUd  extends  the  term  to  include  the  case  of  any  parallelogram  J, 
whether  rectangular  or  not  (Fig.^  354);    and  Hero  of  Alexandria 

*  I  am  well  aware  that  the  debt  of  Greek  science  to  Egypt  and  the  East  is 
vigorously  denied  by  many  scholars,  some  of  whom  go  so  far  as  to  believe  that  the 
Egyptians  never  had  any  science,  save  only  some  "rough  rules  of  thumb  for 
measuring  fields  and  pyramids"  (Burnet's  Greek  Philosophy,  1914,  p.  5). 

t  Categ.  14,  15a,  30:  ?(Ttl  tipo.  av^au 6 fxeva  a  ouk  dWoiourat,  olov  to  Terpdyuvop, 
•)vu)ixouo^  wepiTedevTo^,  rjv^rjrat.  fiiv  dWoidrepov  5e  oudev  yeyiurjTai. 

I  Euclid  (II,  def.  2). 


760  THE  EQUIANGULAR  SPIRAL  [ch. 

specifically  defines  a  gnomon  (as  indeed  Aristotle  had  implicitly 
defined  it),  as  any  figure  which,  being  added  to  any  figure  what- 
soever, leaves  the  resultant  figure  similar  to  the  original.  Included 
in  this  important  definition  is  the  case  of  numbers,  considered 
geometrically;  that  is  to  say,  the  clbrjTLKot  apiOixol,  which  can  be 
translated  into  fwm,  by  means  of  rows  of  dots  or  other  signs  (cf. 
Arist.  Metaph.  1092  b  12),  or  in  the  pattern  of  a  tiled  floor:  all 
according  to   "the  mystical  way  of  Pythagoras,   and  the  secret 


Fig.  354.     Gnomonic  figures. 

magick  ot  numbers."  For  instance,  the  triangular  numbers,  1,  3, 
6,  10  etc.,  have  the  natural  numbers  for  their  "differences";  and 
so  the  natural  numbers  may  be  called  their  gnomons,  because  they 
keep  the  triangular  numbers  still  triangular.  In  hke  manner  the 
square  numbers  have  the  successive  odd  numbers  for  their  gnomons, 
as  follows: 

0+  1-  P 

P  +  3  =  22 

22  +  5  =  32 

32  +  7  =  42    etc. 

And  this  gnomonic  relation  we  may  illustrate  graphically  {axr]fJiaTo- 
ypa<f>€tv)  by  the  dots  whose  addition  keeps  the  annexed  figures 
perfect  squares*: 

•  •     •  •     •     • 

•     •  •     •     • 


There  are  other  gnomonic  figures  more  curious  still.   For  example, 
if  we  make  a  rectangle  (Fig.  355)  such  that  the  two  sides  are  in  the 


*  Cf.  Treutlein,  Ztschr.  /.  Math.  u.  Phya.  {Hist.  litt.  Abth.),  xxviii,  p.  209,  1883. 


XI] 


CONCERNING  GNOMONS 


761 


ratio  of  1 :  a/2,  it  is  obvious  that,  on  doubling  it,  we  obtain  a  similar 
figure;  for  1 :  V2  : :  a/2  :  2;  and  each  half  of  the  figure,  accordingly, 
is  now  a  gnomon  to  the  other.  Were  we  to  make  our  paper  of  such 
a  shape  (say,  roughly,  10  in.  x  7  in.),  we  might  fold  and  fold  it, 
and  the  shape  of  folio,  quarto  and  octavo  pages  would  be  all  the 
same.  For  another  elegant  example,  let  us  start  with  a  rectangle 
(A)  whose  sides  are  in  the  proportion  of  the  "divine"  or  "golden 
section*"  that  is  to  say  as  1  :  J  {Vb  —  1),  or,  approximately,  as 
1:0-618....  The  gnomon  to  this  rectangle  is  the  square  (B) 
erected  on  its  longer  side,  and  so  on  successively  (Fig.  356). 


1 

V2 

1 

V2 

V2 


1 
Fig.  355. 


0 


Fig.  356. 


In  any  triangle,  as  Hero  of  Alexandria  tells  us,  one  part  is  always 
a  gnomon  to  the  other  part.  For  instance,  in  the  triangle  ABC 
(Fig.  357),  let  us  draw  BD,  so  as  to  make  the  angle  CBD  equal  to 
the  angle  A.  Then  the  part  BCD  is  a  triangle  similar  to  the  whole 
triangle  ABC,  and  ABD  is  a  gnomon  to  BCD.  A  very  elegant  case 
is  when  the  original  triangle  ABC  is  an  isosceles  triangle  having 
one  angle  of  36°,  and  the  other  two  angles,  therefore,  each  equal 
to  72°  (Fig.  358).  Then,  by  bisecting  one  of  the  angles  of  the  base, 
we  subdivide  the  large  isosceles  triangle  into  two  isosceles  triangles, 
of  which  one  is  similar  to  the  whole  figure  and  the  other  is  its 
gnomon  f.  There  is  good  reason  to  believe  that  this  triangle  was 
especially  studied  by  the  Pythagoreans;    for  it  lies  at  the  root  of 

*  Euclid,  Ti,  11. 

t  This  is  the  so-called  Dreifachgleichschenkelige  Dreieck;  cf.  Naber,  op.  infra 
cit.     The  ratio  1  :  0-618  is  again  not  hard  to  find  in  this  construction. 


762  THE  EQUIANGULAR  SPIRAL  [ch. 

many  interesting  geometrical  constructions,  such  as  the  regular 
pentagon,  and  its  mystical  "pentalpha,"  and  a  whole  range  of  other 
curious  figures  beloved  of  the  ancient  mathematicians  * :  culminating 

A 


C  B  C  B 

Fig.  357.  Fig.  358. 

in  the  regular,  or  pentagonal,  dodecahedron,  which  symbolised  the 
universe  itself,  and  with  which  Euclidean  geometry  ends. 

If  we  take  any  one  of  these  figures,  for  instance  the  isosceles 
triangle  which  we  have  just  described,  and  add  to  it  (or  subtract 


Fig.  359. 

from  it)  in  succession  a  series  of  gnomons,  so  converting  it  into  larger 
and  larger  (or  smaller  and  smaller)  triangles  all  similar  to  the  first, 
we  find  that  the  apices  (or  other  corresponding  points)  of  all  these 

*  See,  on  the  mathematical  history  of  the  gnomon,  Heath's  Euclid,  i,  passim, 
1908;  Zeuthen,  Theoreme  de  Pythagore,  Geneve,  1904;  also  a  curious  and 
interesting  book,  Das  Theorem  des  Pythagoras,  by  Dr  H.  A.  Naber,  Haarlem,  1908. 


XI]  CONCERNING  GNOMONS  763 

triangles  have  their  locus  upon  a  equiangular  spiral:  a  result  which 
follows  directly  froni  that  alternative  definition  of  the  equiangular 
spiral  which  I  have  quoted  from  Whit  worth  (p.  757). 

If  in  this,  or  any  other  isosceles  triangle,  we  take  corresponding 
median  lines  of  the  successive  triangles,  by  joining  C  to  the  mid-' 
point  (M)  of  AB,  and  D  to  the  mid-point  (N)  of  BC,  then  the  pole 
of  the  spiral,  or  centre  of  similitude  of  ABC  and  BCD,  is  the  point 
of  intersection  of  CM  and  DN*. 

Again,  ,we  may  build  up  a  series  of  right-angled  triangles,  each 
of  which  is  a  gnomon  to  the  preceding  figure;  and  here  again,  an 
equiangular  spiral  is  the  locus  of  corresponding  points  in  these  suc- 
cessive triangles.  And  lastly,  whensoever  we  fill  up  space  with  a 
collection  of  equal  and  similar  figures,  as  in  Figs.  360,  361,  there 
we  can  always  discover  a  series  of  equiangular  spirals  in  their 
successive  multiples*)*. 

Once  more,  then,  we  may  modify  our  definition,  and  say  that: 
"Any  plane  curve  proceeding  from  a  fixed  point  (or  pole),  and  such 
that  the  vectorial  area  of  any  sector  is  always  a  gnomon  to  the 
whole  preceding  figure,  is  called  an  equiangular,  or  logarithmic, 
spiral."  And  we  may  now  introduce  this  new  concept  and  nomen- 
clature into  our  description  of  the  Nautilus  shell  and  other  related 
organic  forms,  by  saying  that:  (1)  if  a  growing  structure  be  built 
up  of  successive  parts,  similar  in  form,  magnified  in  geometrical 
progression,  and  similarly  situated  with  respect'  to  a  centre  of 
similitude,  we  can  always  trace  through  corresponding  points  a 
series  of  equiangular  spirals;    and  (2)  it  is  characteristic  of  the 

*  I  owe  this  simple  but  novel  construction,  like  so  much  else,  to  Dr  G.  T.  Bennett. 

t  In  each  and  all  of  these  gnomonic  figures  we  may  now  recognise  a  never- 
ending  polygon,  with  equal  angles  at  its  corners,  and  with  its  successive  sides  in 
geometrical  progression;  and  such  a  polygon  we  may  look  upon  as  the  natural 
precursor  of  the  equiangular  spiral.  If  we  call  the  exterior  or  "bending"  angle 
of  th'^  polygon  j3,  and  the  ratio  of  its  sides  A,  then  the  vertices  lie  on  an  equiangular 
spiral  of  angle  a,  given  by  logg  A  =  j3  cot  a.  In  the  spiral  of  Fig.  359  the  constant 
angle  is  thus  found  to  be  about  75°  40',  in  "that  of  Fig.  355,  77°  40',  and  in  that  of 
Fig.  356,  72°  50'. 

The  calculation  is  as  follows.     Taking,  for  example,  the  successive  triangles  of 
Fig.  359,  the  ratio  (A)  of  the  sides,  as  BC  :  AC,  is  that  of  the  golden  section,  1 : 1-618. 
The  external  angle  (^),  as  ADB,  is  108°,  or  in  radians  1-885.  Then 
log  1-618=0-209,  from  which  log,  1-618=0-481 

and  cota  =  !^  =  ^^=0-255  =  cot75°4y. 


764  THE  EQUIANGULAR  SPIRAL  [ch. 

growth  of  the  horn,  of  the  shell,  and  of  all  other  organic  forms  in 
which  an  equiangular  spiral  can  be  recognised,  that  each  successive 
increment  of  growth  is  similar,  and  similarly  magnified,  and  similarly 


1 

\\ 

\ 

\ 

. 

\ 

\ 

^ 

\\ 

\ 

V 

\ 

u 

^ 

\ 

\ 

/ 

\ 

\ 

/ 

1 

\ 

\, 

\ 

/ 

/ 

^^ 

\ 

z 

^ 

y 

Fig.  360*.     Logarithmic  spiral  derived  from  corresponding  points  in 
a  system  of  squares. 


Fig.  361.     The  same  in  a  system  of  hexagons.     From  Naber. 

situated  to  its  predecessor,  and  is  in  consequence  a  gnomon  to  the  entire 
pre-existing  structure.     Conversely  (3)  it  follows  that  in  the  spiral 

*  This  diagram  was  at  fault  in  my  first  edition  (p.  512),  as  Dr  G.  T.  Bennett  shews 
me.  The  curve  met  its  chords  at  equal  angles  at  either  end:  whereas  it  ought  to 
meet  the  further  end  at  a  lesser  angle  than  the  other,  and  ought  in  consequence  to 
intersect  the  hues  of  the  coordinate  framework.  The  constant  angle  of  this  spiral 
is  about  66°  11'  (tan  a=7r/2  logg2). 


XI]  CONCERNING  GNOMONS  765 

outline  of  the  shell  or  of  the  horn  we  can  always  inscribe  an  endless 
variety  of  other  gnomonic  figures,  having  no  necessary  relation, 
save  as  a  mathematical  accident,  to  the  nature  or  mode  of  develop- 
ment of  the  actual  structure*.  But  observe  that  the  gnomons  to 
a  square  may  form  increments  of  any  size,  and  the  same  is  true  of 
the  gnomons  to  a  Haliotis-sheW ;  but"  in  the  higher  symmetry  of  a 
chambered  Nautilus,  or  of  the  successive  triangles  in  Fig.  359, 
growth  goes  on  by  a  progressive  series  of  gnomons,  each  one  of 
which  is  the  gnomon  to  another. 


Fig.  362.  A  shell  of  Haliotis,  with  two  of  the  many  hues  of  growtii,  or  generating 
curves,  marked  out  in  black:  the  areas  bounded  by  these  hnes  of  growth  being 
in  all  cases  gnomons  to  the  pre-existing  shell. 

Of  these  three  propositions,  the  second  is  of  great  use  and 
advantage  for  our  easy  understanding  and  simple  description  of 
the  molluscan  shell,  and  of  a  great  variety  of  other  structures  whose 
mode  of  growth  is  analogous,  and  whose  mathematical  properties 
are  therefore  identical.  We  see  that  the  successive  chambers  of  a 
spiral  Nautilus  or  of  a  straight  Orthocems,  each  whorl  or  part  of  a 
whorl  of  a  periwinkle  or  other  gastropod,  each  new  increment  of  the 
operculum  of  a  gastropod,  each  additional  increment  of  an  elephant's 
tusk,  or  each  new  chamber  of  a  spiral  foraminifer,  has  its  leading 
characteristic  at  once  described  and  its  form  so  far  explained  by  the 

*  For  many  beautiful  geometrical  constructions  based  on  the  molluscan  shell, 
see  S.  Colman  and  C.  A.  Coan,  Nature  a  Harmonic  Unity  (ch.  ix,  Conchology), 
New  York,  1912. 


766  THE  EQUIANGULAR  SPIRAL  [ch. 

simple  statement  that  it  constitutes  a  gnomon  to  the  whole  previously 
existing  structure.  And  herein  lies  the  explanation  of  that  "time- 
element"  in  the  development  of  organic  spirals  of  which  we  have 
spoken  already;   for  it  follows  as  a  simple  corollary  to  this  theory 


M^iZ 


Fig.  363.     A  spiral  foraminifer  {Pnlvinulina),  to  shew  how  each  successive  chanibei 
continues  the  symmetry  of,  or  constitutes  a  gnomon  to,  the  rest  of  the  structure. 

of  gnomons  that  we  nmst  never  expect  to  find  the  logarithmic  spiral 
manifested  in  a  structure  whose  parts  are  simultaneously  produced, 
as  for  instance  in  the  maririn  of  a  leaf,  or  among  the  many  curves 
that  make  the  contour  of  a  fish.  But  we  most  look  for  it  wherever 
,   -  '   the  organism  retains,  and  still  presents 

at  a  single  view,  the  successive  phases  of 
preceding  growth :  the  successive  magni- 
tudes attained,  the  successive  outhnes 
occui)ied,  as  growth  pursued  the  even 
tenor  of  its  way.  And  it  follows  from 
this  that  it  is  in  the  hard  parts  of 
organisms,  and  not  the  soft,  fleshy, 
actively  growing  parts,  that  this  spiral 
is  commonly  and  characteristically 
found:  not  in  the  fresh  mobile  tisssue 
whose  form  is  constrained  merely  by 
the  active  forces  of  the  moment;  but 
in  things  like  shell  and  tusk,  and  horn  and  claw,  visibly  composed 


Fig.  364.     Another  spiral  fora-. 
minifer,  Cristellaria. 


XI 


THE  CYMOSE  INFLORESCENCE 


767 


of  parts  successively  and  permanently  laid  down.  The  shell-less 
molluscs  are  never  spiral;  the  snail  is  spiral  but  not  the  slug*.  In 
short,  it  is  the  shell  which  curves  the  sr^ail,  and  not  the  snail  which 
curves  the  shell.  The  logarithmic  spiral  is  characteristic,  not  of  the 
hving  tissues,  but  of  the  dead.  And  for  the  same  reason  if  will 
always  or  nearly  always  be  accompanied,  and  adorned,  by  a  pattern 
formed  of  "hues  of  growth,"  the  lasting  record  of  successive  stages 
of  form  and  magnitude!. 

The  cymose  inflorescences  of  the  botanists  are  analogous  in  a 
curious  and  instructive  way  to  the  equiangular  spiral. 

In  Fig.  365  B  (which  represents  the  Cicinnus  of  Schimper,  or  cyme  unipare 
scorpioide  of  Bravais,  as  seen  in  the  Borage),  we  begin  with  a  primary  shoot 
from  which  is  given  off,  at  a  certain  definite  angle, 
a  secondary  shoot:  and  from  that  in  turn,  on  the 
same  side  and  at  the  same  angle,  another  shoot,  and 
so  on.  The  deflection,  or  curvature,  is  continuous 
and  progressive,  for  it  is  caused  by  no  external 
force  but  only  by  causes  intrinsic  in  the  system. 
And  the  whole  system  is  symmetrical:  the  angles 
at  which  the  successive  shoots  are  given  off  being 
all  equal,  and  the  lengths  of  the  shoots  diminishing 
in  constant  ratio.  The  result  is  that  the  successive 
shoots,  or  successive  increments  of  growth,  are 
tangents  to  a  curve,  and  this  curve  is  a  true 
logarithmic  spiral.  Or  in  other  words,  we  may 
regard  each  successive  shoot  as  forming,  or  defining, 
a  gnomon  to  the  preceding  structure.  While  in 
this  simple  case  the  successive  shoots  are  depicted 
as  lying  in  a  plane,  it  may  also  happen  that,  in  addition  to  their  successive 
angular  divergence  from  one  another  wildiin  that  plane,  they  also  tend  to 


Fig.  365.     A,  a  helicoid; 
B,  jbl  scorpioid  cyme. 


*  Note  also  that  Chiton,  where  the  pieces  of  the  shell  are  disconnected,  shews 
no  sign  of  spirality. 

t  That  the  invert  to  an  equiangular  spiral  is  identical  with  the  original  curve 
does  not  concern  us  in  our  study  of  organic  form,  but  it  is  one  of  the  most  beautiful 
and  most  singular  properties  of  the  curve.  It  was  this  which  led  James  Bernoulli, 
in  imitation  of  Archimedfes,  to  have  the  logarithmic  spiral  inscribed  upon  his  tomb; 
and  on  John  Goodsir's  grave  near  Edinburgh  the  same  symbol  is  reinscribed. 
Bernoulli's  account  of  the  matter  is  interesting  and  remarkable:  "Cum  autem 
ob  proprietatem  tam  singularem  tamque  admirabilem  mire  mihi  placeat  spira 
haec  mirabilis,  sic  ut  ejus  contemplatione  satiari  vix  nequeam:  cogitavi  iUam 
ad  varias  res  symbolice  repraesentandas  non  inconcinne  adhiberi  posse.  Quoniam 
enim  semper  sibi  et  eandera  spiram  gignit,  utcunque  volvatur.  evjlvatur,  radiet, 
hinc  poterit  esse  vei  sobolis  parentibus  per  omnia  similis  Emblema:  Simillima 
Filia  Matri;    vel  (si  rem  aeternae  veritatis  Fidel  mysteriis  accommodare  non  eat 


768  THE  EQUIANGULAR  SPIRAL  [ch. 

diverge  by  successive  equal  angles  from  that  plane  of  reference;  and  by 
this  means,  there  will  be  superposed  upon  the  equiangular  spiral  a  twist  or 
screw.  And,  in  the  particular  case  where  this  latter  angle  of  divergence  is 
just  equal  to  180°,  or  two  right  angles,  the  successive  shoots  will  once  more 
come  to  lie  in  a  plane,  but  they  will  appear  to  come  off  from  one  another  on 
alternate  sides,  as  in  Fig.  365  A.  This  is  the  Schrauhel  or  Bostryx  of  Schimper, 
the  cyme  unipare  helicoide  of  Bravais.  The  equiangular  spiral  is  still  latent 
in  it,  as  in  the  other;  but  is  concealed  from  view  by  the  deformation  resulting 
from  the  helicoid.  Many  botanists  did  not  recognise  (as  the  brothers  Bravais 
did)  the  mathematical  significance  of  the  latter  case,  but  were  led  by  the 
snail-like  spiral  of  the  scorpoid  cyme  to  transfer  the  name  "helicoid"  to  it*. 

The  spiral  curve  of  the  shell  is,  in  a  sense,  a  vector  diagram  of  its 
own  growth;  for  it  shews  at  each  instant  of  time  the  direction, 
radial  and  tangential,  of  growth,  and  the  unchanging  ratio  of 
velocities  in  these  directions.  Regarding  the  actual  velocity  of 
growth  in  the  shell,  we  know  very  little  by  way  of  experimental 
measurement;  but  if  we  make  a  certain  simple  assumption,  then 
we  may  go  a  good  deal  further  in  our  description  of  the  equiangular 
spiral  as  it  appears  in  this  concrete  case. 

Let  us  make  the  assumption  that  similar  increments  are  added 
to  the  shell  in  equal  times;  that  is  to  say,  that  the  amount  of 
growth  in  unit  time  is  measured  by  the  areas  subtended  by  equal 
angles.  Thus,  in  the  outer  whorl  of  a  spiral  shell  a  definite  area 
marked  out  by  ridges,  tubercles,  etc.,  has  very  different  linear 
dimensions  to  the  corresponding  areas  of  an  inner  whorl,  but  the 
symmetry  of  the  figure  imphes  that  it  subtends  an  equal  angle 
with  these;  and  it  is  reasonable  to  suppose  that  the  successive 
regions,  marked  out  in  this  way  by  successive  natural  boundaries 
or  patterns,  are  produced  in  equal  intervals  of  time. 

prohibitum)  ipsius  aeternae  generationis  Filii,  qui  Patris  veluti  Imago,  et  ab  illo  ut 
Lumen  a  Lumine  emanans,  eidem  bixoLovaLo%  existit,  qualiscunque  adumbratio.  Aut, 
si  mavis,  quia  Curva  nostra  mirabilis  in  ipsa  mutatione  semper  sibi  constantissime 
manet  similis  at  numero  eadem,  poterit  esse  vel  fortitudinis  et  constantiae  in 
adversitatibus,  vel  etiam  Carnis  nostrae  post  varias  alterationes  et  tandem  ipsam 
quoque  mortem,  ejusdem  numero  resurrecturae  symbolum:  adeo  quidem,  ut  si 
Archimedem  imitandi  hodiernum  consuetudo  obtineret,  libenter  Spiram  hanc  tumulo 
meo  juberem  incidi,  cum  Epigraphe,  Eadem  numero  mutata  resurgeV;  Acta  Erudi- 
torum,  M.  Mali,  1692,  p.  213.  Cf.  L.  Isely,  Epigraphes  tumulaires  de  mathe- 
maticiens,  Bull.  Soc.  Set.  nat.  Neuchdtel.  xxvii,  p.  171,  1899. 

*  The  names  of  these  structures  have  been  often  confused  and  misunderstood ; 
cf.  S.  H.  Vines,  The  history  of  the  scorpioid  cyme,  Journ.  Bot.  (n.s.),  x,  pp.  3-9, 
1881. 


XI]  ITS  MATHEMATICAL  PROPERTIES  769 

If  this  be  so,  the  radii  measured  from  the  pole  to  the  boundary 
of  the  shell  will  in  each  case  be  proportional  to  the  velocity  of 
growth  at  this  point  upon  the  circumference,  and  at  the  time  when 
it  corresponded  with  the  outer  hp,  or  region  of  active  growth ;  and 
while  the  direction  of  the  radius  vector  corresponds  with  the 
direction  of  growth  in  thickness  of  the  animal,  so  does  the  tangent 
to  bhe  curve  correspond  with  the  direction,  for  the  time  being,  of 
the  animal's  growth  in  length.  The  successive  radii  are  a  measure 
of  the  acceleration  of  growth,  and  the  spiral  curve  of  the  shell 
itself,  if  the  radius  rotate  uniformly,  is  no  other  than  the  hodograph 
of  the  growth  of  the  contained  organism*. 

So  far  as  we  have  now  gone,  we  have  studied  the  elementary 
properties  of  the  equiangular  spiral,  including  its  fundamental 
property  of  continued  si?nilarity;  and  we  have  accordingly  learned 
that  the  shell  or  the  horn  tends  necessarily  to  assume  the  form 
of  this  mathematical  figure,  because  in  these  structures  growth 
proceeds  by  successive  increments  which  are  ailways  similar  in 
form,  similarly  situated,  and  of  constant  relative  magnitude  one 
to  another.  Our  chief  objects  in  enquiring  further  into  the  mathe- 
matical properties  of  the  equiangular  spiral  will  be:  (1)  to  find 
means  of  confirming  and  verifying  the  fact  that  the  shell  (or  other 
organic  curve)  is  actually  an  equiangular  spiral;  (2)  to  learn  how, 
by  the  properties  of  the  curve,  we  may  further  extend  our  knowledge 
or  simphfy  our  descriptions  of  the  shell;  and  (3)  to  understand  the 
factors  by  which  the  characteristic  form  of  any  particular  equiangular 
spiral  is  determined,  and  so  to  comprehend  the  nature  of  the  specific 
or  generic  differences  between  one  spiral  shell  and  another. 

Of  the  elementary  properties  of  the  equiangular  spiral  the 
following  are  those  which  we  may  most  easily  investigate  in  the 
concrete  case  of  the  molluscan  shell:  (1)  that  the  polar  radii  whose 
vectorial  angles  are  in  arithmetical  progression  are  themselves  in 
geometrical  progression;  hence  (2)  that  the  vectorial  angles  are  pro- 
portional to  the  logarithms  of  the  corresponding  radii ;  and  (3)  that 
the  tangent  at  any  point  of  an  equiangular  spiral  makes  a  constant 
angle  (called  the  angle  of  the  spiral)  with  the  polar  radius  vector. 

*  The  hodograph  of  a  logarithmic  spiral  (i.e.  of  a  point  which  lies  on  a  uniformly 
revolving  radius  and  describes  a  logarithmic  spiral)  is  likewise  a  logarithmic  spiral : 
W.  Walton,  Collection  of  Problerha  in  Theoretical  Mechanics  (3rd  ed.),  1876,  p.  296. 


770  THE  EQUIANGULAR  SPIRAL  [ch. 

The  first  of  these  propositions  may  be  written  in  a  simpler  form, 
as  follows:  radii  which  form  equal  angles  about  the  pole  of  the 
equiangular  spira]  are  themselves  continued  proportionals.  That 
is  to  say,  in  Fig.  366,  when  the  angle  ROQ  is  equal 
to  the  angle  QOP,  then  OP:OQ::OQ:  OR. 

A  particular  case  of  this  proposition  is  when  the 
equal  angles  are  each  angles  of  360° :  that  is  to  say 
when  in  each  case  the  radius  vector  makes  a  complete 
revolution,  and  when,  therefore,  P,  Q  and  R  all  he 
upon  tjie  same  radius. 

It  was  by  observing  with  the  help  of  very  careftil 
measurement  this  continued  proportionality,  that 
Moseley  was  enabled  to  verify  his  first  assumption, 
based  on  the  general  appearance  of  the  shell,  that 
the  shell  of  Nautilus  was  actually  an  equiangular 
spiral,  and  this  demonstration  he  was  soon  after- 

Via    ^ftfi 

wards  in  a  position  to  generalise  by  extending  it  to 
all  spiral  Ammonitoid  and  Gastropod  mollusca*.  For,  taking  a 
median  transverse  section  of  a  Nautilus  pompilius,  and  carefully 
measuring  the  successive  breadths  of  the  whorls  (from  the  dark  line 
which  marks  what  was  originally  the  outer  surface,  before  it  was 
covered  up  Tjy  fresh  deposits  on  the  part  of  the  growing  and 
advancing  shell),  Moseley  found  that  "the  distance  of  any  two  of  its 
whorls  measured  upon  a  radius  vector  is  one-third  that  of  the  two 
next  whorls  measured  upon  the  same  radius  vectorf.     Thus  (in 

*  The  Rev.  H.  Moseley,  On  the  geometrical  forms  of  turbinated  and  discoid 
shells,  Phil.  Trans.  1838,  Pt.  i,  pp.  351-370.  Reaumur,  in  describing  the  snail-shell 
(Mem.  Acad,  des  Sci.  1709,  p.  378),  had  a  glimpse  of  the  same  geometrical  law: 
"  Le  diametre  de  chaque  tour  des  spirale,  ou  sa  plus  grande  longueur,  est  a  peu  pres 
double  de  celui  qui  la  precede  et  la  moitie  de  celui  qui  la  suit."  Leslie  (in  his 
Geometry  of  Curved  Lines,  1822,  p.  438)  compared  the  "general  form  and  the 
elegant  septa  of  the  Nautilus'"  to  an  equiangular  spiral  and  a  series  of  its  involutes. 

f  It  will  be  observed  that  here  Moseley,  speaking  as  a  mathematician  and 
considering  the  linear  spiral,  speaks  of  whorls  when  he  means  the  linear  boundaries, 
or  lines  traced  by  the  revolving  radius  vector;  while  the  conchologist  usually 
applies  the  term  whorl  to  the  whole  space  between  the  two  boundaries.  As  con- 
chologists,  therefore,  we  call  the  breadth  of  a  whorl  what  Moseley  looked  updn  as 
the  distance  between  two  consecutive  whorls.  But  this  latter  nomenclature  Moseley 
himself  often  uses.  Observe  also  that  Moseley  gets  a  very  good  approximate  result 
by  his  measurements  "upon  a  radius  vector,"  although  he  has  to  be  content  with 
a  very  rough  determination  of  the  pole. 


XI] 


OF  THE  NAUTILUS  SHELL 


771 


Fig.  367),  ab  is  one-third  of  6c,  de  of  e/,  gh  of  hi,  and  kl  of  Im.     The 
curve  is  therefore  an  equiangular  spiral." 

The  numerical  ratio  in  the  case  of  the  Nautilus  happens  to  be 
one  of  unusual  simplicity.  Let  us  take,  with  ^oseley,  a  somewhat 
more  complicated  example. 


Fig.  367.     Spiral  of  the  Naviilua. 


Fig.  368.  Turritella  dupli- 
cata      (L.),      Moseley's 

.  Turbo  duplicatus.  From 
Chenu.     x  ^. 

From  the  apex  of  a  large  Turritella  {Turbo)  duplicata*  a  line 
was  drawn  across  its  whorls,  and  their  widths  were  measured  upon 
it  in  succession,  beginning  with  the  last  but  one.     The  measure- 

*  In  the  case  of  ''Turbo'",  and  all  other  turbinate  shells,  we  are  dealing  not  with 
a  plane  logarithmic  spiral,  as  in  Nautilus,  but  with  a  "gauche"  spiral,  such  that 
the  radius  vector  no  longer  revolves  in  a  plane  perpendicular  to  the  axis  of  the 
system,  but  is  inclined  to  that  axis  at  some  constant  angle  (j3).  The  figure  still 
preserves  its  continued  similarity,  and  may  be  called  a  logarithmic  spiral  in  space; 
indeed  it  is  commonly  spoken  of  as  a  logarithmic  spiral  wrapped  upon  a  cone,  its  pole 
coinciding  with  the  apex  of  the  cone.  It  follows  that  the  distances  of  successive 
whorls  of  the  spiral  measured  on  the  same  straight  Line  passing  through  the  apex 
of  the  cone  are  in  geometrical  progression,  and  conversely;  just  as  in  the  former 
case.  But  the  ratio  between  any  two  consecutive  interspaces  (i.e.  ^3  -  R2IR2  -  Ri) 
is  now  equal  to  e^'^^i^^cota^^  being  the  semi-angle  of  the  enveloping  cone.  (Cf. 
Moseley,  Phil.  Mag.  xxi,  p.  300,  1842.) 


772  THE  EQUIANGULAR  SPIRAL  [cH. 

ments  were,  as  before,  made  with  a  fine  pair  of  compasses  and  a 
diagonal  scale.  The  sight  was  assisted  by  a  magnifying  glass. 
In  a  parallel  colimin  to  the  following  admeasurements  are  the 
terms  of  a  geometric  progression,  whose  first  term  is  the  width  of 
the  widest  whorl  measured,  and  whose  common  ratio  is  1-1804. 

Turritella  duplicata 

Widths  of  successive  Terms  of  a  geometrical  progression, 

whorls,  measured  in  whose  first  term  is  the  width  of 

inches  and  parts  the  widest  whorl,  and  whose 

of  an  inch  common  ratio  is  1-1804 

1-31  1-310 

1-12  1110 

0-94  0-940 

0-80  0-797 

0-67  0-675 

0-57  0-572 

0-48  0-484 

0-41  0-410 

The  close  coincidence  between  the  observed  and  the  calculated 
figures  is  very  remarkable,  and  is  amply  sufiicient  to  justify  the 
conclusion  that  we  are  here  deaUng  with  a  true  logarithmic  spiral*. 

Nevertheless,  in  order  to  verify  his  conclusion  still  further, 
and  to  get  partially  rid  of  the  inaccuracies  due  to  successive  small 
measurements,  Moseley  proceeded  to  investigate  the  same  shell, 
measuring  not  single  whorls  but  groups  of  whorls  taken  several 
at  a  time:  making  use  of  the  following  property  of  a  geometrical 
progression,  that  ''if  /x  represent  the  ratio  of  the  sum  of  every 
even  number  (m)  of  its  terms  to  the  sum  of  half  that  number  of 
terms,  then  the  common  ratio  (r)  of  the  series  is  represented  by 
the  formula 

2 

r={fjL-  1)^." 

*  Moseley,  writing  a  hundred  years  ago,  uses  an  obsolete  nomenclature  which 
is  apt  to  be  very  misleading.  His  Turbo  duplicatus,  of  Linnaeus,  is  now  Turritella 
duplicata,  the  common  large  Indian  Turritella,  a  slender,  tapering  shell  with  a 
very  beautiful  spiral,  about  six  or  seven  inches  long.  But  the  operculum  which 
he  describes  as  that  of  Turbo  does  indeed  belong  to  that  genus,  sensu  stricto;  it  is 
the  well-known  calcareous  operculum  or  "eyestone"  of  some  such  common  species 
as  Turbo  petholatus.  Turritella  has  a  very  different  kind  of  operculum,  a  thin 
chitinous  disc  in  the  form  of  a  close  spiral  coil,  not  nearly  fiUing  up  the  aperture 
of  the  shell.  Moseley's  Turbo  phasianus  is  again  no  true  Turbo,  but  is  (to  judge 
from  his  figure)  Phasianella  bulimoides  Lam.  =P.  australis  (Gmelin);  and  his 
Buccinum  aubulatum  is  Terebra  subulata  (L.). 


XI]  OF  CERTAIN  OPERCULA  773 

Accordingly,  Moseley  made  the  following  measurements,  begin- 
ning from  the  second  and  third  whorls  respectively: 

Width  of  Ratio  fi 


Six  whorls 

Three  whorls 

5-37 
4-55 

203 
1-72 

2-645 
2-645 

Four  whorls 

Two  whorls 

415 
3-52 

1-74 
1-47 

2-385 
2-394 

"By  the  ratios  of  the  two  first  admeasurements,  the  formula 
gives 

/•=  (1-645)*=  M804. 

By  the  mean  of  the  ratios  deduced  from  the  second  two  admeasure- 
ments, it  gives 

r  =  (l-389)i  =  M806. 

"It  is  scarcely  possible  to  imagine  a  more  accurate  verification 
than  is  deduced  from  these  larger  admeasurements,  and  we  may 
with  safety  annex  to  the  species  Turbo  duplicatus  the  characteristic 
number  M8." 

By  similar  and  equally  concordant  observations,  Moseley  found 
for  Turbo  phasianus  the  characteristic  ratio,  1-75;  and  for  Bucci- 
num  subulatum  that  of  1*13. 

From  the  measurements  of  Turritella  dupUcata  (on  p.  772),  it  is 
perhaps  worth  while  to  illustrate  the  logarithmic  statement  of  the 
same  thing:  that  is  to  say,  the  elementary  fact,  or  corollary,  that  if 
the  successive  radii  be  in  geometric  progression,  their  logarithms  will 
differ  from  one  another  by  a  constant  amount. 

Turritella  duplicata 


Widths  of  suc- 

Logarithms 

Differences 

Ratios  of  suc- 

cessive whorls 

of  do. 

of  logarithms 

cessive  widths 

131 

211727 

— 

— 

112 

2-04922 

0-06805 

1-170 

94 

1-97313 

,    0-07609 

1-191 

80 

1-90309 

0-07004 

1-175 

67 

1-82607 

0-07702 

M94 

57 

1-75587 

0-07020 

1-175 

48 

1-68124 

0-07463 

1-188 

41 

1-61278 

0-06846 

1-171 

Mean    0-07207 

M806 

And  0-07207  is  the  logarithm  of  1-1805. 


774 


THE  EQUIANGULAR  SPIRAL 


[CH. 


Lastly,  we  may  if  we  please,  in  this  simple  case,  reduce  the  whole 
matter  to  arithmetic,  and,  dividing  the  width  of  each  whorl  by 
that  of  the  next,  see  that  these  quotients  are  nearly  identical,  and 
that  their  mean  value,  or  common  ratio,  is  precisely  that  which 
we  have  already  found. 

We  may  shew,  in  the  same  simple  fashion,  by  measurements 
of  Terebra  (Fig.  397),  how  the  relative  widths  of  successive  whorls 
fall  into  a  geometric  progression,  the  criterion  of  a  logarithmic 
spiral.  ^ 

Measurements  of  a  large  specimen  (15-5  cm.)  of  Terebra  maculata, 
along  three  several  tangents  (a,  6,  c)  to  the  whorls.  {After  Chr. 
Peterson,  1921.) 


Width  (mm.) 

Ratio 

25 

1-25 

20 

1-33 

15 

1-25 

12 

Width 


Ratio 


Width 


Ratio 


24-5 

23 

•25 

18^5 

1-32 

17-5 

1-31 

•33 

14 

132 

13-3 

1-31 

•25 

10-75 

130 

9-75 

136 

•33 

8 

1-34 

7-25 

1-34 

Mean 


1-29 


1-32 


1-33 


Mean  ratio,  b31 


The  logarithmic  spiral  is  not  only  very  beautifully  manifested  in 
the  molluscan  shell*,  but  also,  in  certain  cases,  in  the  little  Hd  or 
"operculum"  by  which  the  entrance  to  the  tubular  shell  is  closed 
after  the  animal  has  withdrawn  itself  withinf.  In  the  spiral  shell 
of  Turbo,  for  instance,  the  operculum  is  a  thick  calcareous  structure, 
with  a  beautifully  curved  outhne,  which  grows  by  successive  incre- 
ments applied  to  one  portion  of  its  edge,  and  shews,  accordingly, 
a  spiral  line  of  growth  upon  its  surface.  The  successive  increments 
leave  their  traces  on  the  surface  of  the  operculum  (Fig.  370),  which 
traces  have  the  form  of  curved  lines  in  Turbo,  and  of  straight  hnes 

*  It  has  even  been  proposed  to  use  a  logarithmic  spiral  in  place  of  a  table  of 
logarithms.  Cf.  Ant.  Favaro,  Statique  graphique,  Paris,  1885;  Hele-Shaw,  in 
Brit.  Ass.  Rep.  1892,  p.  403. 

t  Cf.  Fred.  Haussay,  Hecherches  sur  Vopercule,  Diss.,  Paris,  1884. 


XI 


OF  CERTAIN  OPERCULA 


775 


in  (e.g.)  Nerita  (Fig.  371);  that  is  to  say,  apart  from  the  side  con- 
stituting the  outer  edge  of  the  operculum  (which  side  is  always  and 
of  necessity  curved)  the  successive  increments  constitute  curvihnear 


Fig.  369.     Operculum  of  Turbo. 

triangles  in  the  one  case,  and   rectilinear  triangles   in  the  other. 
The  sides  of  these  triangles  are  tangents  to  the  spiral  Une  of  the 


Fig.  370.  Fig.  371. 

Figs.  370,  371.     Opercula  of  Turbo  and  Nerita.     After  Moseley. 

operculum,  and  may  be  supposed  to  generate  it  by  their  consecutive 
intersections. 

In  a  number  of  such  opercula,  Moseley  measured  the  breadths 


776  THE  EQUIANGULAR  SPIRAL  [ch. 

of  tlie  successive  whorls  along  a  radius  vector*,  just  in  the  same 


way  as  he  did  with  the  entire  shell  in  the  foregoing  cases; 
here  is  one  example  of  his  results. 

Operculum  of  Turbo  sp. ;  breadth  {in  inches)  of  successive 
whorls,  mea^redfrom  the  pole 


and 


Distance 

Ratio 

Distance 

Ratio 

Distance 

Ratio 

Distance 

Ratio 

0-24 

016 

0-2 

0-18 

2-28 

2-31 

2-30 

2-30 

0-55 

0-37 

0-6 

0-42 

2-32 

2-30 

2-30 

2-24 

1-28 

0-85 

1-38 

0-94 

The  ratio  is  approximately  constant,  and  this  spiral  also  is, 
therefore,  a  logarithmic  spiral. 

But  here  comes  in  a  very  beautiful  illustration  of  that  property 
of  the  logarithmic  spiral  which  causes  its  whole  shape  to  remain 
unchanged,  in  spite  of  its  apparently  unsymmetrical,  or  unilateral, 
mode  of  growth.  For  the  mouth  of  the  tubular  shell,  into  which 
the  operculum  has  to  fit,  is  growing  or  widening  on  all  sides:  while 
the  operculum  is  increasing,  not  by  additions  made  at  the  same 
time  all  round  its  margin,  but  by  additions  made  only  on  one  side 
of  it  at  each  successive  stage.  One  edge  of  the  operculum  thus 
remains  unaltered  as  it  advances  into  its  new  position,  and  comes  to 
occupy  a  new-formed  section  of  the  tube,  similar  to  but  greater  than 
the  last.  Nevertheless,  the  two  apposed  structures,  the  chamber  and 
its  plug,  at  all  times  fit  one  another  to  perfection.  The  mechanical 
problem  (by  no  means  an  easy  one)  is  thus  solved:  "How  to  shape 
a  tube  of  a  variable  section,  so  that  a  piston  driven  along  it  shall,  by 
one  side  of  its  margin,  coincide  continually  with  its  surface  as  it 
advances,  provided  only  that  the  piston  be  made  at  the  same  time 
continually  to  revolve  in  its  own  plane." 

As  Moseley  puts  it:  "That  the  same  edge  which  fitted  a  portion 
of  the  first  less  section  should  be  capable  of  adjustment,  so  as  to 
fit  a  portion  of  the  next  similar  but  greater  section,  supposes  a 
geometrical  provision  in  the  curved  form  of  the  chamber  of  great 

*  As  the  successive  increments  evidently  constitute  similar  figures,  similarly 
related  to  the  pole  (P),  it  follows  that  their  linear  dimensions  are  to  one  another 
as  the  radii  vectores  drawn  to  similar  points  in  them:  for  instance  as  PPj,  PP2> 
which  (in  Fig.  370)  are  radii  vectores  drawn  to  the  points  where  they  meet  the 
common  boundary. 


XI]  OF  CERTAIN  OPERCULA  777 

apparent  complication  and  difficulty.  But  God  hath  bestowed 
upon  this  humble  architect  the  practical  skill  of  a  learned  geo- 
metrician, and  he  makes  this  provision  with  admirable  precision 
in  that  curvature  of  the  logarithmic  spiral  which  he  gives  to  the 
section  of  the  shell.  This  curvature  obtaining,  he  has  only  to  turn 
his  operculum  shghtly  round  in  its  own  plane  as  he  advances  it 
into  each  newly  formed  portion  of  his  chamber,  to  adapt  one  margin 
of  it  to  a  new  and  larger  surface  and  a  different  curvature,  leaving 
the  space  to  be  filled  up  by  increasing  the  operculum  wholly  on 
the  other  margin."  The  fact  is  that  self-similar  or  gnomonic  growth 
is  taking  place  both  in  the  shell  and  its  operculum;  in  both  of  them 
growth  is  in  reference  to  a  fixed  centre,  and  to  a  fixed  axis  through 
that  centre;  and  in  both  of  them  growth  proceeds  in  geometric 
progression  from  the  centre  while  rotation  takes  place  in  arithmetic 
progression  about  the  axis.  The  same  architecture  which  builds 
the  house  constructs  the  door.  Moreover,  not  only  are  house  and 
door  governed  by  the  same  law  of  growth,  but,  growing  together, 
door  and  doorway  adapt  themselves  to  one  another. 

The  operculum  of  the  gastropods  varies  from  a  more  or  less  close -wound 
spiral,  as  in  Turritella,  Trochus  or  Pleurotomaria,  to  cases  in  which  accretion 
takes  place,  by  concentric  (or  more. or  less  excentric)  rings,  all  round.  But 
these  latter  cases,  so  Mr  Winckworth  tells  me,  are  not  very  common.  Paludina 
and  Ampullaria  come  near  to  having  a  concentric  operculum,  and  so  do  some 
of  the  Murices,  such  as  M.  tribulus,  and  a  few  Turrids,  and  the  genus  Helicina; 
but  even  these  opercula  probably  begin  as  spirals,  adding  on  their  gnomonic 
increments  at  one  end  or  side,  and  only  growing  on  all  sides  later  on.  There 
would  seem  to  be  a  truly  concentric  operculum  in  the  Siphonium  group  of 
Vermetus,  where  the  spiral  of  the  shell  itself  is  lost,  or  nearly  so;  but  it  is 
usually  overgrown  with  Melobesia,  and  hard  to  see. 


Fig.  372. 

One  more  proposition,  an  all  but  self-evident  one,  we  may  make 
passing  mention  of  here:    If  upon  any  polar  radius  vector  OP, 


778  THE  EQUIANGULAR  SPIRAL    '  [ch. 

a  triangle  OPQ  be  drawn  similar  to  a  given  triangle,  the  locus  of 
the  vertex  Q  will*  be  a  spiral  similar  to  the  original  spiral.  We  may 
extend  this  proposition  (as  given  by  Whitworth)  from  the  simple 
case  of  the  triangle  to  any  similar  figures  whatsoever ;  and  see  from 
it  how  every  spot  or  ridge  or  tubercle  repeated  symmetrically  from 
one  radius  vector  (or  one  generating  curve)  to  another  becomes 
part  of  a  spiral  pattern  on  the  shell. 

Viewed  in  regard  to  its  own  fundamental  properties  and  to  those 
of  its  hmiting  cases,  the  equiangular  spiral  is  one  of  the  simplest 
of  all  known  curves ;  aiid  the  rigid  uniformity  of  the  simple  laws  by 
which  it  is  developed  sufficiently  account  for  its  frequent  manifesta- 
tion in  the  structures  built  up  by  the  slow  and  steady  growth  of 
organisms. 

In  order  to  translate  into  precise  terms  the  whole  form  and 
growth  of  a  spiral  shell,  we  should  have  to  employ  a  mathematical 
notation  considerably  more  complicated  than  any  that  I  have 
attempted  to  make  use  of  in  this  book.  But  we  may  at  least  try 
to  describe  in  elementary  language  the  general  method,  and  some  of 
the  variations,  of  the  mathematical  development  of  the  shell.  But 
here  it  is  high  time  to  observe  that,  while  we  have  been  speaking  of 
the  shell  (which  is  a  surface)  as  a  logarithmic  spiral  (which  is  a  line), 
we  have  been  simphfying  the  case,  in  a  provisional  or  preparatory 
way.  The  logarithmic  spiral  is  but  one  factor  in  the  case,  albeit 
the  chief  or  dominating  one.  The  problem  is  one  not  of  plane  but 
of  sohd  geometry,  and  the  solid  in  question  is  described  by  the 
movement  in  space  of  a  certain  area,  or  closed  curve*. 

Let  us  imagine   a   closed  curve  in  space,   whether  circular  or 

eUiptical  or  of  some  other  and  more  complex  specific  form,  not 

necessarily  in  a  plane:    such  a  curve  as  we  see  before  us  when  we 

consider  the  mouth,  or  terminal  orifice,  of  our  tubular  shell.     Let 

*  For  a  more  advanced  study  of  the  family  of  surfaces  of  which  the  Nautilus 
is  a  simple  case,  see  M.  Haton  de  la  Goupilliere  {op.  cit.).  The  turbinate  shells 
represent  a  sub-family,  which  may  be  called  that  of  the  "surfaces  cerithioides " ; 
and  "surfaces  a  front  generateur"  is  a  short  title  of  the  whole  family.  The  form  of 
the  generating  curve,  its  rate  of  expansion,  the  direction  of  its  advance,  and  the 
angle  which  the  generating  front  makes  with  the  directrix,  define,  and  give  a  wide 
extension  to,  the  family.  These  parameters  are  all  severally  to  be  recognised  in  the 
growth  of  the  living  object;  and  they  make  of  a  collection  of  shells  an  unusually 
beautiful  materialisation  of  the  rigorous  definitions  of  geometry. 


XI]  OF  THE  MOLLUSCAN  SHELL  779 

us  call  this  closed  curve  the  "generating  curve";  the  surface  which 
it  bounds  we  may  call  (if  need  arise)  the  "generating  front,"  and 
let  us  imagine  some  one  characteristic  point  within  this  closed 
curve,  such  as  its  centre  of  gravity.  Then,  starting  from  a  fixed 
origin,  let  this  characteristic  point  describe  an  equiangular  spiral 


Fig.  |373.     Melo  ethiopicus  L. 

in  space  about  a  fixed  axis  or  "conductrix"  (namely  the  axis  of  the 
shell),  while  at  the  same  time  the  generating  curve  grows  with  each 
increment  of  rotation  in  such  a  way  as  to  preserve  the  symmetry 
of  the  entire  figure,  with  or  without  a  simultaneous  movement  of 
translation  along  the  axis. 

The  resulting  shell  may  now  be  looked  upon  in  either  of  two  ways. 
It  is,  on  the  one  hand,  an  ensemble  gf  similar  closed  curves,  spirally 
arranged  in  space,  and  gradually  increasing  in  dimensions  in  pro- 
portion to  the  increase  of  their  vector-angle  from  the  pole*.     In 

*  The  plumber,  the  copper-smith  and  the  glass-blower  are  at  pains  to  conserve 
in  every  part  of  their  tubular  constructions,  however  these  branch  or  bend,  the 
constant  form  which  their  cross-sections  ought  to  have.  Throughout  the  spiral 
twisting  of  the  shell,  throughout  the  windings  and  branchings  of  the  blood-vessels, 
the  same  uniformity  is  maintained. 


780  THE  EQUIANGULAR  SPIRAL  [ch. 

other  words,  we  can  imagine  our  shell  cut  up  into  a  system  of  rings, 
following  one  another  in  continuous  spiral  succession,  from  that 
terminal  and  largest  one  which  constitutes  the  hp  of  the  orifice  of 
the  shell.  Or  on  the  other  hand,  we  may  figure  to  ourselves  the 
whole  shell  as  made  up  of  an  ensemble  of  spiral  lines  in  space,  each 
spiral  having  been  traced  out  by  the  gradual  growth  and  revolution 
of  a  radius  vector  from  the  pole  to  a  given  point  on  the  boundary 
of  the  generating  curve. 


-r" 

■^^^ 

-j|KjSp^^ 

M. 

-J,,... 

K2 

m 

\'^^ 

-     '   ' '          ■'■■.'           /   • 

■   i 

WSrWWW^^II^B 

f 

I  2 

Fig.  374.     1,  Harpa;   2,  Dolium.     The  ridges  on  the  shell  correspond 
in  (1)  to  generating  curves,  in  (2)  to  generating  spirals. 

Both  systems  of  fines,  the  generating  spirals  (as  these  latter  may 
be  called)!  and  the  closed  generating  curves  corresponding  to  suc- 
cessive margins  or  hps  of  the  shell,  may  be  easily  traced  in  a  great 
variety  of  cases.  Thus,  for  example,  in  Dolium,  Eburna,  and  a 
host  of  others,  the  generating  spirals  are  beautifully  marked  out 
by  ridges,  tubercles  or  bands  of  colour.  In  Trophon,  Scalaria,  and 
(among  countless  others)  in  the  Ammonites,  it  is  the  successive 
generating  curves  which  more  conspicuously  leave  their  impress  on 


XI]  OF  THE  MOLLUSCAN  SHELL  781 

the  shell.  And  in  not  a  few  cases,  as  in  Harpa,  Dolium  perdix,  etc., 
both  ahke  are  conspicuous,  ridges  and  colour-bands^  intersecting 
one  another  in  a  beautiful  isogonal  system. 

In  ordinary  gastropods  the  shell  is  formed  at  or  near  the  mantle- 
edge.  Here,  near  the  mantle-border,  is  a  groove  Hned  with  a 
secretory  epithehum  which  produces  the  horny  cuticle  or  perio 
stracum  of  the  shell*.  A  narrow  zone  of  the  mantle  just  behind 
this  secretes  Hme  abundantly,  depositing  it  in  a  layer  below  the 
periostracum ;  and  for  some  httle  way  back  more  hme  may  be 
secreted,  and  pigment  superadded  from  appropriate  glands.  Growth 
and  secretion  are  periodic  rather  than  continuous.  Even  in  a  snail- 
shell  it  is  easy  to  see  how  the  shell  is  built  up  of  narrow  annular 
increments;  and  many  other  shells  record,  in  conspicuous  colour- 
patterns,  the  alternate  periods  of  rest  and  of  activity  which  their 
pigment-glands  have  undergone. 

The  periodic  accelerations  and  retardations  in  the  growth  of  a 
shell  are  marked  in  various  ways.  Often  we  have  nothing  more 
than  an  increased  activity  from  time  to  time  at  or  near  the  mantle- 
edge — enough  to  give  rise  to  shght  successive  ridges,  each  corre- 
sponding to  a  "generating  curve"  in  the  conformation  of  the  shell. 
But  in  many  other  cases,  as  in  Murex,  Ranella  and  the  like,  the 
mantle-edge  has  its  alternate  phases  of  rest  and  of  turgescence,  its 
outline  being  plain  and  even  in  the  one  and  folded  and  contorted 
in  the  other;  and  these  recurring  folds  or  pleatings  of  the  edge 
leave  their  impress  in  the  form  of  various  ridges,  ruffles  or  comb-like 
rows  of  spines  upon  the  shell f. 

In  not  a  few  cases  the  colour-pattern  shews,  or  seems  to  shew, 
how  some  play  of  forces  has  fashioned  and  transformed  the  first 
elementary  pattern  of  pigmentary  drops  or  jets.  As  the  book- 
binder drops  or  dusts  a  little  colour  on  a  viscous  fluid,  and  then 
produces  the  beautiful  streamlines  of  his  marbled  papers  by  stirring 

*  That  the  shell  grows  by  accretion  at  the  mantle-edge  was  one  of  Reaumur's 
countless  discoveries  {Mem.  Acad.  Roy.  des  Sc.  1709,  p.  364  seq.).  It  follows 
that  the  mathematical  "generating  curves,"  as  Moseley  chose  them,  correspond  to 
the  material  increments  of  the  shell. 

t  The  periodic  appearance  of  a  ridge,  or  row  of  tubercles,  or  other  ornament 
on  the  growing  shell  is  illustrated  or  even  exaggerated  in  the  delicate  "combs" 
of  Murex  aculeatus.  Here  normal  growth  is.  interrupted  for  the  time  being,  the 
mantle-edge  is  temporarily  folded  and  reflexed,  and  shell-substance  is  poured  out 
into  the  folds. 


782  THE  EQUIANGULAR  SPIRAL  [ch. 

and  combing  the  colloid  mass,  so  we  may  see,  in  the  harp-shells 
or  the  volutes,  how  a  few  simple  spots  or  hnes  have  been  drawn  out 
into  analogous  wavy  patterns  by  streaming  movements  during  the 
formation  of  the  shell. 
'  In  the  complete  mathematical  formula  for  any  given  turbinate 
shell,  we  may  include,  with  Moseley,  factors  for  the  following 
elements:  (1)  for  the  specific  form  of  a  section  of  the  tube,  or 
(as  we  have  called  it)  the  generating  curve;  (2)  for  the  specific  rate 
of  growth  of  this  generating  curve;  (3)  for  its  incHnation  to  the 
directrix,  or  to  the  axis ;  (4)  for  its  specific  rate  of  angular  rotation 
about  the  pole,  in  a  projection  perpendicular  to  the  axis;  and 
(5)  in  turbinate  (as  opposed  to  nautiloid)  shells,  for  its  rate  of 
screw- translation,  parallel  to  the  axis,  as  measured  by  the  angle 
between  a  tangent  to  the  whorls  and  the  axis  of  the  shell*.  It  seems 
a  compHcated  affair;  but  it  is  only  a  pathway  winding  at  a  steady 
slope  up  a  conical  hill.  This  uniform  gradient  is  traced*  by  any 
given  point  on  the  generating  curve  while  the  vector  angle  increases 
in  arithmetical  progression,  and  the  scale  changes  in  geometrical 
progression;  and  a  certain  ensemble,  or  bunch,  of  these  spiral  curves 
in  space  constitutes  the  self-similar  surface  of  the  shell. 

But  after  all  this  is  not  the  only  way,  neither  is  it  the  easiest  way, 
to  approach  our  problem  of  the  turbinate  shell.  The  conchologist 
turned  mathematician  is  apt  to  think  of  the  generating  curve  by 
which  the  spiral  surface  is  described  as  necessarily  identical,  or 
coincident,  with  the  mouth  or  Hp  of  the  shell;  for  this  is  where 
growth  actually  goes  on,  and  where  the  successive  increments  of 
shell-growth  are  visibly  accumulated.  But  it  does  oiot  follow  that 
this  particular  generating  curve  is  chosen  for  the  best  from  the 
mathematical  point  of  view;  and  the  mathematician,  unconcerned 
with  the  physiological  side  of  the  case  and  regardless  of  the  suc- 
cession of  the  parts  in  time,  is  free  to  choose  any  other  generating 
curve  which  the  geometry  of  the  figure  may  suggest  to  him.  We 
are  following  Moseley's  example  (as  is  usually  done)  when  we  think 
of  no  other  generating  curve  but  that  which  takes  the  form  of  a 

*  Note  that  this  tangent  touches  the  curve  at  a  series  of  points,  whorl  by  whorl, 
instead  of  at  one  only.  Observe  also  that  we  may  have  various  tangent-cones, 
all  centred  on  the  apex  of  the  shell.  In  an  open  spiral,  like  a  ram's  horn,  or  a 
half-open  spiral  like  the  shell  Solarium,  w^  have  two  cones,  one  touching  the 
outside,  the  other  the  inside  of  the  shell. 


XI]  OF  THE  MOLLUSCAN  SHELL  783 

frontal  plane,  outlined  by  the  lip,  and  sliding  along  the  axis  while 
revolving  round  it;  but  the  geometer  takes  a  better  and  a  simpler 
way.  For,  when  of  two  similar  figures  in  space  one  is  derived  from 
the  other  by  a  screw-displacement  accompanied  by  change  of  scale — 
as  in  the  case  of  a  big  whelk  and  a  little  whelk — there  is  a  unique 
(apical)  point  which  suffers  no  displacement;  and  if  we  choose  for 
our  generating  curve  a  sectional  figure  centred  on  the  apical  point 
and  passing  through  the  axis  of  rotation,  the  whole  development 
of  the  surface  may  be  simply  described  as  due  to  a  rotation  of  this 
generating  figure  about  the  axis  (z),  together  with  a  change  of  scale 
with  the  point  0  as  centre  of  simihtude.  We  need  not,  and  now 
must  not,  think  of  a  slide  or  shear  as  part  of  the  operation;  the 
translation  along  the  axis  is  merely  part  and  parcel  of  the  magnifica- 
tion of  the  new  generating  curve.  It  follows  that  angular  rotation 
in  arithmetical  progression,  combined  with  change  of  scale  (from  0) 
in  geometrical  progression,  causes  any  arbitrary  point  on  the 
generating  curve  to  trace  a  path  of  uniform  gradient  round  a 
circular  cone,  or  in  other  words  to  describe  a  helico-spiral  or  gauche 
equiangular  spiral  in  space.  The  spiral  curve  cuts  all  the  straight- 
fine  generators  of  the  cone  at  the  same  angle ;  and  it  further  follows 
that  the  successive  increments  are,  and  the  whole  figure  constantly 
remains,  "self -similar"*. 

Apart  from  the  specific  form  of  the  generating  curve,  it  is  the 
ratios  which  happen  to  exist  between  the  various  factors,  the  ratio 
for  instance  between  the  growth-factor  and  the  rate  of  angular 
revolution,  which  give  the  endless  possibihties  -  of  permutation  of 
form.  For  example,  a  certain  rate  of  growth  in  the  generating 
curve,  together  with  a  certain  rate  of  vectorial  rotation,  will  give 
us  a  spiral  shell  of  which  each  successive  whorl  will  just  touch  its 
predecessor  and  no  more;  with  a  slower  growth-factor  the  whorls 
will  stand  asunder,  as  in  a  ram's  horn;  with  a  quicker  growth-factor 

*  The  equation  to  the  surface  of  a  turbinate  shell  is  discussed  by  Moseley  both 
in  terms  of  polar  and  of  rectangular  coordinates,  and  the  method  of  polar  co- 
ordmates  is  used  also  by  Haton  de  la  Goupilliere;  but  both  accounts  are  subject 
to  mathematical  objection.  Dr  G.  T.  Bennett,  choosing  his  generating  curve 
(as  described  above)  in  the  axial  plane  from  which  the  vertical  angles  are  measured 
(the  plane  ^  =0),  would  state  his  equation  in  cylindrical  coordinates,  /  {za^,  ra^)  =0: 
that  is  to  say  in  terms  of  z,  conjointly  with  ordinary  plane  cylindrical  coordi- 
nates. 


784  THE  EQUIANGULAR  SPIRAL  [ch. 

each  will  cut  or  intersect  its  predecessor,  as  in  an  Ammonite  or 
the  majority  of  gastropods,  and  so  on. 

A  similar  relation  of  velocities  suffices  to  determine  the  apical 
angle  of  the  resulting  cone,  and  give  us  the  diflFerence,  for  example, 
between  the  sharp,  pointed  cone  of  Turritella,  the  less  acute  one  of 
Fusus  or  Buccinum,  and  the  obtuse  one  of  Harpa  or^of  Dolium. 
In  short  it  is  obvious  that  all  the  differences  of  form  which  we 
observe  between  one  shell  and  another  are  referable  to  inatters  of 
degree,  depending,  one  and  all,  upon  the  relative  magnitudes  of  the 
various  factors  in  the  complex  equation  to  the  curve.  This  is  an 
immensely  important  thing.  To  learn  that  all  the  multitudinous 
shapes  of  shells,  in  their  all  but  infinite  variety,  may  be  reduced  to 
the  variant  properties  of  a  single  simple  curve,  is  a  great  achieve- 
ment. It  exemphfies  very  beautifully  what  Bacoi^  meant  in  saying 
that  the  forms  or  differences  of  things  are  simple  and  few,  and  the 
degrees  and  coordinations  of  these  make  all  their  variety*.  And 
after  such  a  fashion  as  this  John  Goodsir  imagined  that  the  naturahst 
of  the  future  would  determine  and  classify  his  shells,  so  that 
conchology  should  presently  become,  hke  mineralogy,  a  mathe- 
matical science  I . 

The  paper  in  which,  more  than  a  hundred  years  ago,  Canon  Moseleyf 
gave  a  simple  mathematical  account,  on  fines  fike  tfies^,  of  the 
spiral  forms  of  univalve  shells,  is  one  of  the  classics  of  Natural 
History.  But  other  students  before,  and  sometimes  long  before, 
him  had  begun  to  recognise  the  same  simpficity  of  form  and 
structure.  About  the  year  1818  Reinecke  had  declared  Nautilus 
to  be  a  well-defined  geometrical  figure,  whose  chambers  followed 

*  For  a  discussion  of  this  idea,  and  of  the  views  of  Bacon  and  of  J.  S.  Mill,  see 
J.  M.  Keynes,  op.  cit.  p.  271. 

t  On  the  employment  of  mathematical  modes  of  investigation  in  the  determina- 
tion of  organic  forms^  in  Anatomical  Memoirs,  ii,  p.  205,  1868  (posthumous 
pubhcation). 

X  The  Rev.  Henry  Moseley  (1801-1872),  of  St  John's  College,  Cambridge, 
Canon  of  Bristol,  Professor  of  Natural  Philosophy  in  King's  College,  London,  was 
a  man  of  great  and  versatile  ability.  He  was  father  of  H.  N.  Moseley,  naturalist 
on  board  the  Challenger  and  Professor  of  Zoology  in  Oxford;  and  he  was  grand- 
father of  H.  G.  J.  Moseley  (1887-1915) — Moseley  of  the  Moseley  numbers — whose 
death  at  Gallipoli,  long  ere  his  prime,  was  one  of  the  major  tragedies  of  the  Four 
Years  War. 


XI]  OF  THE  MOLLUSCAN  SHELL  785 

one  another  in  a  constant  ratio  or  continued  proportion*;  and 
Leopold  von  Buch  and  others  accepted  and  even  developed  the 
idea. 

Long  before,  Swammerdam  had  grasped  with  a  deeper  insight 
the  root  of  the  whole  matter;  for,  taking  a  few  diverse  examples, 
such  as  Helix  and  Spirula,  he  shewed  that  they  and  all  other  spiral 
shells  whatsoever  were  referable  to  one  common  type,  namely  to 
that  of  a  simple  tube,  variously  curved  according  to  definite  mathe- 
matical laws;  that  all  manner  of  ornamentation,  in  the  way  of 
spines,  tuberosities,  colour-bands  and  so  forth,  might  be  superposed 
upon  them,  but  the  type  was  one  throughout  and  specific  differences 
were  of  a  geometrical  kind.  "Omnis  enim  quae  inter  eas  anim- 
advertitur  differentia  ex  sola  nascitur  diversitate  gyrationum: 
quibus  si  insuper  externa  quaedam  adjunguntur  ornamenta  pin- 
narum,  sinuum,  anfractuum,  planitierum,  eminentiarum,  profundi- 
tatum,  extensionum,  impressionum,  circumvolutionum,  colorumque : 
. . .  tunc  deinceps  facile  est,  quarumcumque  Cochlearum  figuras 
geometricas,  curvosque,  obliquos  atque  rectos  angulos,  ad  unicam 
omnes  speciem  redigere :  ad  oblongum  videhcet  tubulum,  qui 
vario  modo  curvatus,  crispatus,  extrorsum  et  introrsum  flexus, 
ita  concrevitf." 

Nay  more,  we  may  go  back  yet  another  hundred  years  and  find 
Sir  Christopher  Wren  contemplating  the  architecture  of  a  snail-shell, 
and  finding  in  it  the  logarithmic  spiral.  For  AValhsf,  after  defining 
and  describing  this  curve  with  great  care  and  simplicity,  tells  us 
that  Wren  not  only  conceived  the  spiral  shell  to  be  a  sort  of  cone 
or  pyramid  coiled  round  a  vertical  axis,  but  also  saw  that  on  the 
magnitude  of  the  angle  of  the  spire  depended  the  specific  form  of 
the  shell:  "Hanc  ipsam  curvam . . . fcontemplatus  est  Wrennius 
noster.     Nee  tantum  curvae  longitudinem,  partiumque  ipsius,  et 

*  J.  C.  M.  Reinecke,  Maris  protogaei  Nautilos,  etc.,  Coburg,  1818,  p.  17:  "In 
eius  forma,  quae  canalis  spiram  convoluti  formam  et  proportiones  simul  sub- 
ministrat,  totius  testae  forma  quoddammodo  data  est.  Restaret  solum  scire, 
quota  cuj usque  anfractus  pars  sequent!  inclusa^it,  ut  testam  geometrice  construere 
possimus."  Cf.  Leopold  von  Buch,  Ueber  die  Ammoniten  in  den  alteren  Gebirgs- 
schichten,  Ahh.  Berlin.  Akad.,  Phys.  Kl.  1830,  pp.  135-158;  Ann.  Sc.  Nat.  xxvin, 
pp.  5-43,  1833;  cf.  Elie  de  Beaumont,  Sur  I'enroulement  des  Ammonites,  Soc. 
Philom.,  Pr.  verb.  1841,  pp.  45-4^. 

t  Biblia  Naturae  sive  Historia  Insectorum,  Leydae,.1737,  p.  152. 

X  Job.  Wallis,  Tractatus  duo,  de  Cydoide,  etc.,  Oxon.,  1659,  pp.  107,  108. 


786  THE  EQUIANGULAR  SPIRAL  [ch. 

magnitudinem  adjacentis  plani;  sed  et,  ipsius  ope,  Limacum  et 
Conchiliorum  domunculos  metitur.  Existimat  utique,  magna  veri- 
similitudine,  domunculos  hosce  non  alios  esse  quam  Pjrramides 
convolutas:  quarum  Axis  sit,  istiusmodo  Spiralis:  non  quidem  in 
piano  jacens,  sed  sensim  in  convolutione  (circa  erectum  axim) 
assurgens:  pro  variis  autem  curvae,  sive  ad  rectam  circumductam 
sive  ad  subjacens  planum, "  angulis,  variae  Conchiliorum  formae 
enascantur.  Atque  hac  hjrpothesi,  mensurata  Pyramide,  metitur 
etiam  ea  conchiliorum  spatia." 

For  some  years  after  the  appearance  of  Moseley's  paper,  a  number 
of  writers  followed  in  his  footsteps,  and  attempted  in  various  ways 
to  put  his  conclusions  to  practical  use.     For  instance,  d'Orbigny 


^ —                    ^—^^-^^^  '''■• 

\  \.  ^ 

^^,^ 

j^^^^^^ 

■ ''|i'iTi|irir|iiii|i'iii(iiiiji 

iijiii()iiiimH 

Mntimi! 

IITTTImitii 

iiiiiliii 

iiiiiliii 

1 — ' — 1 — ^ — 1 — ^ 

°                     o                     « 

L       L 

J — 1 — - 

— \ — ^—-\ — ■ — 1 

?       % 

Fig.  375.     d'Orbigny's  helicometer. 

devised  a  very  simple  protractor,  which  he  called  a  Helicometer*, 
and  which  is  represented  in  Fig.  375.  By  means  of  this  little 
instrument  the  apical  angle  of  the  turbinate  shell  was  immediately 
read  off,  and  could  then  be  used  as  a  specific  and  diagnostic  character. 
By  keeping  one  limb  of  the  protractor  parallel  to  the  side  of  the 
cone  wjiile  the  other  was  brought  into  hne  with  the  suture  between 
two  adjacent  whorls,  another  specific  angle,  the  '^sutural  angle," 
could  in  like  manner  be  recorded.  And,  by  the  hnear  scale  upon 
the  instrument,  the  relative  breadths  of  the  consecutive  whorls, 
and  that  of  the  terminal  chamber  to  the  rest  of  the  shell,  might 

*  Alcide  d'Orbigny,  Bull,  de  la*soc.  geol.  Fr.  xiii,  p.  200,  1842;  Cours  elem. 
de  PaUontologie,  ii,  p.  5,  1851.  A  somewhat  similar  instrument  was  described  by 
Boubee,  in  Bull.  soc.  geol.  i,  p.  232,  1831.  Naumann's  conchyliometer  {Poggend, 
Ann.  Liv,  p.  544,  1845)  was  an  application  of  the  screw-micrometer;  it  was  provided 
also  with  a  rotating  stage  for  angular  measurement.  It  was  adapted  for  the 
study  of  a  discoid  or  ammonitoid  shell,  while  d'Orbigny 's  instrument  was  meant 
for  the  study  of  a  turbinate  shell. 


XI]  OF  THE  MOLLUSCAN  SHELL  787 

ako,  though  somewhat  roughly,  be  determined.  For  instance,  in 
Terebra  dimidiata  the  apical  angle  was  found  to  be  13°,  the  sutural 
angle  109°,  and  so  forth. 

It  was  at  once  obvious  that,  in  such  a  shell  as  is  represented  in 
Figs.  369  and  375  the  entire  outUne  (always  excepting  that  of 
the  immediate  neighbourhood  of  the  mouth)  could  be  restored  from 
a  broken  fragment.  For  if  we  draw  our  tangents  to  the  cone,  it 
follows  from  the  symmetry  of  the  figure  that  we  can  continue  the 
projection  of  the  sutural  hne,  and  so  mark  off  the  successive  whorls, 
by  simply  drawing  a'  series  of  consecutive  parallels,  and  by  then 
filling  into  the  quadrilaterals  so  marked  off  a  series  of  curves  similar 
to  one  another,  and  to  the  whorls  which  are  still  intact  in  the  broken 
shell.  But  the  use  of  the  hehcometer  soon  shewed  that  it  was 
by  no  means  universally  the  case  that  one  and  the  same  cone  was 
tangent  to  all  the  turbinate  whorls;  in  other  words,  there  was  not 
always  one  specific  apical  angle  which  held  good  for  the  entire 
system.  In  the  great  majority  of  cases,  it  is  true,  the  same  tangent 
touches  all  the  whorls,  and  is  a  straight  Hne.  But  in  others,  as  in 
the  large  Cerithium  nodosum,  such  a  hne  is  sKghtly  concave  to  the 
axis  of  the  shell;  and  in  the  short  spire  of  Doliumj  for  instance, 
the  concavity  is  marked,  ^nd  the  apex  of  the  spire  is  a  distinct 
cusp.  On  the  other  hand,  in  Pupa  and  Clausilia  the  conmion 
tangent  is  convex  to  the  axis  of  the  shell. 

So  also  is  it,  as  we  shall  presently  see,  among  the  Ammonites: 
where  there  are  some  species  in  which  the  ratio  of  whorl  to  whorl 
remains,  to  all  appearance,  perfectly  constant;  others  in  which 
it  gi^dually  though  only  shghtly  increases;  and  others  again  in 
which  it  slightly  and  gradually  falls  away.  It  is  obvious  that, 
among  the  manifold  possibihties  of  growth,  such  conditions  as 
these  are  very  easily  conceivable.  It  is  much  more  remarkable 
that,  among  these  shells,  the  relative  velocities  of  growth  in  various 
dimensions  should  be  as  constant  as  they  are  than  that  there 
should  be  an  occasional  departure  from  perfect  regularity.  In  these 
latter  cases  the  logarithmic  law  of  growth  is  only  approximately 
true.  The  shell  is  no  longer  to  be  represented  simply  as  a  cone  which 
has  been  rolled  up,  but  as  a  cone  which  (while  rolling  up)  had  grown 
trumpet-shaped,  or  conversely  whose  mouth  had  narrowed  in,  and 
which  in  longitudinal  section  is  a  curvilinear  instead  of  a  rectihnear 


788 


THE  EQUIANGULAR  SPIEAL 


[CH. 


triangle.  But  all  that  has  happened  is  that  a  new  factor,  usually 
of  small  or  all  but  imperceptible  magnitude^  has  been  introduced 
into  the  case ;  so  that  the  ratio,  log  r  =  6  log  a,  is  no  longer  con- 
stant but  varies  shghtly,  and  in  accordance  with  some  simple  law. 
Some '  writers,  such  as  Naumann*  and  Grabau,  maintained  that 
the  moUuscan  spiral  was  no  true  logarithmic  spiral,  but  differed 
from  it  specifically,  and  they  gave  it  the  name  of  Conchospiral. 
They  said  that  the  logarithmic  spiral  originates  in  a  mathematical 
point,  while  the  molluscan  shell  starts  with  a  Httle  embryonic,  shell, 
or  central  chamber  (the  "protoconch"  of  the  conchologists),  around 
which  the  spiral  is  subsequently  wrapped.  But  this  need  not  affect 
the  logarithmic  law  of  the  shell  as  a  whole ;  indeed  we  have  already 
allowed  for  it  by  writing  our  equation  in  the  form  r  =  ma^.  And 
Grabauf,  while  he  clung  to  Naumann's  conchospiral  against 
Moseley's  logarithmic  spiral,  confessed  that  they  were  so  much  ahke 
that  ordinary  measurements  would  seldom  shew  a  difference  between 
them. 

There  would  seem,  by  the  way,  to  be  considerable  confusion  in  the  books 
with  regard  to  the  so-called  "protoconch."  In  many  cases  it  is  a  definite 
structure,  of  simple  form,  representing  the  more  or 
less  globular  embyyonic  shell  before  it  began  to 
elongate  into  its  conical  or  spiral  form.  But  in 
many  cases  what  is  described  as  the  "protoconch" 
is  merely  an  empty  space  in  the  middle  of  the  spiral 
coil,  resulting  from  the  fact  that  the  actual  spiral 
.shell  must  have  some  magnitude  to  begin  with,  and 
that  we  cannot  follow  it  down  to  its  vanishing  point 
in  infinity.  For  instance,  in  the  accompanying 
figure,  ■  the  large  space  a  is  styled  the  protoconch, 
but  it  is  the  little  bulbous  or  hemispherical  chamber 
within  it,  at  the  end  of  the  spire,  which  is  the  real 
beginning  of  the  tubular  shell.  The  form  and  mag- 
nitude of  the  spa^ce  a  are  determined  by  the  "angle  of  retardation,"  or  ratio 
of  rate  of  growth  between  the  inner  and  outer  curves  of  the  spiral  shell.     They 


Fig.  376. 


*  C.  F.  Naumann,  Beitrag  zur  Konchyliometrie,  Poggend.  Ann.  l,  p.  223,  1840; 
Ueber  die  Spiralen  der  Ammoniten,  ibid,  li,  p.  245,  1840;  ibid,  liv,  p.  541,  1845;  etc. 
(See  also  p.  755.)  Cf.  also  Lehmann,  Die  von  Seyfriedsche  Konchyliensammlung 
und  das  Windungsgesetz  von  einigen  Planorben,  Constanz,  1855. 

f  A.  H.  Grabau,  Ueber  die  Naumannsche  Conchospirale,  und  ihre  Bedeutung 
fiir  die  Conchyliometrie,  Inauguraldiss.,  Leipzig,  1872;  Ueber  die  Spiralen  der 
Conchylien,  etc.,  Leipzig  Progr.  No.  502,  1880;  cf.  Sb.  naturf.  Gesellsch.  Leipzig, 
1881,  pp.  23-32. 


XI] 


ITS  MATHEMATICAL  PROPERTIES 


789 


are  independent  of  the  shape  and  size  of  the  embryo,  and  depend  only  (as  we 
shall  see  better  presently)  on  the  direction  and  relative  rate  of  growth  of  the 
double  contour  of  the  shell*. 


rd0 


Now  that  we  have  dealt,  m  a.  general  way,  with  some  of  the  more 
obvious  properties  of  the  equiangular  or  logarithmic  spiral,  let  us 
consider  certain  of  them  a  httle  more  particularly,  keeping  in  view 
as  our  chief  object  of  study  the  range  of  variation  of  the  molluscan 
shell. 

There  is  yet  another  equation  to  the  logarithmic 
spiral,  very  commonly  employed,  and  without  the 
help  of  which  we  cannot  get  far.     It  is  as  follows : 

This  follows  directly  from  the  fact  that  the  angle 
a  (the  angle  between  the  radius  vector  and  the 
tangent  to  the  curve)  is  constant. 

For  then, 

tan  a  (=  tan  (f>)  =  rdd/dr; 
therefore  dr/r  =  dO  cot  a, 

and,  integrating,  log  r  =  ^  cot  a. 


or 


Fig.  377. 


It  is  easy  to  see  (we  might  indeed  have  noted  it  before)  that  the 
logarithmic  spiral  is  but  a  plotting  in  polar  coordinates  of  increase 
by  compound  interest.  For  if  A  be  the  "amount"  of  £1  in  one  year 
(A  =  1  +  a,  where  a  is  the  rate  of  interest),  and  PA  the  amount 
of  P  in  one  year,  then  the  whole  amount,  M,  in  t  years  is  M  =  PA^\ 
this,  provided  that  interest  is  payable  once  a  year.  But,  as  we  are 
taught  by  algebra,  and  as  we  have  seen  in  our  study  of  growth, 
this  formula  becomes  Pe"*  when  the  intervals  of  time  between  the 
payments  of  interest  decrease  without  limit,  that  is  to  say,  wh^n  we 
may  consider  growth  to  be  continuous.  And  this  formula  PeP-^  is 
precisely  that  of  our  logarithmic  spiral,  when  we  represent  the  time 

*  J.  F.  Blake  (cf.  infra,  p.  793)  says  of  Naumann's  formula:  "By  such  a 
modification  he  hoped  to  bring  the  measurements  of  actual  shells  more  into 
harmony  with  calculation.  The  errors  of  observation,  however,  are  always  greater 
than  this  change  would  correct — if  founded  on  fact,  which  is  doubtful;  and  all 
practical  advantage  is  lost  by  the  compUcation  of  the  equations." 


790 


THE  EQUIANGULAR  SPIRAL 


[CH. 


by  a  vector  angle  d,  and  when  for  a,  the  particular  rate  of  interest 
in  the  cas.e,  we  write  cot  a,  the  constant  measure  of  growth  of  the 
particular  spiral. 

As  we  have  seen  throughout  our  preliminary  discussion,  the  two 
most  important  constants  (or  "specific  Characters,"  as  the  naturalist 
would  say)  in  an  equiangular  or  logarithmic  spiral  are  (1)  the  magni- 
tude of  the  angle  of  the  spiral,  or  "constant  angle"  a,  and  (2)  the 
rate  of  increase  of  the  radius  vector  for  any  given  angle  of  revolution,  6. 
But  our  two  magnitudes,  that  of  the  constant  angle  and  that  of  the 
ratio  of  the  radii  or  breadths  of  whorl,  are  directly  related  to  one 
another,  so  that  we  may  determine  either  of  them  by  measurement 
and  calculate  the  other. 


Fig.  378. 

In  any  complete  spiral,  such  as  that,  of  Nautilus,  it  is  (as  we 
have  seen)  easy  to  measure  any  two  radii  (r),  or  the  breadths  in 
a  radial  direction  of  any  two  whorls*  (If).  We  have  then  merely 
to  apply  the  formula 


^"+1  ^  g^cota 


W. 


or 


«+l  ^  g^cota 


which  we  may  simply  write  r  =  e^^°*",  etc.,  when  one  radius  or  whorl 
is  regarded,  for  the  purpose  of  comparison,  as  equal  to  unity. 

Thus,  in  Fig.   378,  OC/OE,  or  EF/BD,   or  DC/'EF,  being  in 
each  case  radii,  or  diameters,  at  right  angles  to  one  another,  are 

-  * 

all  equal  to  e^"^"  ".     While  in  like  manner,  EO/OF,  EG/FH,  or 

GO/HO,  all  equal  e^^o^a.   ^nd  BC/BA,  or  CO/OB  =  e2^cota^ 


XI]  ITS  MATHEMATICAL  PROPERTIES  791 

As  soon,  then,  as  we  have  prepared  tables  for  these  values,  the 
determination  of  the  constant  angle  a  in  a  particular  shell  becomes 
a  very  simple  matter. 

A  complete  table  would  be  cumbrous,  and  it  will  be  sufficient 
to  deal  with  the  simple  case  of  the  ratio  between  the  breadths  of 
adjacent,  or  immediately  succeeding,  whorls. 

Here  we  have  r  =  e27rcota^  qp  log/ =  log  e  x  27r  x  cot  a,  from 
which  we  obtain  the  following  figures*: 

The  shape  of  a  nautiloid  spiral 


Ratio  of  breadth  of  each 

whorl  to  the  next  preceding 

Constant  angl< 

r/1 

a 

11 

89°    8' 

1-25 

87  58 

1-5 

86   18 

2-0 

83  42 

2-5 

81   42 

30 

80     5 

3-5 

78  43 

40 

77  34 

4-5 

76  32 

50 

75  38 

100 

69  53 

200 

64  31 

500 

58     5 

1000 

53  46 

1000-0 

42  17 

10,000 

34   19 

100,000 

28  37 

1,000,000 

24  28 

10,000,000 

21    18 

100,000,000 

18  50 

1,000,000,000 

16  52 

We  learn  several  interesting  things  from  this  short  table.  We 
see,  in  the  first  place,  that  where  each  whorl  is  about  three  times 
the  breadth  of  its  neighbour  and  predecessor,  as  is  the  case  in 
Nautilus,  the  constant  angle  is  in  the  neighbourhood  of  80°;  and 
hence  also  that,  in  all  the  ordinlary  ammonitoid  shells,  and  in  all 
the  t3rpically  spiral  shells  of  the  gastropods  f,  the  constant  angle 
is  also  a  large  one,  being  very  seldom  less  than  80°,  and  usually 
between  80°  and  85°.     In  the  next  place,  we  see  that  with  smaller 

*  It  is  obvious  that  the  ratios  of  opposite  whorls,  or  of  radii  180°  apart,  are 
represented  by  the  square  roots  of  these  values ;  and  the  ratios  of  whorls  or  radii 
90°  apart,  by  the  square  roots  of  these  again. 

■j"  For  the  correction  to  be  applied  in  the  case  of  the  helicoid,  or  "turbinate" 
shells,  see  p.  816. 


792  THE  EQUIANGULAR  SPIRAL  [ch. 

angles  the  apparent  form  of  the  spiral  is  greatly  altered,  and  the 
very  fact  of  its  being  a  spiral  soon  ceases  to  be  apparent  (Figs.  379, 
380).  Suppose  one  whorl  to  be  an  inch  in  breadth,  then,  if  the 
angle  of  the  spiral  were  80°,  the  next  whorl  would  (as  we  have  just 
seen)  be  about  three  inches  broad;  if  it  were  70°,  the  next  whprl 
would  be  nearly  ten  inches,  and  if  it  were  60°,  the  next  whorl  would 
be  nearly  four  feet  broad.  If  the  angle  were  28°,  the  next  whorl 
would  be  a  mile  and  a  half  in  breadth;  and  if  it  were  17°,  the  next 
would  be  spme  15,000  miles  broad. 

In  other  words,  the  spiral  shells  of  gentle  curvature,  or  of  small 
constailt  angle,  such  as  Dentalium  or  Cristellaria,  are  true  equi- 
angular spirals,  just  as  are  those  of  Nautilus  or  Rotalia:    from 


Fig.  379. 


Fig.  380. 


which  they  differ  only  in  degree,  in  the  magnitude  of  an  angular 

constant.     But  this  diminished  magnitude  of  the  angle  causes  the 

spiral  to  dilate  with  such  immense  rapidity  that,   so  to  speak, 

it  never  comes 'round;   and  so,  in  such  a  shell  as  Dentalium,  we 

never  see  but  a  small  portion  of  a  single  whorl. 

We  might  perhaps  be  inclined  to  suppose  that,  in  such  a  shell  as  Dentalium^ 
the  lack  of  a  visible  spiral  convolution  was  only  due  to  our  seeing  but  a  small 
portion  of  the  curve,  at  a  distance  from  the  pole,  and  when,  therefore,  its 
curvature  had  already  greatly  diminished.  That  is  to  say  we  might  suppose 
that,  however  small  the  angle  a,  and  however  rapidly  the  whorls  accordingly 
increased,  there  would  nevertheless  be  a  manifest  spiral  convolution  in  the 
immediate  neighbourhood  of  the  pole,  as  the  starting  point  of  the  curve. 
But  it  is  easy  to  see  that  it  is  not  so.  It  is  not  that  there  cease  to  be  con- 
volutions of  the,  spiral  round  the  pole  when  a  is  a  small  angle ;  on  the  contrary, 
there  are  infinitely  many,  mathematically  speaking.  But  as  a  diminishes, 
and  cot  a  increases  towards  infinity,  the  ratio  between  the  breadth  of  one 
whorl  and  the  next  increases  very  rapidly.  Our  taTble  shews  us  that  even 
when  a  is  no  less  than  40°,  and  our  shell  stiU  looks  strongly  curved,  one  whorl 
is  a  thousandt'i  part  of  the  breadth  of  the  next,  and  a  thousandfold  that 


XI]  ITS  MATHEMATICAL  PROPERTIES  793 

of  the  one  before ;  we  cannot  expect  to  see  either  of  them  under  the  materialised 
conditions  of  the  actual  shell.  Our  shells  of  small  constant  angle  and  gentle 
curvature,  such  as  Dentalium,  are  accordingly  as  much  as  we  can  ever  expect 
to  see  of  their  respective  spirals. 

The  spiral  whose  constant  angle  is  45°  is  both  a  simple  case  and 
a  mathematical  curiosity;  for,  since  the  tangent  of  45°  is  unity, 
we  need  merely  write  r  =  e^\  which  is  as  much  as  to  say  that  the 
natural  logarithms  of  the  radii  give  us,  without  more  ado,  the 
vector  angles.  In  this  spiral  the  ratio  between  the  breadths  of 
two  consecutive  whorls  becomes  r  =  e^^  ■=  e2x3-i4i6_  Reducing  this 
from  Naperian  to  common  logs,  we  have  log  r  =  2-729;  whicH  tells 
us  (by  our  tables)  that  the  radius  vector  is  multiphed  about  535J 
times  after  a  whole  polar  revolution;  it  is  doubled  after  turning 
through  a  polar  angle  of  less  than  40°.  Spirals  of  so  low  an  angle 
as  45°  are  common  enough  in  tooth  and  claw,  but  rare  among 
molluscan  shells;  but  one  or  two  of  the  more  strongly  curved 
Dentaliums,  like  D.  elejphantinum,  come  near  the  mark.  It  is  not 
easy  to  determine  the  pole,  nor  to  measure  the  constant  angle,  in 
forms  like  these. 

Let  us  return  to  the  problem  of  how  to  ascertain,  by  direct 
measurement,  the  spiral  angle  of  any  particular  shell.  The  method 
aL-eady  employed  is  only  applicable  to  complete  spirals,  that  is  to 
say  to  those  in  which  the  angle  of  the  spiral  is  large,  and  further- 
more it  is  inapplicable  to  portions,  or  broken  fragments,  of  a  shell. 
In  the  case  of  the  broken  fragment,  it  is  plain  that  the  determination 
of  the  angle  is  not  merely  of  theoretic  interest;  but  may  be  of  great 
practical  use  to  the  conchologist  as  the  one  and  only  way  by  which 
he  may  restore  the  outline  of  the  missing  portions.  We  have  a  con- 
siderable choice  of  methods,  which  have  been  summarised  by,  and 
are  partly  due  to,  a  very  careful  student  of  the  Cephalopoda,  the 
late  Rev.  J.  F.  Blake*. 

(1)  When  an  equiangular  spiral  rolls  on  a  straight  line,  the  pole 
traces  another  straight  line  at  an  ^ngle  to  the  first  equal  to  the 
complement  of  the  constant  angle  of  the  spiral;  for  the  contact 
point  is  the  instantaneous  centre  of  the  rotational  movement,  and 
the  line  joining,  it  to  the  pole  of  the  spiral  is  normal  to  the  roulette 
path  of  that  point.     But  the  difficult^  of  determining  the  pole 

*  On  the  measurement  of  the  curves  formed  by  Cephalopods  and  other  MoUusks, 
Phil.  Mag.  (5),  vi,  pp.  241-263,  1878. 


794  THE  EQUIANGULAR  SPIRAL  [ch. 

(which  is  indeed  asymptotic)  makfes  this  of  Httle  use  as  a  method  of 
determining  the  constant  angle.  It  is,  however,  a  beautiful  property 
of  the  curve,  and  all  the  more  interesting  that  Clerk  Maxwell  dis- 
covered it  when  he  was  a  boy*. 

(2)  The  following  method  is  useful  and  easy  when  we  have  a 
portion  of  a  single  whorl,  such  as  to  shew  both  its  inner  and  its 
outer  edge.  A  broken  whorl  of  an  Ammonite,  a  curved  shell  such 
as  Dentalium,  or  a  horn  of  similar  form  to  the  latter,  will  fall  under 

this  head.  We  have  merely  to  draw  a  tangent, 
GEH,  to  the  outer  whorl  at  any  point  E;  then 
draw  to  the  inner  whorl  a  tangent  parallel  to  GEH, 
touching  the  curve  in  some  point  F.  The  straight 
hne  joining  the  points  of  contact,  EF,  must 
evidently  pass  through  the  pole :  and^  accordingly, 
the  angle  GEF  is  the  angle  required.  In  shells 
which  bear  longitudinal  striae  or  other  ornaments, 
any  pair  of  these  will  suffice  for  our  purpose, 
instead  of  thfe  actual  boundaries  of  the  whorl. 
But  it  is  obvious  that  this  method  will  be  apt  to 
fail  us  when  the  angle  a  is  very  small;  and 
when,  consequently,  the  points  E  and  F  are  very 
remote. 

(3)  In  shells  (or  horns)  shewing  rings  or  other  transverse 
ornamentation,  we  may  take  it  that  these  ornaments  are  set  at 


Fig.  381. 


Fig.  382.  An  Ammonite,  to 
shew  corrugated  surface- 
pattern. 

*  Clerk  Maxwell,  On  the  theory  of  rolling  curves.  Trans.  B.S.E.  xvi,  pp.  519- 
540,  1849;    Sci.  Papers,  i,  pp.  4-29. 


XI]  ITS  MATHEMATICAL  PROPERTIES  795 

a  constant  angle  to  the  spire,  and  therefore  to  the  radii.  The  angle 
(6)  between  two  of  them,  as  AC,  BD,  is  therefore  equal  to  the 
angle  9  between  the  polar  radii  from  A  and  B,  or  from  C  and  D; 
and  therefore  BD/AC  =  e^cota^  which  gives  us  the  angle  a  in  terms 
of  known  quantities. 

(4)  If  only  the  outer  edge  be  available,  we  have  the  ordinary- 
geometrical  problem — given  an  arc  of  an  equiangular  spiral,  to  find 
its  pole  and  spiral  angle.  The  methods  we  may  employ  depend 
(i)  on  determining  directly  the  position  of  the  pole,  and  (ii)  on 
determining  the  radius  of  curvature. 

The  first  method  is  theoretically  simple,  but  difficult  in  practice; 
for  it  requires  great  accuracy  in  determining  th'fe  points.  Let  AD, 
DB  be  two  tangents  drawn  to  the 
curve.  Then  a  circle  drawn  through 
the  points  A,  B,  D  will  pass  through 
the  pole  0,  since  the  angles  OAT>, 
QBE  (the  supplement  of  OBD)  are 
equal.  The  point  0  may  be  deter- 
mined by  the  intersection  of  two 
such  circles;  and  the  angle  DBO  is 
then  the  angle,  a,  required. 

Or  we  may  determine  graphically,        '  p-    ^^ 

at  two  points,  the  radii  of  curvature 

P1P2.     Then,  if  s  be  the  length  of  the  arc  between  them  (which  may 

be  determined  with  fair  accuracy  by  rolling  the  margin  of  the  shell 

along  a  ruler), 

cot  a  =  (pi  —  P2)is. 

The  following  method*,  given  by  Blake,  will  save  actual  determination  of 
the  radii  of  curvature. 

Measure  along  a  tangent  to  the  curve  the  distance,  AC,  at  which  a  certain 
small  offset,  CD,  is  made  by  the  curve;    and  from  another  point  B,  measure 
the  distance  at  which  the  curve  makes  an  equal  offset.     Then,  calling  the 
offset  jLt;  the  arc  AB,  s;  and  AC,  BE,  respectively  x^,  x^,  we  have 
a;  2^_    2 
Pj_  =  ^  ,  approximately, 

/^ 

and  cota  = 


2ixs 

Of  all  these  methods  by  which  the  mathematical  constants,  or 
specific  characters,  of  a  given  spiral  shell  ipay  be  determined,  the 
*  For  an  example  of  this  method,  see  Blake,  loc.  cit.  p.  251. 


796  THE  EQUIANGULAR  SPIRAL  [ch. 

only  one  of  which  much  use  has  been  made  is  that  which  Moseley 
first  employed,  namely,  the  simple  method  of  determining  the 
relative  breadths  of  the  whorl  at  distances  separated  by  some 
convenient  vectorial  angle  such  as  90°,  180°,  or  360°. 

Very  elaborate  measurements  of  a  number  of  Ammonites  have 
been  made  by  Naumann*,  by  Grabau,  by  Sandberger  f,  and  by 
Miiller,  among  which  we  may  choose  a  couple  of  cases  for  considera- 
tion f.     In  the  following  table  I  have  taken  a  portion  of  Grabau's 

Ammonites  intuslabiatus 
Ratio  of  breadth  of 


Ith  of  whorls 

successive  whorls 

The  angle  (a) 

J0°  apart) 

(360°  apart) 

as  calculated 

0-30  mm. 

— 

— 

— 

0-30 

1-333 

87° 

23' 

0-40 

1-500 

86 

19 

0-45 

1-500 

86 

19 

0-60 

1-444 

86 

39 

0-65 

1-417 

86 

49 

0-85 

1-692 

85 

13 

MO 

1-588 

85 

47 

1-35 

1-545 

86 

2 

1-70 

1-630 

85 

33 

2-20 

1-441 

86 

40 

2-45 

1-432 

86 

43 

315 

1-735 

85 

0 

4-25 

1-683 

85 

16 

5-30 

1-482 

86 

25 

6-30 

1-519 

86 

12 

8-05 

1-635 

85 

32 

10-30 

1-416 

86 

50 

11-40 

1-252 

87 

57 

12-90 

Mean 

— 

— 

86° 

15' 

*  C.  F.  Naumann,  Ueber  die  Spiralen  von  Conchylien,  Ahh.  k.  sacks.  Ges.  1846, 
pp.  153-196;  Ueber  die  cyclocentrische  Conchospirale  u.  iiber  das  Windungsgesetz 
von  Planorhis  corneus,  ibid,  i,  pp.  171-195,  1849;  Spirale  von  Nautilus  u.  Ammonites 
galeatus,  Ber.  k.  sacks.  Ges.  ir,  p.  26,  1848;  Spirale  von  Amm.  Ramsaueri,  ibid,  xvi, 
p.  21,  1864.  Oken,  reviewing  Naumann's  work  (in  Isis,  1847,  p.  867)  foretold  how 
some  day  the  naturalist  and  the  mathematician  would  each  learn  of  the  other: 
"Um  die  Sache  zu  Vollendung  zu  bringen  wird  der  Mathematiker  Zoolog  und 
Physiolog,  und  dieae  Mathematiker  werden  miissen." 

I  G.  Sandberger,  Clymenia  subnautilina,  Jakresber.  d.  Ver.  f.  Naturk.  im  Herzogtk. 
Nassau,  1855,  p.  127;  Spiralen  des  Ammonites  Amaltkeus,  A.  Gaytani  und  Goniatites 
intumescens,  Ztschr.  d.  d.  Geolog.  Gesellsch.  x,  pp.  446-^49,  1858.  Also  Miiller, 
Beitrag  zur  Konchyliometrie,  Poggend.  Ann.  lxxxvi,  p.  533,  1850;  ibid,  xc,  p.  323, 
1853.    These  two  authors  upheld  the  logarithmic  law  against  Naumann  and  Grabau. 

J  See  also  Chr.  Petersen,  Das  Quotientengesetz,  eine  biologisch-statistische  Unter- 
suckung,  119  pp.,  Copenhagen,  1921;  E.  Sporn,  Ueber  die  Gesetzmassigkeit  im 
Baue  der  Muschelgehaiiser,  Arch.  f.  Entw.  Meek,  cviii,  pp.  228-242,  1926. 


XI]  OF  SHELLS  GENERALLY  797 

determinations  of  the  breadth  of  the  whorls  in  Ammonites  (Arcestes) 
intuslabiatus;  these  measurements  Grabau  gives  for  every  45°  of 
arc,  but  I  have  only  set  forth  successive  whorls  measured  along 
one  diameter  on  both  sides  of  the  pole.  The  ratio  between  alternate 
measurements  is  therefore  the  same  ratio  as  Moseley  adopted, 
namely  the  ratio  of  breadth  between  contiguous  whorls  along  a 
radius  vector.  I  have  then  added  to  these  observed  values  the 
corresponding  calculated  values  of  the  angle  a,  as  obtained  from 
our  usual  formula. 

There  is  considerable  irregularity  in  the  ratios  derived  from  these 
measurements,  but  it  will  be  seen  that  this  irregularity  only  implies 
a  variation  of  the  angle  of  the  spiral  between  about  85°  and  87°; 
and  the  values  fluctuate  pretty  regularly  about  the  mean,  which 
is  86°  15'.  Considering  the  difficulty  of  measuring  the  whorls, 
especially  towards  the  centre,  and  in  particular  the  difficulty  of 
determining  with  precise  accuracy  the  position  of  the  pole,  it  is 
clear  that  in  such  a  case  as  this  we  are  not  justified  in  asserting  that 
the  law  of  the  equiangular  spiral  is  departed  from. 

Ammonites  tornatus 


Ratio  of  breadth  of 

The  spiral 

eadth  of  whorls 

successive  whorls 

angle  (a)  as 

(180'  apart) 

(360"  apart) 

calculated 

0-25  mm. 

— 

-^  — . 

0-30 

1-400 

86°  56' 

0-35 

1-667 

85   21 

0-50 

2-000 

83   42 

0-70 

2-000 

83   42 

1-00 

2-000 

83   42 

1-40 

2-100 

83    16 

2-10 

2-lW 

82   56 

305 

2-238 

82   42 

4-70 

2-492 

81    44 

7-60 

2-574 

81    27 

1210 

2-546 

81    33 

19-35 

— 



Mean    2-11 

83°  22' 

In  some  cases,  however,  it  is  undoubtedly  departed  from.  Here 
for  instance  is  another  table  from  Grabau,  shewing  the  corre- 
sponding ratios  in  an  Ammonite  of  the  group  of  Arcestes  tornatus. 
In  this  case  we  see  a  distinct  tendency  of  the  ratios  to  increase  as 
we  pass  from  the  centre  of  the  coil  outwards,  and  consequently  for 
the  values  of  the  angle  a  to  diminish.     The  case  is  comparable  to 


7.98  THE  EQUIANGULAR  SPIRAL  [ch. 

that  of  a  cone  with  shghtly  curving  sides :  in  which,  that  is  to  say, 
there  is  a  sHght  acceleration  of  growth  in  a  transverse  as  compared 
with  the  longitudinal  direction. 

In  a  tubular  spiral,  whether  plane  or  hehcoid,  the  consecutive 
whorls  may  either  be  (1)  isolated  and  remote  from  one  another; 
or  (2)  they  may  precisely  meet,  so  that  the  outer  border  of  one 
and  the  inner  border  of  the  next  just  coincide;  or  (3)  they  may 
overlap,  the  vector  plane  of  each  outer  whorl  cutting  that  of  its 
immediate  predecessor  or  predecessors. 

Looking,  as  we  have  done,  upon  the  spiral  shell  as  being  essentially 
a  cone  roUed  up*,  it  is  plain  that,  for  a  given  spiral  angle,  intersection 
or  non-intersection  of  the  successive  whorls  will  depend  upon  the 
apical  angle  of  the  original  cone.  For  the  wider  the  cone,  the  more 
will  its  inner  border  tend  to  encroach  on  the  preceding  whorl.  But 
it  is  also  plain  that  the  greater  the  apical  angle  of  the  cone,  and  the 
broader,  consequently,  the  cone  itself,  the  greater  difference  will 
there  be  between  the  total  lengths  of  its  inner  and  outer  borders. 
And,  since  the  inner  and  outer  borders  are  describing  precisely 
the  same  spiral  about  the  pole,  we  may  consider  the  inner  border 
as  being  retarded  in  growth  as  compared  with  the  outer,  and  as 
being  always  identical  with  a  smaller  and  earUer  part  of  the  latter. 

If  A  be  the  ratio  of  growth  between  the  outer  and  the  inner  curve, 
then,  the  outer  curve  being  represented  by 

r  =  ae^ootcc^ 

the  equation  to  the  inner  one  will  >^e 

/  =  aAe^cota^ 

or  r'  =  ae^^-y^^^^"", 


*  To  speak  of  a  cone  "rolling  up,"  and  becoming  a  nautiloid  spiral  by  doing  so, 
is  a  rough  and  non-mathematical  description;  nor  is  it  easy  to  see  how  a  cone  of 
wide  angle  could  roll  up,  and  yet  remain  a  cone.  But  if  (i)  the  centre  of  a  sphere 
move  along  a  straight  line  and  its  radius  keep  proportional  to  the  distance  the 
centre  has  moved,  the  sphere  generates  as  its  envelope  a  circular  cone  of  which 
the  straight  line  is  the  axis;  and  so,  similarly,  if  (ii)  the  centre  of  a  sphere  move 
along  an  equiangular  spiral  and  its  radius  keep  proportional  to  the  arc-distance 
along  the  spiral  back  to  the  pole,  the  sphere  generates  as  its  envelope  a  self-similar 
shell-surface,  or  nautiloid  spiral. 


XI]  OF  SHELLS  GENERALLY  799 

and  y  may  then  be  called  the  angle  of  retardation,  to  which  the 
inner  curve  is  subject  by  virtue  of  its  slower  rate  of  growth. 

Dispensing  with  mathematical  formulae,  the  several  conditions 
may  be  illustrated  as  follows : 

In  the  diagrams  (Fig.  385),  OF-^F^F^,  etc.  represents  a  radius, 
on  which  P^,  F^,  Pg  are  the  points  attained  by  the  outer  border 
of  the  tubular  shell  after  as  many  entire  consecutive  revolutions. 
And  Pi',  P2',  P3'  are  the  points  similarly  intersected  by  the  inner 
border ;  OFjOF'  being  always  =  A,  which  is  the  ratio  of  growth, 
or  ' '  cutting-down  factor. ' '    Then,  obviously,  ( 1 )  when  OP^  is  less  than 


Fig.  385. 

OP2'  the  whorls  will  be  separated  by  an  interspace  (a);  (2)  when 
OF^  =  OP2'  they  will  be  in  contact  (6),  and  (3)  when  OF^  is  greater 
than  OF^  there  will  be  a  greater  or  less  extent  of  overlapping, 
that  is  to  say  of  concealment  of  the  surfaces  of  the  earher  by  the 
later  whorls  (c).  And  as  a  further  case  (4),  it  is  plain  that  if  A  be 
very  large,  that  is  to  say  if  OF-^  be  greater,  not  only  than  OF^ 
but  also  than  OP3',  OP4',  etc.,  we  shall  have  complete,  or  all  but 
complete,  concealment  by  the  last  formed  whorl  of  the  whole  of 
its  predecessors.  This  latter  condition  is  completely  attained  in 
Nautilus  pompilius,  and  approached,  though  not  quite  attained,  in 
N.  umbilicatus]  and  the  difference  between  these  two  forms,  or 
"species,"  is  constituted  accordingly  by  a  difference  in  the  value 
of  A.     (5)   There  is  also  a  final  case,   not  easily  distinguishable 


800  THE  EQUIANGULAR  SPIRAL  [ch. 

externally  from  (4),  where  P'  lies  on  the  opposite  side  of  the  radius 
vector  to  P,  and  is  therefore  imaginary.  This  final  condition  is 
exhibited  in  Argonauta. 

The  Hmiting  values  of  A  are  easily 
ascertained. 

In  Fig.  386  we  have  portions  of 
two  successive  whorls,  whose  corre- 
sponding points  on  the  same  radius 
vector  (as  R  and  R)  are,  therefore, 
^^*       '  at  a  distance  apart  corresponding  to 

277-.  Let  r  and  r'  refer  to  the  inner,  and  R,  R'  to  the  outer  sides  of 
the  two  whorls.     Then,  if  we  consider 

it  follows  that         R'  =  ae<^+2;r)  cot  a^ 

and  r'  =  Aae^^+^^^^^o*^  =  j^g(^f27r-7)cota_ 

Now  in  the  three  cases  (a,  b,  c)  represented  in  Fig.  385,  it  is 
plain  that  r'  =  R,  respectively.     That  is  to  say, 

Xae^0+2n)COta  j  ae^cota^ 

and  Ae^^^ota  |  i^ 

The  case  in  which  Ae^'^^^*'^  =  1,  or  —  log  A  =  277- cot  a  log  e,  is 
the  case  represented  in  Fig.  385,  b:  that  is  to  say,  the  particular 
case,  for  each  value  of  a,  where  the  consecutive  whorls  just  touch, 
without  interspace  or  overlap.  For  such  cases,  then,  we  may 
tabulate  the  values  of  A  as  follows: 


Cdiistaiit  angle  a 

Uatio  (A)  of 

rate 

of  growth  of  inner  border  of  tube. 

of  siiiral 

as  coiiiji 

ared 

with  that  of  tlie  outer  l)order 

89° 

0-896 

88 

0-803 

87 

0-720 

86 

0-645 

85 

0-577 

80 

0-330 

75 

0-234 

70 

0-1016 

65 

0-0534 

We  see,  accordingly,  that  in  plane  spirals  whose  constant  angle 
lies,  say,  between  65°  and  70°,  we  can  only  obtain  contact  between 


XI] 


OF  SHELLS  GENERALLY 


801 


consecutive  whorls  if  the  rate  of  growth  of  the  inner  border  of  the 
tube  be  a  small  fraction — a  tenth  or  a  twentieth — of  that  of  the 
outer  border.  In  spirals  whose  con- 
stant angle  is  80°,  contact  is  attained 
when  the  respective  rates  of  growth 
are,  approximately,  as  3  to  1;  while 
in  spirals  of  constant  angle  from  about 
85°  to  89°,contact  is  attained  when  the 
rates  of  growth  are  in  the  ratio  of  from 
about  f  to  j%. 

If  on  the  other  hand  we  have,  for 
any  given  value  of  a,  a  value  of  A 
greater  or  less  than  the  value  given 
in  the  above  table,  then  we  have, 
respectively,  the  conditions  of  separa- 
tion or  of  overlap  which  are  exempUfied 
in  Fig.  385,  a  and  c.  And,  just  as  we 
have  constructed  this  table  for  the 
particular  case  of  simple  contact,  so  we  could  construct  similar  tables 
for  various  degrees  of  separation  or  of  overlap. 

For  instance,  a  case  which  admits  of  simple  solution  is  that  in 
which  the  interspace  between  the  whorls  is  everywhere  a  mean  pro- 
portional between  the  breadths  of  the  whorls  themselves  (Fig.  387). 
In  this  case,  let  us  call  OA  =  R,  OC  =  R^,  and  OB  =  r.  We  then 
have 


Fig.  387. 


i^2=OC  =  ae(^+2'^>^^*^ 


And 


r2=  (1/A)2  .  €2^«0t«, 


whence,  equating. 


1/A  =  e 


TTCota 


*  It  has  been  pointed  out  to  me  that  it  does  not  follow  at  once  and  obviously 
that,  because  the  interspace  AB  i&  &  mean  proportional  between  the  breadths  of 
the  adjacent  whorls,  therefore  the  whole  distance  OB  is  a  mean  proportional 
between  OA  and  OC.  This  is  a  corollary  which  requires  to  be  proved;  but  the 
proof  is  easy.. 


802  THE  EQUIANGULAR  SPIRAL  [ch. 

The  corresponding  values  of  A  are  as  follows : 

Ratio  (\)  of  rates  of  growth  of  outer  and  inner 

border,  such  as  to  produce  a  spiral  with  interspaces 

between  the  wliorls,  the  breadth  of  which 

intei-spaces  is  a  mean  proportional  between  the 

Constant  angle  (a)  breadths  of  the  whorls  themselves 

90°  1-00  (imaginary) 

89  0-95 

88  0-89 

87  0-85 

86  0-81 

85  0-76 

80  0-57 

75  0-43 

70  0-32 

65  0-23 

60  018 

55  013 

50  0090 

45  0-063 

40  0042 

35  0026 

30  0016   . 

As  regards  the  angle  of  retardation,  y,  in  the  formula 
/  =  Ae^^o*«,  or  /  =  e(^-^)cota^ 
and  in  the  case 

r'  =  e(2^-y)cota^   ^j.  _  log  A  -  (277  -  y)  cot  a, 

it  is  evident  that  when  y  =  27t,  that  will  mean  that  A  =  1.  In 
other  words,  the  outer  and  inner  borders  of  the  tube  are  identical, 
and  the  tube  is  constituted  by  one  continuous  hne. 

When  A  is  a  very  small  fraction,  that  is  to  say  when  the  rates 
of  growth  of  the  two  borders  of  the  tube  are  very  diverse,  then 
y  wfll  tend  towards  infinity — tend  that  is  to  say  towards  a  condition 
in  which  the  inner  border  of  the  tube  never  grows  at  all.  This 
condition  is  not  infrequently  approached  in  nature.  I  take  it  that 
Cyfraea  is  such  a  case.  But  the  nearly  parallel-sided  cone  of 
Dentalium,  or  the  widely  separated  whorls  of  Lituites,  are  cases 
where  A  nearly  approaches  unity  in  the  one  case,  and  is  still  large 
in  the  other,  y  being  correspondingly  small ;  while  we  can  easily  find 
cases  where  y  is  very  large,  and  A  is  a  small  fraction,  for  instance 
in  Haliotis,  in  Calyptraea,  or  in  Gryphaea. 

For  the  purposes  of  the  morphologist,  thoji,  the  main  result  of 
this  last  general  investigation  is  to  shew  that  all  the  various  types 
of  ''open"  and  "closed"  spirals,  all  the  various  degrees  of  separation 


XI]  OF  SHELLS  GENERALLY  803 

or  overlap  of  the  successive  whorls,  are  simply  the  outward  ex- 
pression of  a  varying  ratio  in  the  rate  of  growth  of  the  outer  as 
compared  with  the  inner  border  of  the  "tubular  shell. 

The  foregoing  problem  of  contact,  or  intersection,  of  successive 
whorls  is  a  very  simple  one  in  the  case  of  the  discoid  shell  but 
a  more  complex  one  in  the  turbinate.  For  in  the  discoid  shell 
contact  will  evidently  take  place  when  the  retardation  of  the  inner 
as  compared  with  the  outer  whorl  is  just  360°,  and  the  shape  of 
the  whorls  need  not  be  considered. 

As  the  angle  of  retardation  diminishes  from  360°,  the  whorls  stand 
further  and  further  apart' in  an  open  coil;  as  it  increases  beyond  360°, 
they  overlap  more  and  more ;  and  when  the  angle  of  retardation  is 
infinite,  that  is  to  say  when  the  true  inner  edge  of  the  whorl  does 
not  grow  at  all,  then  the  shell  is  said  to  be  completely  involute.  Of 
this  latter  condition  we  have  a  striking  example  in  Argonauta,  and 
one  a  Httle  more  obscure  in  Nautilus  pompilius. 

In  the  turbinate  shell  the  problem  of  contact  is  twofold,  for  v.\, 
have  to  deal  with  the  possibihties  of  contact  on  the  same  side  of 
the  axis  (which  is  what  we  have  dealt  with  in  the  discoid)  and  also 
with  the  new  possibiHty  of  contact  or  mtersection  on  the  opposite 
side;  it  is  this  latter  case  which  will  .determine  the  presence  or 
absence  of  an  open  umbilicus.  It  is  further  obvious  that,  in  the 
case  of  the  turbinate,  the  question  of  contact  or  no  contact  will 
depend  on  the  shape  of  the  generating  curve ;  and  if  we  take  the 
simple  case  where  this  generating  curve  may  be  considered  as  an 
eUipse,  then  contact  will  be  found  to  depend  on  the  angle  which 
the  major  axis  of  this  elhpse  makes  with  the  axis  of  the  shell.  The 
question  becomes  a  compHcated  one,  and  the  student  will  find  it 
treated  in  Blake's  paper  already  referred  to. 

When  one  whorl  overlaps  another,  so  that  the  generating  curve 
cuts  its  predecessor  (at  a  distance  of  27r)  on  the  same  radius  vector, 
the  locus  of  intersection  will  follow^  a  spiral  fine  upon  the  shell, 
whjch  is  called  the  "suture"  by  conchologists.  It  is  one  of  that 
ensemble  of  spiral  lines  in  space  of  which,  as  we  have  seen,  the 
whole  shell  may  be  conceived  to  be  constituted ;  and  we  might  call 
it  a  "contact-spiral,"  or  "spiral  of  intersection."  In  discoid  shells, 
such  as  an  Ammonite  or  a  Planorbis,  or  in  Nautilus  umbilicatus, 
there  are  obviously  two  such  contact-spirals,  one  on  each  side  of 


804  THE  EQUIANGULAR  SPIRAL  [ch. 

the  shell,  that  is  to  say  one  on  each  side  of  a  plane  perpendicular 
to  the  axis.  In  turbinate  shells  such  a  condition  is  also  possible, 
but  is  somewhat  rare.  We  have  it  for  instance  in  Solarium  per- 
spectivum,  where  the  one  contact-spiral  is  visible  on  the  exterior  of 


Fig.  388.     Solarium  perspectivum. 

the  shell,  and  the  other  lies  internally,  winding  round  the  open 
cone  of  the  umbihcus*;  but  this  second  contact-spiral  is  usually 
imaginary,  or  concealed  within  the  whorls  of  the  turbinated  shell. 


Fig.  389. 


Haliotis  tuberculata  L. ;  the  ormer, 
or  ear  shell. 


Fig.  390.  Scalaria 
pretiosa  L. ;  the 
wentletrap.  From 
Cooke's  Spirals. 


Again,  in  Haliotis,  one  of  the  contact-spirals  is  non-existent,  because 
of  the  extreme  obUquity  of  the  plane  of  the  generating  curve.     In 

*  A  beautiful  construction:   stupendum  Naturae  artijicium,  Linnaeus. 


XI] 


OF  SHELLS  GENERALLY 


805 


Scalaria  pretiosa  and  in  Spirula*  there  is  no  contact-spiral,  because 
the  growth  of  the  generating  curve  has  been  too  slow  in  comparison 
with  the  vector  rotation  of  its  plane.  In  Argonauta  and  in  Cypraea 
there  is  no  contact-spiral,  because  the  growth  of  the  generating 
curve  has  been  too  quick.  Nor,  of  course,  is  there  any  contact- 
spiral  in  Patella  or  in  Dentalium,  because  the  angle  a  is  too  small 


Fig.  392.  Turhinella  napus 
Lam.;  an  Indian  chank- 
shell.     From  Chenu. 


Fig.  391.     Thatcheria  mirabilis  Angas; 
from  a  radiograph  by  Dr  A.  Miiller. 

ever  to  give  us  a  complete  revolution  of  the  spire.  Thatcheria 
mirabilis  is  a  peculiar  and  beautiful  shell,  in  which  the  outhne  of 
the  Up  is  sharply  triangular,  instead  of  being  a  smooth  curve :  with 
the  result  that  the  apex  of  the  triangle  forms  a  conspicuous  "gene- 
rating spiral",  which  winds  round  the  shell  and  is  more  conspicuous 
than  the  suture  itself. 

In  the  great  majority  of  hehcoid  or  turbinate  shells  the  innermost 


*  "It  [Spirula]  is  curved  so  as  its  roundness  is  kept,  and  the  Parts  do  not  touch 
one  another":   R.  Hooke,  Posthumoics  Works,  1745,  p.  284. 


806  THE  EQUIANGULAR  SPIRAL  [ch. 

or  axial  portions  of  the  whorls  tend  to  form  a  soHd  axis  or 
"columella";  and  to  this  is  attached  the  columellar  muscle  which 
on  the  one  hand  withdraws  the  animal  within  its  shell,  and  on  the 
other  hand  provides  the  controlhng  force  or  trammel,  by  which  (in 
the  gastropod)  the  growing  shell  is  kept  in  its  spiral  course.  This 
muscle  is  apt  to  leave  a  winding  groove  upon  the  columella  (Fig.  373) ; 
now  and  then  the  muscle  is  spht  into  strands  or  bundles,  and  then 
it  leaves  parallel  grooves  with  ridges  or  pleats  between,  and  the 
number  of  these  folds  or  pleats  may  vary  with  the  species,  as  in  the 
Volutes,  or  even  with  race  or  locahty.  Thus,  among  the  curiosities 
of  conchology,  the  chank-shells  on  the  Trincomah  coast  have  four 
columellar  folds  or  ridges;  but  all  those  from  Tranquebar,  just  north 
of  Adam's  Bridge,  have  only  three  (Fig.  392)*. 

The  various  forms  of  straight  or  spiral  shells  among  the  Cephalo- 
pods,  which  we  have  seen  to  be  capable  of  complete  definition  by 
the  help  of  elementary  mathematics,  have  received  a  very  com- 
phcated  descriptive  nomenclature  from  the  palaeontologists.  For 
instance,  the  straight  cones  are  spoken  of  as  orthoceracones  or 
bactriticories,  the  loosely  coiled  forms  as  gyroceracones  or  mimo- 
ceracones,  the  more  closely  coiled  shells,  in  which  one  whorl  overlaps 
the  other,  as  nautilicones  or  ammoniticones,  and  so  forth.  In  such 
a  series  of  forms  the  palaeontologist  sees  undoubted  and  unquestioned 
evidence  of  ancestral  descent.  For  instance  We  read  in  Zittel's 
Palaeontologyf :  "  The  bactriticone  obviously  represents  the  primitive 
or  primary  radical  of  the  Ammonoidea,  and  the  mimoceracone  the 
next  or  secondary  radical  of  this  order  " ;  while  precisely  the  opposite 
conclusion  was  drawn  by  Owen,  who  supposed  that  the  straight 
chambered  shells  of  such  fossil  Cephalopods  as  Orthoceras  had  been 
produced  by  the  gradual  unwinding  of  a  coiled  nautiloid  shell  {. 
The  mathematical  study  of  the  forms  of  shells  lends  no  support  to  these 

*  Cf.  R.  Winckworth,  Proc.  Malacol  Soc.  xxiii,  p.  345,  1939. 

t  English  edition,  1900,  p.  537.  The  chapter  is  revised  by  Professor  Alpheus 
Hyatt,  to  whom  the  nomenclature  is  largely  due.  For  a  more  copious  terminology, 
see  Hyatt,  Phytogeny  of  an  Acquired  Characteristic,  1894,  p.  422  seq.  Cf.  also 
L.  F.  Spath,  The  evolution  of  the  Cephalopoda,  Biol.  Reviews,  viii,  pp.  418-462, 
1933 . 

X  Th  is  latter  conclusion  is  adopted  by  Willey,  Zoological  Results,  1902,  p.  747. 
Cf.  also  Graham  Kerr,  on  Spirula:   Dana  Reports,  No.  8,  Copenhagen,  1931. 


XI]  OF  VARIOUS  CEPHALOPODS  807 

or  any  stichlihe  phylogenetic  hypotheses*.  If  we  have  two  shells 
in  which  the  constant  angle  of  the  spire  be  respectively  80°  and 
60°,  that  fact  in  itself  does  not  at  all  justify  an  assertion  that  the 
one  is  more- primitive,  more  ancient,  or  more  "ancestral"  than  the 
other.  Nor,  if  we  find  a  third  in  which  the  angle  happens  to  be 
70°,  does  that  fact  entitle  us  to  say  that  this  shell  is  intermediate 
between  the  other  two,  in  time,  or  in  blood  relationship,  or  in 
any  other  sense  whatsoever  save  only  the  strictly  formal  and 
mathematical  one.  For  it  is  evident  that,  though  these  particular 
arithmetical  constants  manifest  themselves  in  visible  and  recog- 
nisable differences  of  form,  yet  they  are  not  necessarily  more 
deep-seated  or  significant  than  are  those  which  manifest  themselves 
only  in  difference  of  magnitude;  and  the  student  of  phylogeny 
scarcely  ventures  to  draw  conclusions  as  to  the  relative  antiquity 
of  two  allied  organisms  on  the  ground  that  one  happens  to  be 
bigger  or  less,  or  longer  or  shorter,  than  the  other. 

At  the  same  time,  while.it  is  obviously  unsafe  to  rest  conclusions 
upon  such  features  as  these,  unless  they  be  strongly  supported 
and  corroborated  in  other  ways — for  the  simple  reason  that  there 
is  unlimited  room  for  coincidence,  or  separate  and  independent 
attainment  of  this  or  that  magnitude  or  numerical  ratio — yet  on 
the  other  hand  it  is  certain  that,  in  particular  cases,  the  evolution 
of  a  race  has  actually  involved  gradual  increase  or  decrease  in 
some  one  or  more  numerical  factors,  magnitude  itself  included — 
that  is  to  say  increase  or  decrease  in  some  one  or  more  of  the 
actual  and  relative  velocities  of  growth.  When  we  do  meet  with 
a  clear  and  unmistakable  series  of  such  progressive  magnitudes  or 
ratios,  manifesting  themselves  in  a  progressive  series  of  ''aUied" 
forms,  then  we  have  the  phenomenon  of  "orthogenesis."'  For 
orthogenesis  is  simply  that  phenomenon  of  continuous  Unes  or 
series  of  form  (and  also  of  functional  or  physiological  capacity), 

*  Phylogenetic  speculation,  fifty  years  ago  the  chief  preoccupation  of  the 
biologist,  has  had  its  caustic  critics.  Cf.  {int.  al.)  Rhumbler,  in  Arch.  f.  Entw. 
Mech.  VII,  p.  104,  1898:  "  Phylogenetische  Speculationen . . ,  werden  immer  auf 
Anklang  bei  den  Fachgenossen  rechnen  dlirfen,  sofern  nicht  ein  anderer  Fachgenosse 
auf  demselben  Gebiet  rait  gleicher  Kenntniss  der  Dinge  und  mit  gleicher  Scharfainn 
zufallig  zu  einer  anderen  Theorie  gekommen  ist. . .  .Die  Richtigkeit  'guter'  phylo- 
genetischer  Schliisse  lasst  sich  im  schlimmsten  Falle  anzweifeln,  aber  direkt 
widerlegen  lasst  sich  in  der  Regel«iicht." 


808  THE  EQUIANGULAR  SPIRAL  [ch. 

which  was  the  foundation  of  the  Theory  of  Evolution,  aUke  to 
Lamarck  and  to  Darwin  and  Wallace;  and  which  we  see  to  exist 
whatever  be  our  ideas  of  the  "origin  of  species,"  or  of  the  nature 
and  origin  of  "functional  adaptations."  And  to  my  mind,  the 
mathematical  (as  distinguished  from  the  purely  physical)  study  of 
morphology  bids  fair  to  help  us  to  recognise  this  phenomenon  of 
orthogenesis  in  many  cases  where  it  is  not  at  once  patent  to  the 
eye;  and,  on  the  other  hand,  to  warn  us  in  many  other  cases  that 
even  strong  and  apparently  complex  resemblances  in  form  may  be 
capable  of  arising  independently,  and  may  sometimes  signify  no 
more  than  the  equally  accidental  numerical  coincidences  which  are 
manifested  in  identity  of  length  or  weight  or  any  other  simple 
magnitudes. 

I  have  already  referred  to  the  fact  that,  while  in  general  a  very 
great  and  remarkable  regularity  of  form  is  characteristic  of  the 
molluscan  shell,  yet  that  complete  regularity  is  apt  to  be  departed 
from.  We  have  clear  cases  of  such  a  departure  in  Pujpa,  Clausilia 
and  various  Bulimi,  where  the  spire  is  not  conical,  but  its -sides  are 
curved  and  narrow  in. 

The  following  measurements  of  three  specimens  of  Clausilia  shew 
a  gradual  change  in  the  ratio  to  one  another  of  successive  whorls,  or 
in  other  words  a  marked  departure  from  the  logarithmic  law: 

Clausilia  lamellosa.     (From  Chr.  Petersen*.) 


Width  of  successive 

Ratios,  or  ' 

'quotients' 

'of 

whorls  (mm. 

) 

successive 

A 

whorls 

T~ 

II 

III 

I 

II 

III 

Mean 

a 

2-42 

2-51 

2-49 

ajb 

1-43 

1-45 

1-42 

1-44 

b 

1-69 

1-72 

1-75 

b/c 

1-36 

1-33 

1-31 

1-33 

c 

1-24 

1-30 

1-33 

cjd 

1-21 

1-29 

1-23 

1-24 

d 
e 

102 
0-83 

100 
0-83 

1-08 
0-86 

die 

1-22 

1-20 

1-26 

1-23 

In  many  ammonites,  where  the  helicoid  factor  does  not  enter  into 
the  case,  we  have  a  clear  illustration  of  how  gradual  and  marked 

*  From  Chr.  Petersen,  Das  Quotientengesetz,  p.  36.  After  making  a  careful 
statistical  study  of  1000  Clausilias,  Peterson  found  the  following  mean  ratios  of  the 
successive  whorls,  aJb,  b/c,  etc.:  1-37,  1-33,  1-27,  1-24,  1-22,  119. 


XI]  OF  VARIOUS  CEPHALOPODS  809 

changes  in  the  spiral  angle  may  be  detected  even  in  ammonites  which 
present  nothing  abnormal  to  the  eye.  But  let  us  suppose  that  the 
spiral  angle  increases  somewhat  rapidly;  we  shall  then  get  a  spiral 
with  gradually  narrowing  whorls,  which  condition  is  characteristic  of 
Oekotraustes,  a  subgenus  of  Ammonites.  If  on  the  other  hand,  the 
angle  a  gradually  diminishes,  and  even  falls  away  to  zero,  we  shall 
have  the  spiral  curve  opening  out,  as  it  does  in  Scaphites,  Ancyloceras 


Fig.  393.     An  ammonitoid  shell  (Macroscaphites)  to  shew  change  of 
curvature. 

and  Lituites,  until  the  spiral  coil  is  replaced  by  a  spiral  curye  so 
gentle  as  to  seem  all  but  straight.  Lastly,  there  are  a  few  cases, 
such  as  BelleropJion  expansus  and  some  Goniatites,  where  the  outer 
spiral  does  not  perceptibly  change,  but  the  whorls  become  more 
"embracing"  or  the  whole  shell  more  involute.  Here  it  is  the 
angle  of  retardation,  the  ratio  of  growth  between  the  outer  and 
inner  parts  of  the  whorl,  which  undergoes  a  gradual  change. 

In  order  to  understand  the  relation  of  a  close-coiled  shell  to  its 
straighter  congeners,  to  compare  (for  example)  an  Ammonite  with 
an  Orthoceras,  it  is  necessary  to  estimate  the  length  of  the  right 
cone  which  has,  so  to  speak,  been  coiled  up  into  the  spiral  shell.  Our 
problem  is,  to  find  the  length  of  a  plane  equiangular  spiral,  in 
terms  of  the  radius  and  the  constant  angle  a.  Then,  if  OP  be  a 
radius  vector,  OQ  a  line  of  reference  perpendicular  to  OP,  and 
PQ  a  tangent  to  the  curve,  PQ,  or  sec  a,  is  equal  in  length  to  the 
spiral  arc  OP.  In  other  words,  the  arc  measured  from  the  pole  is 
equal  to  the  polar  tangent*.     And  this  is  practically  obvious:  for 

*  Descartes  made  this  discovery,  and  records  it  in  a  letter  to  Mersenne,  1638. 
The  equiangular  spiral  was  thus  the  first  transcendental  curve  to  be  "rectified." 


810  THE  EQUIANGULAR  SPIRAL  [ch. 

PP'/PR'  =  dsjdr  =  sec  a,  and  therefore  sec  a  =  sjr,  or  the  ratio  of 
arc  to  radius  vector. 


Fig.  394. 

Accordingly,  the  ratio  of  I,  the  total  length,  to  r,  the  radius 
vector  up  to  which  the  total  length  is  to  be  measured,  is  expressed 
by  a  simple  table  of  secants ;   as  follows : 


a 

Hr 

5° 

1004 

10 

1015 

20 

1064 

30 

1165 

40 

1-305 

50 

1-56 

60 

20 

70 

2-9 

75 

3-9 

80 

5-8 

85 

11-5 

86 

14-3 

a 

llT 

87° 

191 

88 

28-7 

89 

57-3 

89°  10' 

68-8 

20 

85-9 

30 

114-6 

40 

171-9 

50 

343-8 

55 

687-5 

59 

3437-7 

90 

Infinite 

Putting  the  same  table  inversely,  so  as  to  shew  the  total  length 
in  terms  of  the  radius,  we  have  as  follows : 


Total  length  (in  terms 

of  the  radius) 

Constant  angl< 

2 

60° 

3 

70  3r 

4 

75    32 

5 

78   28 

10 

84    16 

20 

87     8 

30 

88     6 

40 

88   34 

50 

88    51 

100 

89   26 

1000 

89   56' 36" 

10,000 

89   59  30 

XI]  OF  VARIOUS  CEPHALOPODS  811 

Accordingly,  we  see  that  (1),  when  the  constant  angle  of  the 
spiral  is  small,  the  shell  (or  for  that  matter  the  tooth,  or  horn  or 
claw)  is  scarcely  to  be  distinguished  from  a  straight  cone  or  cylinder; 
and  this  remains  pretty  much  the  case  for  a  considerable  increase  of 
angle,  say  from  0°  to  20°  or  more;  (2)  for  a  considerably  greater 
increase  of  the  constant  angle,  say  to  50°  or  rnore,  the  shell  would 
still  only  have  the  appearance  of  a  gentle  curve;  (3)  the  charac- 
teristic close  coils  of  the  Nautilus  or  Ammonite  would  be  typically 
represented  only  when  the  constant  angle  lies  within  a  few  degrees 
on  either  side  of  about  80°.  The  coiled  up  spiral  of  a  Nautilus, 
with  a  constant  angle  of  about  80°,  is  about  six  times  the  length 
of  its  radius  vector,  or  rather  more  than  three  times  its  own 
diameter ;  while  that  of  an  Ammonite,  with  a  constant  angle  of,  say, 
from  85°  to  88°,  is  from  about  six  to  fifteen  times  as  long  as  its  own 
diameter.  And  (4)  as  we  approach  an  angle  of  90°  (at  which  point 
the  spiral  vanishes  in  a  circle),  the  length  of  the  coil  increases  with 
enormous  rapidity.  Our  spiral  would  soon  assume  the  appearance 
of  the  close  coils  of  a  Nummuhte,  and  the  successive  increments 
of  breadth  in  the  successive  whorls  would  become  inappreciable  to 
the  eye. 

The  geometrical  form  of  the  shell  involves  many  other  beautiful 
properties,  of  great  interest  to  the  mathematician  but  which  it  is 
not  possible  to  reduce  to  such  simple  expressions  as  we  have  been 
content  to  use.  For  instance,  we  may  obtain  an  equation  which 
shall  express  completely  the  surface  of  any  shell,  in  terms  of  polar 
or  of  rectangular  coordinates  (as  has  been  done  by  Moseley  and 
by  Blake),  or  in  Hamiltonian  vector  notation*.  It  is  likewise  pos- 
sible (though  of  little  interest  to  the  naturalist)  to  determine  the 
area  of  a  conchoid al  surface  or  the  volume  of  a  conchoidal  solid, 
and  to  find  the  centre  of  gravity  of  either  surface  or  solid  f.  And 
Blake  has  further  shewn,  with  considerable  elaboration,  how  we  may 
deal  with  the  symmetrical  distortion  due  to  pressure  which  fossil 
shells  are  often  found  to  have  undergone,  and  how  we  may  re- 
constitute by  calculation  their  original  undistorted  form — a  problem 
which,  were  the  available  methods  only  a  Kttle  easier,  would  be 

*  Cf.  H.  W.  L.  Hime's  Outlines  of  Quaternions,  1894,  pp.  171-173. 
t  See  Moseley,  op.  cit.  p.  361  seq.     Also,  for  more  complete  and  elaborate  treat- 
ment, Haton  de  la  Goupilliere,  op.  cit.  1908,  pp.  5-46,  69-204. 


812  THE  EQUIANGULAR  SPIRAL  [ch 

very  helpful  to  the  palaeontologist;  for,  as  Blake  himself  has  shewn, 
it  is  easy  to  mistake  a  symmetrically  distorted  specimen  of  (for 
instance)  an  Ammonite  for  a  new  and  distinct  species  of  the  same 
genus.  But  it  is  evident  that  to  deal  fully  with  the  mathematical 
problems  contained  in,  or  suggested  by,  the  spiral  shell,  would  require 
a  whole  treatise,  rather  than  a  single  chapter  of  this  elementary  book. 
Let  us  then,  leaving  mathematics  aside,  attempt  to  summarise,  and 
perhaps  to  extend,  what  has  been  said  about  the  general  possibilities 
of  form  in  this  class  of  organisms. 

The  univalve  shell:  a  summary 

The  surface  of  any  shell,  whether  discoid  or  turbinate,  may  be 
imagined  to  be  generated  by  the  revolution  about  a  fixed  axis  of 
a  closed  curve,  which,  remaining  always  geometrically  similar  to 
itself,  increases  its  dimensions  continually:  and,  since  the  scale  of 
the  figure  increases  in  geometrical  progression  while  the  angle 
of  rotation  increases  in  arithmetical,  and  the  centre  of  similitude 
remains  fixed,  the  curve  traced  in  space  by  corresponding  points 
in  the  generating  curve  is,  in  all  such  cases,  an  equiangular  spiral.  In 
discoid  shells,  the  generating  figure  revolves  in  a  plane  perpendicular 
to  the  axis,  as  in  the  Nautilus,  the  Argonaut  and  the  Ammonite.  In 
turbinate  shells,  it  follows  a  skew  path  with  respect  to  the  axis  of 
revolution,  and  the  curve  in  space  generated  by  any  given  point  makes 
a  constant  angle  to  the  axis  of  the  enveloping  cone,  and  partakes, 
therefore,  of  the  character  of  a  helix,  as  well  as  of  a  logarithmic  spiral.; 
it  may  be  strictly  entitled  a  helico-spiral.  Such  turbinate  or  helico- 
spiral  shells  include  the  snail,  the  periwinkle  and  all  the  common 
typical  Gastropods. 

When  the  envelope  of  the  shell  is  a  right  cone — and  it  is  seldom  far  from 
being  so — then  our  helico-spiral  is  a  loxodromic  curve,  and  is  obviously 
identical  with  a  projection,  parallel  with  the  axis,  of  the  logarithmic  spiral 
of  the  base.  As  this  spiral  cuts  all  radii  at  a  constant  angle,  so  its  orthogonal 
projection  on  the  surface  intersects  all  generatrices,  and  consequently  all 
parallel  circles,  under  a  constant  angle*  this  being  the  definition  of  a  loxodromic 
curve  on  a  surface  of  revolution.  Guido  Grandi  describes  this  curve  for  the 
first  time  in  a  letter  to  Ceva,  printed  at  the  end  of  his  Demonstratio  theorematum 
Hugenianorum  circa. .  .logarithmicam  lineam,  1701  *. 

*  See  R.  C.  Archibald,  op.  cit.  1918.  Ohvier  discussed  it  again  {Rev.  de  geom. 
descriptive,   1843)   calling  it  a   "conical  equiangular"   or  "conical  logarithmic" 


XI]  OF  VARIOUS  UNIVALVES  813 

The  generating  figure  may  be  taken  as  any  section  of' the  shell, 
whether  parallel,  normal,  or  otherwise  incHned  to  the  axis.  It  is  very 
commonly  assumed  to  be  identical  with  the  mouth  of  the  shell;  in 
which  case  it  is  sometimes  a  plane  curve  of  simple  form ;  in  other  and 
more  numerous  cases,  it  becomes  compHcated  in  form  and  its  boun- 
daries do  not  he  in  one  plane:  but  in  such  cases  as  these  we  may 
replace  it  by  its  "trace,"  on  a  plane  at  some 
definite  angle  to  the  direction  of  growth,  for 
instance  by  its  form  as  it  appears  in  a  section 
through  the  axis  of  the  helicoid  shell.  The 
generating  curve  is  of  very  various  shapes. 
It  is  circular  in  Scalaria  or  Cyclostoma,  and 
in  Spirula ;  it  may  be  considered  as  a  segment 
of  a  circle  in  Natica  or  in  Planorbis.  It  is 
triangular  in  Conus  or  Thatcheria,  and 
rhomboidal  in  Solarium  or  Potamides.  It 
is  very  commonly  more  or  less  elliptical :  the 
long  axis  of  the  ellipse  being  parallel  to  the 
axis  of  the  shell  in  Oliva  and  Cypraea;  all 
but  perpendicular  to  it  in  many  Trochi ;  and 
oblique  to  it  in  many  well-marked  cases,^such 
as  Stomatella,  Lamellaria,  Sigaretus  halio- 
toides  (Fig.  396)  and  Haliotis.  In  Nautilus 
pomjpilius  it  is  approximately  a  semi-ellipse,  Fig.  395.  Section  of  a  spiral 
and  in  A^.  umbilicatus  rather  more  than  a     ^^^^^^>  Triton  corrugatus 

Lam.   jrom  Woodward. 

semi-ellipse,  the  lo'ng  axis  lying  in  both  cases 

perpendicular  to  the  axis  of  the  shell*.     Its  form  is  seldom  open  to 

easy  mathematical  expression,  save  when  it  is  an  actual  circle  or 

spiral.  Paul  Serret  (Th.  nouv. .  .des  lignes  a  double  courbure,  1860,  p.  101)  called 
it  ""helice  cylindroconique'" ;  Haton  de  la  Goupilliere  calls  it  a  '' conhelice.'"  It  has 
also  been  studied  by  {int.  at.)  Tissot,  Nouv.  ann.  de  mathem.  1852;  G.  Pirondini, 
Mathesis,  xix,  pp.  153-8,  1899;   etc. 

*  In  Nautilus,  the  "hood"  has  somewhat  dififerent  dimensions  in  the  two 
sexes,  and  these  differences  are  impressed  upon  the  shell,  that  is  to  say  upon  its 
"generating  curve."  The  latter  constitutes  a  somewhat  broader  ellipse  in  the 
male  than  in  the  female.  But  this  difference  is  not  to  be  detected  in  the  young; 
in  other  words,  the  form  of  the  generating  cu'rve  perceptibly  alters  with  advancing 
age.  Somewhat  similar  differences  in  the  shells  of  Ammonites  were  long  ago 
suspected,  by  d'Orbigny,  to  be  due  to  sexual  differences.  (Cf.  Willey,  Natural 
Science,  vi,  p.  411,  1895;  Zoological  Results,  1902,  p.  742.) 


814  THE  EQUIANGULAR  SPIRAL  [ch. 

ellipse;  but  an  exception  to  this  rule  may  be  found  in  certain 
Ammonites,  forming  the  group  "  Cordati,"  where  (as  Blake  points  out) 
the  curve  is  very  nearly  represented  by  a  cardioid,  whose  equation 
is  r  =  a  (1  +  cos  6). 

When  the  generating  curves  of  successive  whorls  cut  one  another, 
the  hne  of  intersection  forms  the  conspicuous  heHco-spiral  or 
loxodromic  curve  called  the  suture  by  conchologists. 

The  generating  curve  may  grow  slowly  or  quickly;  its  growth- 
factor  is  very  slow  in  Dentalium  or  Turritella,  very  rapid  in  Nerita, 
or  Pileopsis,  or  Haliotis  or  the  Limpet.  It  may  contain  the  axis 
in  its  plane,  as  in  Nautilus ;  it  may  be  parallel  to  the  axis,  as  in  the 
majority  of  Gastropods;  or  it  may  be  inclined  to  the  axis,  as  it  is  in 
a  very  marked  degree  in  Haliotis,     In  fact,  in  Haliotis  the  generating 


B 


A 

Fig.  396.     A,  Lanydlaria  perspicua;   B,  Sigaretus  haliotoides. 
After  Woodward. 

curve  is  so  oblique  to  the  axis  of  the  shell  that  the  latter  appears 
to  grow  by  additions  to  one  margin  only  (cf.  Fig.  362),  as  in  the 
case  of  the  opercula  of  Turbo  and  Nerita  referred  to  on  p.  775; 
and  this  is  what  Moseley  supposed  it  to  do. 

The  general  appearance  of  the  entire  shell  is  determined  (apart 
from  the  form  of  its  generating  curve)  by  the  magnitude  of  three 
angles;  and  these  in  turn  are  determined,  as  has  been  sufficiently 
explained,  by  the  ratios  of  certain  velocities  of  growth.  These 
angles  are  (1)  the  constant  angle  of  the  equiangular  spiral  (a);  (2)  in 
turbinate  shells,  the  enveloping  angle  of  the  cone,  or  (taking  half 
that  angle)  the  angle  (p)  which  a  tangent  to  the  whorls  makes  with 
the  axis  of  the  shell;  and  (3)  an  angle  called  the  "angle  of  retarda- 
tion" (y),  which  expresses  the  retardation  in  growth  of  the  inner 
as  compared  with  the  outer  part  of  each  whorl,  and  therefore 
measures  the  extent  to  which  one  whorl  overlaps,  or  the  extent  to 
which  it  is  separated  from,  another. 


XI]  OF  VARIOUS  UNIVALVES  815 

The  spiral  angle  (a)  is  very  small  in  a  limpet,  where  it  is  usually- 
taken  as  =0°;  but  it  is  evidently  of  a  significant  amount, 
though  obscured  by  the  shortness  of  the  tubular  shell.  In 
Dentalium  it  is  still  small,  but  sufficient  to  give  the  appearance 
of  a  regular  curve;  it  amounts  here  probably  to  about  30°  to 
40°.  In  Haliotis  it  is  from  about  70°  to  75°;  in  Nautilus  about 
80°;  and  it  lies  between  80°  and  85°  or  even  more,  in  the  majority 
of  Gastropods*. 

The  case  of  Fissurella  is  curious.  Here  we  have,  apparently, 
a  conical  shell  with  no  trace  of  spiral  curvature,  or  (in  other  words) 
with  a  spiral  angle  which  approximates  to  0°;  but  in  the  minute 
embryonic  shell  (as  in  that  of  the  limpet)  a  spiral  convolution  is 
distinctly  to  be  seen.  It  would  seem,  then,  that  what  we  have  ^to 
do  with  here  is  an  unusually  large  growth-factor  in  the  generating 
curve,  causing  the  shell  to  dilate  into  a  cone  of  very  wide  angle, 
the  apical  portion  of  which  has  become  lost  or  absorbed,  and  the 
remaining  part  of  which  is  too  short  to  show  clearly  its  intrinsic 
curvature.  In  the  closely  allied  Emarginula,  there  is  hkewise  a 
well-marked  spiral  in  the  embryo,  which  however  is  still  manifested 
in  the  curvature  of  the  adult,  nearly  conical,  shell.  In  both  cases 
we  have  to  do  with  a  very  wide-angled  cone,  and  with  a  high 
retardation-factor  for  its  inner,  or  posterior,  border.  The  series  is 
continued,  from  the  apparently  simple  cone  to  the  complete  spiral, 
through  such  forms  as  Calyptraea. 

The  angle  a,  as  we  have  seen,  is  not  always,  nor  rigorously, 
a  constant  angle.  In  some  Ammonites  it  may  increase  with  age, 
the  whorls  becoming  closer  and  closer;  in  others  it  may  decrease 
rapidly  and  even  fall  to  zero,  the  coiled  shell  then  straightening 
out,  as  in  Lituites  and  similar  forms.  It  diminishes  somewhat,  also, 
in  many  Orthocerata,  which  are  slightly  curved  in  youth  but  straight 
in  age.  It  tends  to  increase  notably  in  some  common  land-shells, 
the  Pupae  and  Bulimi ;  and  it  decreases  in  Succinea. 

Directly  related  to  the  angle  a  is  the  ratio  which  subsists  between 
the  breadths  of  successive  whorls.     The  following  table  gives  a  few 

*  What  is  sometimes  called,  as  by  Leslie,  the  angle  of  deflection  is  the  complement 
of  what  we  have  called  the  spiral  angle  (a),  or  obliquity  of  the  spiral.  When  the 
angle  of  deflection  is  6°  17'  41",  or  the  spiral  angle  83''  42'  19",  the  radiants,  or 
breadths  of  successive  whorls,  are  doubled  at  each  entire  circuit. 


816  THE  EQUIANGULAR  SPIRAL  [ch. 

illustrations  of  this  ratio  in  particular  cases,  in  addition  to  those 
which  we  have  already  studied. 


Ratio  of  breadth  of  consecutive  whorls 


Pointed  Turbinates 

Telescopium  fuscum            ...  1-14 

Terebra  suhulata       ...         ...  1-16 

*  Turritella  terebellata            ...  1-18 
*Turritella  imbricata...         ...  1-20 

Cerithium  palustre   ...         ...  1  "22 

Turritella  duplicata ...         ...  1-23 

Melanopsis  terebralis           ...  1-23 

Cerithium  nodulosum          ...  1-24 

*  Turritella  car  inata   ...         ...  1-25 

Terebra  crenulata      ...         ...  1-25 

Terebra  maculata  (Fig.  397)  1-25 

*Cerithium  lignitarum     '     ...  1-26 

Terebra  dimidiata     1-28 

Cerithium  sulcatum ...         ...  1-32 

Fusus  longissimus    ...         ...  1*34 

*Pleurotomaria  conoidea       ...  1-34 

Trochus  niloticus  (Fig.  398)  1-41 

Mitra  episcopalis     ...         ...  1-43 

Fusus  antiquus        ...         ...  1  -50 

Scalaria  pretiosa      ...         ...  1-56 

Fusus  colosseus        ...         ...  1-71 

PhasiaTvella  australis           ...  1-80 

Helicostyla  polychroa           ...  2-00 


Obtuse  Turbinates  and  Discoids 

Conus  virgo ...         ...  ...  1-25 

XClymenia  laevigata ...  ...  1-33 

Conus  litteratus       ...  ...  1*40 

Conus  betulinus       ...  ...  1-43 

XClymenia  arietina   ...  ...  1-50 

%Goniatites  bifer        ...  ...  1-50 

*  Helix  nemoralis       ...  ...  1-50 

*  Solarium  perspectivum  ...  1*50 
Solarium  trochleare  ...  1-62 
Solarium  magnificum  ...  1-75 

*Natica  aperta           ...  ...  2-00 

Euomphalus  pentangulatus  2-00 

Planorbis  corneus    ...  ...  2-00 

Solaropsis  pellis-serpentis  ...  2-00 

Dolium  zondtum      ...  ...  2-10 

XGoniatites  carinatus  ...  2-50 

*Natica  glaucina       ...  ...  3-00 

Nautilus  pompilius  ...  3-00 

Haliotis  excavatus  ...  ...  4-20 

Hal^otis  parvus       ...  ...  6*00 

Delphinula  atrata    ...  ...  6-00 

Haliotis  rugoso-plicata  ...  9-30 

Haliotis  viridis        ...  ...  10-00 


Those  marked  *  from  Naumann;   J  from  Miiller;   the  rest  from  Macalisterf. 


In  the  case  of  turbinate  shells,  we  muSt  take  into  account  the 
angle  y^,  in  order  to  determine  the  spiral  angle  a  from  the  ratio 
of  the  breadths  of  consecutive  whorls;  for  the  short  table  given 
on  p.  791  is  only  applicable  to  discoid  shells,  in  which  the  angle  fi 
is  an  angle  of  90°.  Our  formula,  as  mentioned  on  p.  771,  now 
becomes 

J^  _  ^27rsin/?cota 

For  this  formula,  I  have  worked  out  the  following  table. 


f  Alex.  Macahster,  Observations  on  the  mode  of  growth  of  discoid  and  turbinated 
shells,  Proc.  R.S.  xviii,  pp.  529-532,  1870;  Ann.  Mag.  N.H.  (6),  iv,  p  160,  1870. 
Cf.  also  his  Law  of  Symmetry  as  exemplified  in  animal  form,  Journ.  B.  Dublin  Soc. 
1869,  p.  327. 


XI] 


OF  VARIOUS  UNIVALVES 


817 


r 


t^OiOt^iOCOeCMOiCiOOiMO 
O5l>?DC0^O5Q0l>COiOO5Tt<t>CO 


§ 


00»0'-^^000»0»0»0»0>-0  0i0 


Ot^r-iO©0»00©000»00 
o  (M  CO  w      ici      r-i      ^  c^      eo<N 

O    o 

^    lO  00  05  |>  O -^ -^  00  CO  Tj*  lO  O  »0  CO 

ooi>cou:)kO-<*'^coeoco(N(M'-*i-t 


o  Xi-HOOCO«<Jt^»00»CiO»000 

»o        o  eo  (M  lo  eo  eo  CO      Tf*  <n  <n      u5 

II  o 

-_  ot^coooocoeO'HOoocooxo 

^«-  00X>iOc0C0<NC^<N<M^^>-H 


818  THE  EQUIANGULAR  SPIRAL  [ch. 

From  this  table,  by  interpolation,  we  may  easily  fill  in  the 
approximate  values  of  a,  as  soon  as  we  have  determined  the  apical 
angle  ^  and  measured  the  ratio  R\  as  follows: 


Turritella  sp. 

112 

7° 

81' 

Cerithium  nodvloaum 

1-24 

15 

82 

Conus  virgo 

1-25 

70 

88 

Mitra  episcopalis . . . 

143 

16 

78 

Scalaria  pretiosa  ... 

1-56 

26 

81 

Phasianella  australis 

1-80 

26 

80 

Solarium  perspectivum 

1-50 

53 

85 

Natica  aperta 

2-00 

70 

83 

Planorbis  corneus 

2-00 

90 

84 

Euomphalus  pentangviat 

us                2-00 

90 

84 

We  see  from  this  that  shells  so  different  in  appearance  as  Cerithium, 
Solarium,  Natica  and    Planorbis  differ  very  httle  indeed  in  the 

•  magnitude  of  the  spiral  angle  a,  that  is 
to  say  in  the  relative  velocities  of  radial 
and  tangential  growth.  It  is  upon  the 
angle  /?  that  the  difference  in  their  form 
mainly  depends. 

The  angle,  or  rather  semi-angle  (/?),  of 
the  tangent  cone  may  be  taken  as  90° 
in  the  discoid  shells,  such  as  Nautilus 
and  Planorbis.  It  is  still  a  large  angle, 
of  70°  or  75°,  in  Conus  or  in  Cymba, 
somewhat  less  in  Cassis,  Harpa,  Dolium 
or  Natica ;  it  is  about  50°  to  55°  in  the 
various  species  of  Solarium,  about  35° 
in  the  typical  Trochi,  such  as  T.  niloticus 
or  T.  zizyphinus,  and  about  25°  or  26°  in 
Scalaria  pretiosa  and  Phasianella  bul- 
-  hides',   it  becomes  a  very  acute  angle. 

Fig.  397.     TerebramacuJutal..      ^^   ^^o ^    ^^o ^    ^^    ^^^^    j^^^^    -^    ^^^^.^^^ 

Turritella  or  Cerithium.  The  British  species  of '  Fusus '  form  a  series 
in  which  the  apical  angle  ranges  from  about  28°  in  F.  antiquus, 
through  F.  Norvegicus,  F.  berniciensis,  F.  Turtoni,  F.  Islandicus, 
to  about  17°  in  F.  gracilis.  It  varies  much  among  the  Cones;  and 
the  costly  Conus  gloria-maris,  one  of  the  great  treasures  of  the 
conchologist,  differs  from  its  congeners  in  no  important  particular 


XI]  OF  VARIOUS  UNIVALVES  819 

save  in  the  somewhat  "produced"  spire,  that  is  to  say  in  the  com- 
paratively low  value  of  the  angle  p. 

A  variation  with  advancing  age  of  ^  is  common,  but  (as  Blake 
points  out)  it  is  often  not  to  be  distinguished  or  disentangled  from 
an  alteration  of  a.  Whether  alone,  or  combined  with  a  cKange  in  a, 
we  find  it  in  all  those  many  gastropods  whose  whorls  cannot  all  be 
touched  by  the  same  enveloping  cone,  and  whose  spire  is  accordingly 
described  as  concave  or  convex.     The  former  condition,  as  we  have 


Fig.  398.     Trochus  niloticus  L. 

it  in  Cerithium,  and  in  the  cusp-Hke  spire  of  Cassis,  Dolium  and 
some  Cones,  is  much  the  commoner  of  the  two*. 

In  the  vast  majority  of  spiral  univalves  the  shell  winds  to  the 
right,  or  turns  clockwise,  as  we  look  along  it  in  the  direction  in  which 
the  animal  crawls  and  puts  out  its  head.  The  thread  of  a  carpenter's 
screw  (except  in  China)  runs  the  same  way,  and  we  call  it  a  "right- 
handed  screw."     Save  that  it  takes  a  right-handed  movement  to 

*  Many  measurements  of  the  linear  dimensions  of  univalve  shells  have  bgen 
made  of  late  years,  and  studied  by  statistical  methods  in  order  to  detect  local  races 
and  other  instances  of  variation  and  variability.  But  conchological  statisticians 
seem  to  be  content  with  some  arbitrary  linear  ratio  as  a  measure  of  "squatness" 
or  the  reverse;  and  the  measurements  chosen  give  Uttle  or  no  help  towards  the 
determination  either  of  the  apical  or  of  the  spiral  angle.  Cf,  (e.g.)  A.  E.  Boycott, 
Conchometry,  Proc.  Malacol.  Soc.  xvn,  p.  8,  1928;  C.  Price- Jones,  ibid,  xix, 
p.  146,  1930;  etc.  See  also  G.  Duncker,  Methode  der  Variations-Statistik,  Arch, 
f.  Entw.  Mech.  vm,  pp.  112-183,  1899. 


820  THE  EQUIANGULAR  SPIRAL  [ch. 

drive  in  a  "right-handed"  screw,  the  terms  right-handed  and  left- 
handed  are  purely  conventional;  and  the  mathematicians  and  the 
naturalists,  unfortunately,  use  them  in  opposite  ways.  Thus  the 
mathematicians  call  the  snail-shell  or  the  joiner's  screw  kiotropic; 
and  Listing  for  one  has  much  to  say  about  lack  of  precision  or  even 
confusion  on  the  part  of  the  conchologists  and  the  botanists,  from 
Linnaeus  downwards,  in  their  attempts  to  deal  with  right-handed 
and  left-handed  spirals  or  screws*.  The  convolvulus  twines  to  the 
right,  the  hop  to  the  left;  vine- tendrils  are  said  to  be  mostly  right- 
handed.  At  any  rate,  Clerk  Maxwell  spoke  of  hop-spirals  and  vine- 
^irals,  trying  to  avoid  the  confusion  or  ambiguity  of  left  and  right. 
Some  climbing  plants  are  one  and  some  the  other;  and, the  architect 
shews  httle  preference,  but  builds  his  spiral  staircases  or  twisted 
columns  either  way.  But  in  all  these,  shells  and  all,  the  spiral  runs 
one  way;  it  is  isotropic,  while  the  fir-cone  shews  spirals  running  both 
ways  at  once,  and  we  call  them  heterotropic,  or  diadroinic. 

When  we  find  a  "reversed  shell,"  a  whelk  or  a  snail  winding  the 
wrong  way,  we  describe  it  mathematically  by  the  simple  statement 
that  the  apical  angle  (^)  has  changed  sign.  Such  left-handed  shells 
occur  as  a  well-known  but  rare  abnormality;  and  the  men  who 
handle  snails  in  the  Paris  market  or  whelks  in  Billingsgate  keep 
a  sharp  look-out  for  them.  In  rare  instances  they  become  common. 
While  left-handed  whelks  (Buccinum  or  Neptunea)  are  very  rare 
nowadays,  it  was  otherwise  in  the  epoch  of  the  Red  Crag;  for 
Neptunea  was  then  extremely  common,  but  right-handed  specimens 
were  as  rare  as  left-handed  are  today.  In  the  beautiful  genus 
Ampullaria,  or  apple-snails,  which  inhabit  tropical  and  sub-tropical 
rivers,  there  is  unusual  diversity;  for  the  spire  turns  to  the  right 
in  some  species,  and  to  the  left  in  others,  and  again  some  are  flat 
or  "discoid,"  with  no  spire  at  all;  and  there  are  plenty  of  half-way 
stages,  with  right  and  left-handed  spires  of  varying  steepness  or 
acutenessf;  in  short,  within  the  limits  of  this  singular  genus  the 
apical  angle  (y^)  may  vary  from  about  ±  35°  to  ±  125°.  But  we 
need  not  imagine  that  the  direction  of  gK)wth  actually  changes 
over  from  right-handed  to  left-handed;    it  is  enough  to  suppose 

*  See  Listing's  Topologie,  p.  36;  and  cf.  Clerk  Maxwell's  Electricity  and 
Magnetism,  i,  p.  24. 

t  See  figures  in  Arnold  Lang's  Comparative  Anatomy  (English  translation), 
II,  p.  ICl,  1902. 


XI]  OF  VARIOUS  UNIVALVES  821 

that  the  ^ew  movement  along  the  axis  has  changed  its  direction. 
For  if  I  take  a  roll  of  tape  and  push  the  core  out  to  one  side  or  to 
the  other,  or  if  I  keep  the  centre  of  the  roll  fixed  and  push  the  rim 
to  the  one  side  or  to  the  other,  I  thereby  convert  the  flat  roll  into  a 
hollow  cone,  or  (in  other  words)  a  plane  into  a  gauche  spiral.  Whether 
we  push  one  way  or  other,  whether  the  spiral  coil  be  plane  or  gauche, 
positively  or  negatively  deformed,  it  remains  right-handed  or  left- 
handed  as  the  case  may  be ;  but  it  does  change  its  direction  as  soon 
as  we  turn  it  upside  down,  or  as  soon  as  the  animal  does  so  in  assuming 
its  natural  attitude.  The  linear  spirals  within  and  without  the  cone 
may  change  places  but  must  remain  congruent  with  one  another; 
for  they  are  merely  the  two  edges  of  the  ribbon,  and  as  such  are 
inseparable  and  identical  twins.  But  of  the  shell  itself  we  may 
reasonably  say  that  a^right-handed  has  given  place  to  a  left-handed 
spiral.  Of  these,  the  one  is  a  mirror-image  of  the  other;  and  the 
passing  from  one  to  the  other  through  the  plane  of  symmetry 
(which  has  no  "handedness*')  is  an  operation  which  Listing  called 
perversion.  The  flat  or  discoid  apple-snails  are  Uke  our  roll  of  tape, 
which  can  be  converted  into  a  conical  spire  and  perverted  in  one 
direction  or  the  other;  and  in  this  genus,  by  a  rare  exception,  it 
seems  wellnigh  as  easy  to  depart  one  way  as  the  other  from  the 
plane  of  sjmametry.  But  why,  in  the  general  run  of  shells,  all  the 
world  over,  in  the  past  and  in  the  present,  one  direction  of  twist  is 
so  overwhelmingly  commoner  than  the  other,  no  man  knows. 

The  phenomenon  of  reversal,  or  "sinistrality,"  has  an  interest  of 
its  own  from  j:he  side  of  development  and  heredity.  For  careful 
study  of  certain  pond-snails  has  shewn  that  dextral  and  sinistral 
varieties  appear,  not  one  by  one,  but  by  whole  broods  of  the  one 
sort  or^the  other;  a  discovery  which  goes  some  way  to  account 
for  the  predominant  left-handedness  of  Fusus  amhiguus  in  the 
Red  Crag.  The  right-handed,  or  ordinary  form,  is  found  to  be 
"dominant"  to  the  other;  but  the  Mendelian  heredity  is  of  a  curious 
and  complicated  kind.  For  the  direction  of  the  twist  appears  to 
be  predetermined  in  the  germ  even  prior  to  its  fertilisation;  and 
a  left-handed  pond-snail  will  produce  a  brood  of  left-handed  young 
even  when  fertilised  by  a  normal,  or  right-handed,  individual*. 

*  See  A,  E.  Boycott  and  others,  Abnormal  forms  of  Limnaea peregra .  .  .and  their 
inheritance,  Phil.  Trans.  (B),  ccxxix,  p.  51,  1930;   and  other  papers. 


822  THE  EQUIANGULAR  SPIRAL  [ch. 

The  angle  of  retardation  (y )  is  very  small  in  .  Dentalium  and 
Patella ;  it  is  very  large  in  Haliotis ;  it  becomes  infinite  in  Argonauta 
and  in  Cypraea.  Connected  with  the  angle  of  retardation  are  the 
various  possibilities  of  contact  or  separation,  in  various  degrees, 
between  adjacent  whorls  in  the  discoid  shell,  and  between  both 
adjacent  and  opposite  whorls  in  the  turbinate.  But  with  these 
phenomena  we  have  already  dealt  sufficiently. 

The  beautiful  shell  of  the  paper-nautilus  {Argonauta  argo  L.)  differs 
in  sundry  ways  both  from  the  Nautilus  and  from  ordinary  univalves. 
Only  the  female  Argonaut  possesses  it;  it  is  not  attached  to  its 
owner,  but  is  (so  to  speak)  worn  loose;  it  is  rather  a  temporary 
cradle  for  the  young  than  a  true  shell  or  bodily  covering;  and  it 
is  not  secreted  in  the  usual  way,  but  is  plastered  on  from  the  outside 
by  two  of  the  eight  arms  of  the  Uttle  Octopus  to  which  it  belongs. 
The  shell  shews  a  single  whorl,  or  but  Httle  more;  and  the  spiral 
is  hard  to  measure,  for  this  reason.  It  ha«  been  supposed  by  some 
to  obey  a  law  other  than  the  logarithmi-c  spiral.  For  my  part  I  have 
made  no  special  study  of  it,  nor  has  any  one  else,  to  my  knowledge, 
of  recent  years;  but  the  simple  fact  that  it  conserves  its  shape  as  it 
grows,  or  that  each  increment  is  a  gnomon  to  the  rest,  is  enough  to 
shew  that  this  dehcate  and  beautiful  shell  is  mathematically,  though 
not  morphologically,  homologous  with  all  the  others. 

Of  bivalve  shells 

Hitherto  we  have  dealt  only  with  univalve  shells,  and  it  is  in 
these  that  all  the  mathematical  problems  connected  .with  the  spiral, 
or  hehco-spiral,  configuration  are  best  illustrated.  But  the  case  of 
the  bivalve  shell,  whether  of  the  lamelHbranch  or  the  brachiopod, 
presents  no  essential  difference,  save  only  that  we  have  here  to  do 
with  two  conjugate  spirals,  whose  two  axes  have  a  definite  relation 
to  one  another,  and  some  independent  freedom  of  rotatory  movement 
relatively  to  one  another. 

The  bivalve  or  lamelHbranch  moUusca  are  very  different  creatures 
from  the  rest.  The  univalves  or  gastropods,  hke  their  cousins  the 
cephalopods,  go  about  their  business  and  get  their  living  in  an 
ordinary  way;  but  the  bivalves  are  linintelHgent,  "acephalous" 
animals,  and  imbibe  the  invisible  plankton-food  which  ciliary 
currents  bring  automatically  to  their  mouths.     There  is  something 


XI] 


OF  THE  ARGONAUT 


823 


<§■ 


OS 
CO 


824  THE  EQUIANGULAR  SPIRAL  [ch. 

to  be  said  for  withdrawing  them,  as  brachiopods  and  others  have 
been  withdrawn,  from  Cuvier's  great  class  of  the  Mollusca.  But 
whether  bivalves  and  univalves  be  near  relations  or  no  is  not  the 
question.  Both  of  them  secrete  a  shell,  and  in  both  the  shell 
grows  by  the  successive  addition  of  similar  parts,  gnomon  after 
gnomon;  so  that  in  both  the  equiangular  spiral  makes,  and  is 
bound  to  make,  its  appearance.  There  is  a  mathematical  analogy 
between  the  two;  but  it  has  no  more  bearing  on  zoological  classi- 
fication than  has  the  still  closer  Hkeness  between  Nautilus  and  the 
nautiloid  Foraminifera. 

The  generating  curve  is  particularly  well  seen  in  the  bivalve, 
where  it  simply  constitutes  what  we  call  "the  outline  of  the  shell." 
It  is  for  the  most  part  a  plane  curve,  but  not  always ;  for  there  are 
forms  such  as  Hippopus,  Tridacna  and  many  Cockles,  or  Rhynchonella 
and  Spirifer  among  the  Brachiopods,  in  which  the  edges  of  the  two 
valves  interlock,  and  others,  such  as  Pholas,  Mya,  etc.,  where  they 
gape  asunder.  In  such  cases  as  these  the  generating  curves,  though 
not  plafie,  are  still  conjugate,  having  a  similar  relation,  but  of 
opposite  sign,  to  a  median  plane  of  reference  or  of  projection.  There 
are  a  few  exceptional  cases,  e.g.  Area  (Parallelepipedon)  tortuosa,  where 
there  is  no  median  plane  of  symmetry,  but  the  generating  curve, 
and  therefore  the  outline  of  the  shell  itself,  is  a  tortuous  curve  in 
three  dimensions. 

A  great  variety  of  form  is  exhibited  among  the  bivalves  by  these 
generating  curves.  In  many  cases  the  curve  or  outline  is  all  but 
circular,  as  in  Anomia,  Sphaerium,  Artemis,  Isocardia]  it  is  nearly 
semicircular  in  Argiope;  it  is  approximately  elliptical  in  Anodon, 
Lutraria,  Orthis;  it  may  be  called  semi-elliptical  in  Spirifer;  it  is 
a  nearly  rectilinear  triangle  in  Lithocardium,  and  a  curvilinear 
triangle  in  Mactra.  Many  apparently  diverse  but  more  or  less 
related  forms  may  be  shewn  to  be  deformations  of  a  common  type, 
by  a  simple  application  of  the  mathematical  theory  of  "trans- 
formations," which  we  shall  have  to  study  in  a  later  chapter.  In 
such  a  series  as  is  furnished,  for  instance,  by  Gervillea,  Perna, 
Avicula,  Modiola,  Mytilus,  etc.,  a  "simple  shear"  accounts  for  most, 
if  not  all,  of  the  apparent  differences. 

Upon  the  surface  of  the  bivalve  shell  we  usually  see  with  great 
clearness  the   "lines  of  growth"   which  represent  the  successive 


XI] 


OF  BIVALVE  SHELLS 


825 


margins  of  the  shell,  or  in  other  words  the  successive  positions 
assumed  during  growth  by  the  growing  generating  curve;  and  we 
have  a  good  illustration,  accordingly,  of  how  it  is  characteristic  of 
the  generating  curve  that  it  should  constantly  increase,  while  never 
altering  its  geometric  similarity. 

Underlying  these  lines  of  growth,  which  are  so  characteristic 
of  a  moUuscan  shell  (and  of  not  a  few  other  organic  formations), 
there  is,  then,  a  law  of  growth  which  we  may  attempt  to  enquire 
into  and  which  may  be  illustrated  in  various  ways.  The  simplest 
cases  are  those  in  which  we  can  study  the  lines  of  growth  on  a  more 
or  less  flattened  shell,  such  as  the  one  valve  of  an  oyster,  a  Pecten 
or  a  Tellina,  or  some  such  bivalve  mollusc.  Here  around  an  origin, 
the  so-called  "'umbo"  of  the  shell,  we  have  a  series  of  curves,  some- 
times nearly  circular,  sometimes  elliptical,  often  asymmetrical;  and 
such  curves  are  obviously  not  "concentric,"  though  we  are  often  apt 
to  call  them  so,  but  have  a  common  centre  of  similitude.  This 
arrangement  may  be  illustrated  by  various  analogies.  We  might 
for  instance  compare  it  to  a  series  of  waves,  radiating  outwards 
from  a  point,  through  a  medium  which  offered  a  resistance  increasing, 
with  the  angle  of  divergence,  according  to  some  simple  law.  We 
may  find  another  and  perhaps  a  simpler  illustration  as  follows: 

In  a  simple  and  beautiful  theorem,  Galileo  shewed  that,  if  we 
imagine  a  number  of  inclined  planes,  or  gutters,  sloping  downwards 
(in  a  vertical  plane)  at  various  angles 
from  a  common  starting-point,  and  if 
we  imagine  a  number  of  balls  rolling 
each  down  its  own  gutter  under  the 
influence  of  gravity  (and  without 
hindrance  from  friction),  then,  at  any 
given  instant,  the  locus  of  all  these 
moving  bodies  is  a  circle  passing 
through  the  point  of  origin.  For  the 
acceleration  along  any  one  of  the 
sloping  paths,  for  instance  AB  (Fig. 
400),  is  such  that 

AB  =  ig  cos  d.t^ 


Therefore 


^ig.AB/AC.t^. 
t^=2lg.AC. 


826  THE  EQUIANGULAR  SPIRAL    .  [ch. 

That  is  to  say,  all  the  balls  reach  the  circumference  of  the 
circle  at  the  same  moment  as  the  ball  which  drops  vertically  from 
^  toC. 

Where,  then,  as  often  happens,  the  generating  curve  of  the  shell 
is  approximately  a  circle  passing  through  the  point  of  origin,  we 
may  consider  the  acceleration  of  growth  along  various  radiants  to 
be  governed  by  a  simple  mathematical  law,  closely  akin  to  that 
simple  law  of  acceleration  which  governs  the  movements  of  a  faUing 
body.  And,  mutatis  mutandis,  a  similar  definite  law  underlies  the 
cases  where  the  generating  curve  is  continually  elliptical,  or  where 
it  assumes  some  more  complex,  but  still  regular  and  constant 
form. 

It  is  easy  to  extend  the  proposition  to  the  particular  case  where 
the  lines  of  growth  may  be  considered  elliptical.  In  such  a  case 
we  have  x^/a^  +  y^jb^  =  1,  where  a  and  b  are  the  major  and  minor 
axes  of  the  ellipse. 

Or,  changing  the  origin  to  the  vertex  of  the  figure, 

x^      2aj      1/2      ^        .  .         (x—  a)^      y^ 

Then,  transferring  to  polar  coordinates,  where  r .  cos  6  =  x, 
r .  sin  6  =  y,  we  have 

r .  cos^  6      2  cos  6      r .  sin^  0 

which  is  equivalent  to 

2a62  cos  d 


r  = 


b^cos^d  +  a^sin^d' 
or,  simpHfying,  by  eliminating  the  sine-function, 

2ab^  cos  d 


r  = 


(62_a2)cos2<9  +  a2* 


Obviously,  in  the  case  when  a  =  b,  this  gives  us  the  circular 
system  which  we  have  already  considered.  For  other  values,  or 
ratios,  of  a  and  6,  and  for  all  values  of  6,  we  can  easily  construct 
a  table,  of  which  the  following  is  a  sample : 


XI 


OF  BIVALVE  SHELLS 


827 


Chords  of  an  ellipse,  whose  major  and  minor  axes  {a,  b) 
are  in  certain  given  ratios 


0° 
10 
20 
30 
40 
50 
60 
70 
80 
90 


a/6  =  1/3 

10 

101 

105 

1115 

1-21 

1-34 

1-50 

1-59 

1-235 

00 


1/2 

10 

101 

103 

1065 

Ml 

1145 

1142 

101^ 

0-635 

0-0 


2/3 

10 

1-002 

1005 

1-005 

0-995 

0-952 

0-857 

0-670 

0-375 

0-0 


1/1 

1-0 

0-985 

0-940 

0-866 

0-766 

0-643 

0-500 

0-342 

0-174 

0-0 


3/2 

10 

0-948 

0-820 

0-666 

0-505 

0-372 

0-258 

0-163 

0-078 

0-0 


2/1 

1-0 

0-902 

0-695 

0-495 

0-342 

0-232 

0-152 

0-092 

0-045 

0-0 


3/1 

1-0 

0-793 

0-485 

0-289 

0-178 

0-113 

0-071 

0-042 

0-020 

0-0 


The  ellipses  whicli  we  then  draw,  from  the  values  given  in  the 
table,  are  such  as  are  shewn  in  Fig.  401  for  the  ratio  a/6  =  f,  and 
in  Fig.  402  for  the  ratio  a/6  =  J ;  these  are 
fair  approximations  to  the  actual  outlines,  and 
to  the  actual  arrangement  of  the  lines  of  growth, 
in  such  forms  as  Solecurtus  or  Cultellus,  and  in 
Tellina  or  Psammobia.  It  is  not  difficult  to  in- 
troduce a  constant  into  our  equation  to  meet  the 
case  of  a  shell  which  is  somewhat  unsymmetrical 
on  either  side  of  the  median  axis.  It  is  a  some- 
what more  troublesome  matter,  however,  to 
bring  these  configurations  into  relation  with  a 
"law  of  growth,"  as  was  so  easily  done  in  the 
case  of  the  circular  figure:   in  other  words,  to  °° 

formulate  a  law  of  acceleration  according  to  which  ^^^'  ^^^  • 

points  starting  from  the  origin  0,  and  moving  along  radial  lines, 
would  all  lie,  at  any  future  epoch,  on  an  ellipse  passing  through  0 ; 
and  this  calculation  we  n^ed  not  enter  into. 

All  that  we  are  immediately  concerned  with  is  the  simple  fact 
that  where  a  velocity,  such  as  our  rate  of  growth,  varies  with  its 
direction — varies  that  is  to  say  as  a  function  of  the  angular  divergfence 
from  a  certain  axis — then,  in  a  certain  simple  case,  we  get  lines  of 
growth  laid  down  as  a  system  of  coaxial  circles,  and,  in  some- 
what less  simple  cases,  we  obtain  a  system  of  ellipses  or  of 
other  more  complicated  coaxial  figures,  which  may  or  may  not 
be  symmetrical  on  either  side  of  the  axis.  Among  our  bivalve 
mollusca  we  shall  find  the  Unes  of  growth  to  be  approximately  circular 
in,  for  instance,  Anomia]    in  Lima  (e.g.  L.  subauriculata)  we  have 


828  THE  EQUIANGULAR  SPIRAL  [ch. 

a  system  of  nearly  symmetrical  ellipses  with  the  vertical  axis  about 
twice  the  transverse;  in  Solen  pellucidus,  we  have  again  a  system 
of  lines  of  growth  which  are  not  far  from  being  symmetrical  eUipses, 


Ct       10 
Fig.  402. 

in  which  however  the  transverse  is  between  three  and  four  times 
as  great  as  the  vertical  axis.  In  the  great  majority  of  cases,  we 
have  a  similar  phenomenon  with  the  further  complication  of  slight, 
but  occasionally  very  considerable,  lateral  asymmetry. 

In  the  above  account  of  the  mathematical  form  of  the  bivalve  shell,  we 
have  supposed,  for  simplicity's  sake,  that  the  pole  or  origin  of  the  system  is 
at  a  point  where  all  the  successive  curves  touch  one  another.  But  such  an 
arrangement  is  neither  theoretically  probable,  nor  is  it  actually  the  case; 
for  it  would  mean  that  in  a  certain  direction  growth  fell,  not  merely  to  a 
minimum,  but  to  zero.  As  a  matter  of  fact,  the  centre  of  the  system  (the 
"umbo"  of  the  conchologists)  lies  not  at  the  edge  of  the  system,  but  very 
near  to  it;  in  other  words,  there  is  a  certain  amount  of  growth  all  round. 
But  to  take  account  of  this  condition  would  involve  more  troublesome  mathe- 
matics, and  it  is  obvious  that  the  foregoing  illustrations  are  a  sufficiently  near 
approximation  to  the  actual  case. 

In  certain  little  Crustacea  (of  the  genus  Estheria)  the  carapace 
takes  the  form  of  a  bivalve  shell,  closely  simulating  that  of  a 
lamellibranchiate  mollusc,  and  bearing  lines  of  growth  in  all  respects 
analogous  to  or  even  identical  with  those  of  the  latter.  The  explana- 
tion is  very  curious  and  interesting.  In  ordinary  Crustacea  the 
carapace,  like  the  rest  of  the  chitinised  and  calcified  integument,  is 
shed  off  in  successive  moults,  and  is  restored  again  as  a  whole. 
But  in  Estheria  (and  one  or  two  other  small  Crustacea)  the  moult  is 


OF  BIVALVE  SHELLS 


829 


XI] 

incomplete:  the  old  carapace  is  retained,  and  the  new,  growing  up 
underneath  it,  adheres  to  it  like  a  lining,  and  projects  beyond  its 
edge:  so  that  in  course  of  time  the  margins  of  successive  old 
carapaces  appear  as  "lines  of  growth"  upon  the  surface  of  the  shell. 
In  this  mode  of  formation,  then  (but  not  in  the  usual  one),  we  obtain 
a  structure  which  "is  partly  old  and  partly  new,"  and  whose  suc- 
cessive increments  are  all  similar,  similarly  situated,  and  enlarged 


Fig.  403.     Hemicardium  inver- 
sum  Lam,     From  Chenu. 


Fig.  405.  Section  of  Productus 
'{Strophonema)  sp.  From 
Woods. 


Fig.  404.     Caprinella  adversa. 
After  Woodward. 

in  a  continued  progression.  We  have,  in  short,  all  the  conditions 
appropriate  and  necessary  for  the  development  of  a  logarithmic 
spiral;  and  this  logarithmic  spiral  (though  it  is  one  of  small  angle) 
gives  its  own  character  to  the  structure,  and  causes  the  little  carapace 
to  partake  of  the  characteristic  conformation  of  the  molluscan  shell. 
Among  the  bivalves  the  spiral  angle  (a)  is  very  small  in  the 
flattened  shells,  such  as  Orthis,  Lingula  or  Anomia.  It  is  larger, 
as  a  rule,  in  the  LameUibranchs  than  in  the  Brachiopods,  but  in 
the  latter  it  is  of  considerable  magnitude  among  the  Pentameri. 


830  THE  EQUIANGULAR  SPIRAL  [ch. 

Among  the  Lamellibranclis  it  is  largest  in  such  forms  as  Isocardia 
and  Diceras,  and  in  the  very  curious  genus  Caprinella;  in  all  of 
these  last-named  genera  its  magnitude  leads  to  the  production  of 
a  spiral  shell  of  several  whorls,  precisely  as  in  the  univalves.  The 
angle  is  usually  equal,  but  of  opposite  sign,  in  the  two  valves  of 
the  LamelHbranch,  and  usually  of  opposite  sign  but  unequal  in 
the  two  valves  of  the  Brachiopod.  It  is  very  unequal  in  many 
Ostreidae,  and  especially  in  such  forms  as  Gryphaea,  or  in  Caprinella, 
which  is  a  kind  of  exaggerated  Gryphaea;  in  the  cretaceous  genus 
Requienia,  the  two  valves  of  the  shell  closely  resemble  a  turbinate 
gastropod  with  its  flat  calcified  operculum.  Occasionally  it  is  of  the 
same  sign  in  both  valves  (that  is  to  say,  both  valves  curve  the  same 
way)  as  we  see  sometimes  in  Anomia,  and  better  in  Productus  or 
Strophonema. 

It  will  be  observed,  and  it  may  not  be  difficult  to  explain,  that 
the  more  the  bivalve  shell  curves  in  the  one  direction  the  more  it 
curves  in  the  other;  each  valve  tends, to  be  spheroidal,  or  ellipsoidal, 
rather  than  cylindroidal.  The  cyhndroidal  form  occurs,  excep- 
tionally, in  Solen.  But  Pecten,  Gryphaea,  Terebratula  are  all  cases 
of  bivalve  shells  where  one  valve  is  flat  and  the  other  curved  from 
side  to  side ;  and  the  flat  valve  tends  to  remain  flat  in  the  longitudinal 
direction  also,  while  the  curved  valve  grows  into  its  logarithmic 
spiral. 

In  the  genus  Gryphaea,  an  oyster-hke  bivalve  from  the  Jurassic, 
the  creature  lay  on  its  side  with'  its  left  valve  downward,  as  oysters 
and  scallops  also  do;  and  this  valve  adhered  to  the  ground  while 
the  animal  was  young.  The  upper  valve  stays  flat,  and  looks  Uke 
a  mere  operculum;  but  the  lower  or  deep  valve  grows  into  a  more 
or  less  pronounced  spiral.  So  is  it  also  in  the  neighbouring  genus 
Pecten,  where  P.  Jacobaeus  has  its  under-valve  much  deeper  and 
more  curved  than,  say,  P.  opercularis ',  but  Gryphaea  incurva  is 
more  spirally  curved  than  any  of  these,  and  G.  arcuata  has  a  spiral 
angle  very  near  to  that  of  Nautilus  itself.  In  both  the  spiral  is  a 
typical  equiangular  one,  built  up  of  a  succession  of  gnomonic  incre- 
ments, which  in  turn  depend  on  a  constant  ratio  between  the 
expansion  of  a  generating  figure  and  its  rotation  about  a  centre 
of  simihtude.  Rate  of  growth  is  at  the  root  of  the  whole  matter. 
Now  Gryphaea,  Hke  some  Ammonites  of  which  we  spoke  before,  is 


XI]  OF  BIVALVE  SHELLS  831 

one  of  those  cases  in  which  not  only  does  the  form  of  the  shell 
vary,  but  geologists  recognise,  now  and  then,  a  trend,  or  progressive 
sequence  of  variation,  from  one  stratum  or  one  "horizon"  to 
anotlier.  In  short,  as  time  goes  on,  we  seem  to  see  the  shell  growing 
thicker  or  widey,  or  more  and  more  spirally  curved,  before  our 
eyes.  What  meaning  shall  we  give,  what  importance  should  we 
assign,  to  these  changes,  and  what  sort  or  grade  of  evolution  do 
they  imply?  Some  hold  that  these  palaeontological  features  are 
"strictly  comparable  with  those  on  which  the  geneticist  bases  his 
factorial  studies";  and  that  as  such  they  may  shew  "linkage  of 
characters,"  as  when  "in  the  evolution  of  Gryjphaea  the  area  of 
attachment  retrogresses  as  the  arching  progresses"*.  These  are 
debatable  matters.  But  in  so  far  as  the  changes  depend  on  mere 
gradations  of  magnitude,  they  lead  indeed  to  variety  but  fall  short 
of  the  full  concept  of  evolution.  For  to  quote  Aristotle  once  again 
(though  we  need  not  go  to  Aristotle  to  learn  it) :  "some  things  shew 
increase  but  suffer  no  alteration ;  because  increase  is  one  thing  and 
alteration  is  another." 

The  so-called  "spiral  arms"  of  Spirifer  and  many  other  Brachio- 
pods  are  not  difficult  to  explain.  They  begin  as  a  single  structure, 
in  the  form  of  a  loop  of  shelly  substance, 
attached  to  the  dorsal  valve  of  the  shell, 
in  the  neighbourhood  of  the  hinge,  and 
forming  a  skeletal  support  for  two  cihate 
and  tentaculate  arms.  These  grow  to  a 
considerable  length,  coiling  up  within  the 
shell  that  they  may  do  so.  In  Terehratula 
the  loop  reilaains  short  and  simple,  and  is 
merely  flattened  and  distorted  somewhat 
by  the  restraining  pressure  of  the  ventral 
valve ;  but  in  Spirifer,  Atrypa,  Athyris  and 

•x  /  ^^^'^  Fig.  406.  Skeletal  loop  of  Terg- 

many  more  it  forms  a  watchsprmg  coil  on        j,^^^^     p^^^  ^^^^^^ 
either  side,  corresponding  to  the  close- 
coiled  arms  of  which  it  was  the  support  and  skeleton.     In  these 
curious  and  characteristic  structures  we  see  no  sign  of  progressive 

*  H.  H.  Swinnerton,  Unit  characters  in  fossils,  Biol.  Reviews,  vii,  pp.  321-335, 
1932;    of.  A.  E.  Truman,  Oeol.  Mag.  lix,  p.  258,  lxi,  p.  358,  1922-24. 


832  THE  EQUIANGULAR  SPIRAL  [ch. 

growth,  no  successional  increments,  no  "gnomons,"  no  self- 
similarity  in  the  figm:e.  In  short  it  has  nothing  to  do  with  a 
logarithmic  or  equiangular  spiral,  but  is  a  mere  twist,  or  tapering 
helix,  and  it  points  now  one  way,  now  another.  The  cases  in 
which  the  heUcoid  spires  point  towards,  or  point  away  from,  the 
middle  line  are  ascribed,  in  zoological  classification,  to  particular 
"families"  of  Brachiopods,  the  former  condition  defining  (or 
helping  to  define)  the  Atrypidae  and  the  latter   the   Spiriferidae 


Fig.  407.     Spiral  arms  of 
Spirifer. 

Fig.  408.  Inwardly  directed 
spiral  arms  of  Atrypa. 

and  Athyridae.  It  is  obvious  that  the  incipient  curvature  of  the 
arms,  and  consequently  the  form  and  direction  of  the  spirals,  will 
be  influenced  by  the  surrounding  pressures,  and  these  in  turn  by 
the  general  shape  of  the  shell.  We  shall  expect,  accordingly,  to 
find  the  long  outwardly  directed  spirals  associated  with  shells  which 
are  transversely  elongated,  as  Spirifer  is;  while  the  more  rounded 
Atrypa  will  tend  to  the  opposite  condition.  In  a  few  cases,  as  in 
Cyrtina  or  Reticularia*  where  the  shell  is  comparatively  narrow  but 
long,  and  where  the  uncoiled  basal  support  of  the  arms  is  long  also, 
the  coils  into  which  the  latter  grow  are  turned  backwards,  in  the 
direction  where  there  is  most  room  for  them.  And  in  the  few  cases 
where  the  shell  is  very  considerably  flattened,  the  spirals  (if  they 
find  room  to  grow  at  all)  will  be  constrained  to  do  so  in  a  discoid 
or  nearly  discoid  fashion,  and  this  is  actually  the  case  in  such 
flattened  forms  as  Koninckina  or  Thecidium. 

The  shells  of  Pteropods 

While  mathematically  speaking  we  are  entitled  to  look  upon 
the  bivalve  shell  of  the  Lamellibranch  as  consisting  of  two  distinct 
elements,  each  comparable  to  the  entire  shell  of  the  univalve,  we 


XI 


THE  SHELLS  OF  PTEROPODS 


833 


have  no  biological  grounds  for  such  a  statement ;  for  the  shell  arises 
from  a  single  embryonic  origin,  and  afterwards  becomes  spHt  into 
portions  which  constitute  the  two  separate  valves.  \¥e  can  perhaps 
throw  some  indirect  light  upon  this  phenomenon,  and  upon  several 
other  phenomena  connected  with  shell-growth,  by  a  consideration 
of  the  simple  conical  or  tubular  shells  of  the  Pteropods.  The  shells 
of  the  latter  are  in  few  cases  suitable  for  simple  mathematical 
investigation,  but  nevertheless  they  are  of  very  considerable  interest 
in  connection  with  our  general  problem.  The  morphology  of  the 
Pteropods  is  bv  no  means  well  understood,  and  in  speaking  of  them 


B 

Fig.  410.     Diagrammatic  transverse  sections,  or 

outlines  of  the  mouth,  in  certain  Pteropod  shells: 

Fig.    409.      Pteropod    shells:  ^'   ^'  ^^^^^''^  australis;    C,  C.  pyramidalis; 

(1)  Cuvierimi    columnella;  D,  CMUintium;  E,  C.  cuspidata.    After  Boas. 

(2)  Cleodora      chierchiae; 

(3)  C.  pygmaea.  After  Boas. 

I  will  assume  that  there  are  still  grounds  for  beheving  (in  spite  of 
Boas'  and  Pelseneer's  arguments)  that  they  are  directly  related  to, 
or  may  at  least  be  directly  compared  with,  the  Cephalopoda*. 

The  simplest  shells  among  the  Pteropods  have  the  form  of  a  tube, 
more  or  less  cylindrical  {Cuvierinay,  more  often  conical  (Creseis, 
Clio) ;  and  this  tubular  shell  (as  we  have  already  had  occasion  to 
remark,  on  p.  416),  frequently  tends,  when  it  is  very  small  and 
delicate,  to  assume  the  character  of  an  unduloid.  (In  such  a  case 
it  is  more  than  likely  that  the  tiny  shell,  or  that  portion  of  it  which 

*  We  need  not  assume  a  close  relationship,  nor  indeed  any  more  than  such  a 
one  as  permits  us  to  compare  the  shell  of  a  Nautilus  with  that  of  a  Gastropod. 


834  THE  EQUIANGULAR  SPIRAL  '[ch. 

constitutes  the  unduloid,  lias  not  grown  by  successive  increments 
or  ''rings  of  growth,"  but  has  developed  as  a  whole.)  A  thickened 
"rib"  is  often,  perhaps  generally,  present  on  thef  dorsal  side  of  the 
little  conical  shell.  In  a  few  cases  (Limacina,  Peraclis)  the  tube 
becomes  spirally  coiled,  in  a  normal  equiangular  spiral  or  helico- 
spiral. 

In  certain  cases  (e.g.  Cleodora,  Hyalaea)  the  tube  or  cone  is  curiously 
modified.  In  the  first  place,  its  cross-section,  originally  circular 
or  nearly  so,  becomes  flattened  or  compressed  dorsoventrally ;  and 


3  -'     4 

Fig.  411.     Shells  of  thecosome  Pteropods  (after  Boas).     (1)  Cleodora  cuspidata; 
(2)  Hyalaea  trispinosa;   (3)  H.  globulosa;   (4)  H.  uncinata;   (5)  H.  inflexa. 

the  angle,  or  rather  edge,  where  dorsal  and  ventral  walls  meet, 
becomes  more  and  more  drawn  out  into  a  ridge  or  keel.  Along  the 
free  margin,  both  of  the  dorsal  and  the  ventral  portion  of  the  shell, 
growth  proceeds  with  a  regularly  varying  velocity,  so  that  these 
margins,  or  lips,  of  the  shell  become  regularly  curved  or  markedly 
sinuous.  At  the  same  time,  growth  in  a  transverse  direction  pro- 
ceeds with  an  acceleration  which  manifests  itself  in  a  curvature  of 
the  sides,  replacing  the  straight  borders  of  the  original  cone.  In 
other  words,  the  cross-section  of  the  cone,  or  what  we  have  been 
calling  the  generating  curve,  increases  its  dimensions  more  rapidly 
than  its  di^ance  from  the  pole. 


XI] 


THE  SHELLS  OF  PTEROPODS 


835 


In  the  above  figures,  for  instance  in  that  of  Cleodora  cicspidata, 
the  markings  of  the  shell  which  represent  the  successive  edges  of 
the  lip  at  former  stages  of  growth  furnish  us  at  once  with  a  ''graph" 


'Fig.  412.     Cleodora  ctispidata. 

of  the  varying  velocities  of  growth  as  measured,  radially,  from  the 
apex.  We  can  reveal  more  clearly  the  nature  of  these  variations 
in  the  following  way,  which  is  simply  tantamount  to  converting  our 
radial  into  rectangular  coordinates.  Neglecting  curvature  (if  any) 
b  c  d  e    f  g  Y 


X  O  X 

Fig.  413.     Curves  obtained  by  transforming  radial  ordinates,  as  in  Fig.  412,  into 
vertical  equidistant  ordinates.     1,  Hyalaea  trispinosa;   2,  Cleodora  cicspidata. 

of  the  sides  and  treating  the  shell  (for  simpUcity's  sake)  as  a  right 
cone,  we  lay  off  equal  angles  from  the  apex  0,  along  the  radii  Oa, 
Ob,  etc.     If  we  then  plot,  as  vertical  equidistant  ordinates,  the 


836 


THE  EQUIANGULAR  SPIRAL 


[CH. 


magnitudes  Oa,  Oh  ...  OY,  and  again  on  to  Oa' ,  we  obtain  a  diagram 
such  as  follows  in  Fig.  413:  by  help  of  which  we  not  only  see 
more  clearly  the  way  in  which  the  growth-rate  varies  from  point 
to  point,  but  we  also  recognise  better  than  before  the  nature  of  the 
law  which  governs  this  variation  in  the  different  species. 

Furthermore,  the  young  shell  having  become  differentiated  into 
a  dorsal  and  a  ventral  part,  marked  off  from  one  another  by  a  lateral 
edge  or  keel,  and  the  inequality  of  growth  being  such  as  to  cause 
each  portion  to  increase  most  rapidly  in  the  median  line,  it  follows 
that  the  entire  shell  will  appear  to  have  been  split  into  a  dorsal 
and  a  ventral  plate,  both  connected  with,  and  projecting  from, 
what  remains  of  the  original  undivided  cone.  Putting  the  same 
thing  in  other  words,  we  may  say  that  the  generating  figure^  which 


Fig.  414.     Development  of  the  shell  of  Hyalaea  (Cavolinia)  tridentata  Forskal: 
the  earlier  stages  being  the  ''' Pleuropits  longifiUs''  of  Troschel.     After  Tesch. 

lay  at  first  in  a  plane  perpendicular  to  the  axis  of  the  cone,  has 
now,  by  unequal  growth,  been  sharply  bent  or  folded,  so  as  to  lie 
approximately  in  two  planes,  parallel  to  .the  anterior  and  posterior 
faces  of  the  cone.  We  have  only  to  imagine  the  apical  connecting 
portion  to  be  further  reduced,  and  finally  to  disappear  or  rupture, 
and  we  should  have  a  bivalve  shell  developed  out-  of  the  original 
simple  cone. 

In  its  outer  and  growing  portion,  the  shell  of  our  Pteropod  now 
consists  of  two  parts  which,  though  still  connected  together  at  the 
apex,  may  be  treated  as  growing  practically  independently.  The 
shell  is  no  longer  a  simple  tube,  or  simple  cone,  in  which  regular 
inequalitiies  of  growth  will  lead  to  the  development  of  a  spiral ;  and 
this  for  the  simple  reason  that  we  have  now  two  opposite  maxima 


XI] 


THE  SHELLS  OF  PTEROPODS 


837 


of  growth,  instead  of  a  maximum  on  the  one  side  and  a  minimum 
on  the  other  side  of  our  tubular  shell.  As  a  matter  of  fact,  the 
dorsal  and  the  ventral  plate  tend  to  curve  in  opposite  directions, 
towards  the  middle  line,  the  dorsal  curving  ventrally  and  the  ventral 
curving  towards  the  dorsal  side. 

In  the  case  of  the  Lamelhbranch  or  the  Brachiopod,  it  is  quite 
possible  for  both  valves  to  grow  into  more  or  less  pronounced  spirals, 
for  the  simple  reason  that  they  are  hinged  upon  one  another;  and 
each  growing,  edge,  instead  of  being  brought  to  a  standstill  by  the 
growth  of  its  opposite  neighbour,  is  free  to  move  out  of  the  way, 
by  the  rotation  about  the  hinge  of  the  plane  in  which  it  Hes. 

But  where  there  is  no  such  hinge,  as  in  the  Pteropod,  the  dorsal 
and  ventral  halves  of  the  shell  (or  dorsal  and  ventral  valves,  if  we 


Fig.  415.     Pteropod  shells,  from  the  side:    (1)  Cleodora  cnspidafa;   (2)  Hyalaea 
longirostris;   (3)  H.  trispinosa.     After  Boas. 


may  call  them  so)  would  soon  interfere  with  one  another's  progress 
if  they  curved  towards  one  another  (as  they  do  in  a  cockle), 
and  the  development  of  a  pair  of  conjugate  spirals  would  become 
impossible.  Nevertheless,  there  is  obviously,  in  both  dorsal  and 
ventral  valve,  a  tendency  to  the  development  of  a  spiral  curve,  that 
of  the  ventral  valve  being  paore  marked  than  that  of  the  larger  and 
overlapping  dorsal  one,  exactly  as  in  the  two  unequal  valves  of 
Terebratula,  In  many  cases  (e.g.  Cleodora  cuspidata),  the  dorsal 
valve  or  plate,  strengthened  and  stiffened  by  its  midrib,  is  nearly 
straight,  while  the  curvature  of  the  other  is  well  displayed.  But 
the  case  will  be  materially  altered  and  simplified  if  growth  be  arrested 
or  retarded  in  either  half  of  the  shell.  Suppose  for  instance  that 
the  dorsal  valve  grew  so  slowly  that  after  a  while,  in  comparison 
with  the  other,  we  might  speak  of  it  as  being  absent  altogether: 


838  THE  EQUIANGULAR  SPIRAL  [ch. 

or  suppose  that  it  merely  became  so  reduced  in  relative  size  as  to 
form  no  impediment  to  the  continued  growth  of  the  ventral  one; 
the  latter  would  continue  to  grow  in  the  direction  of  its  natural 
curvature,  and  would  end  by  forming  a  complete  and  coiled 
logarithmic  spiral.  It  would  be  precisely  analogous  to  the  spiral 
shell  of  Nautilus,  and,  in  regard  to  its  ventral  position,  concave 
towards  the  dorsal  side,  it  would  even  deserve  to  be  called  directly 
homologous  with  it.  Suppose,  on  the  other  hand,  that  the  ventral 
valve  were  to  be  greatly  reduced,  and  even  to  disappear,  the  dorsal 
valve  would  then  pursue  its  unopposed  growth;  and,  were  it  to  be 
markedly  curved,  it  would  come  to  form  a  logarithmic  spiral,  concave 
towards  the  ventral  side,  as  is  the  case  in  the  shell  of  Spirula*. 
Were  the  dorsal  valve  to  be  destitute  of  any  marked  curvature  (or 
in  other  words,  to  have  but  a  low  spiral  angle),  it  would  form  a 
simple  plate,  as  in  the  shells  of  Sepia  or  Loligo.  Indeed,  in  the 
shells  of  these  latter,  and  especially  in  that  of  Sepia,  we  seem  to 
recognise  a  manifest  resemblance  to  the  dorsal  plate  of  the  Pteropod 
shell,  as  we  have  it  (e.g.)  in  Cleodora  or  Hyalaea ;  the  Httle  "  rostrum  " 
of  Sepia  is  but  the  apex  of  the  primitive  cone,  and  the  rounded 
anterior  extremity  has  grown  according  to  a  law  precisely  such  as 
that  which  has  produced  the  curved  margin  of  the  dorsal  valve  in 
the  Pteropod.  The  ventral  portion  of  the  original  cone  is  nearly, 
but  not  wholly,  wanting ;  it  is  represented  by  the  so-called  posterior 
wall  of  the  "siphuncular  space."  In  many  decapod  cuttle-fishes 
also  (e.g.  Todarodes,  Illex,  etc.)  we  still  see  at  the  posterior  end  of 
the  "pen"  a  vestige  of  the  primitive  cone,  whose  dorsal  ^nargin 
only  has  continued  to  grow;  and  the  same  phenomenon,  on  an 
exaggerated  scale,  is  represented  in  the  Belemnites. 

It  is  not  at  all  impossible  that  we  may  explain  on  the  same  lines 
the  development  of  the  curious  "operculum"  of  the  Ammonites. 
This  consists  of  a  single  horny  plate  (Anaptychus),  or  of  a  thicker, 
more  calcified  plate  divided  into  two  symmetrical  halves  (Aptychi), 
often  found  inside  the  terminal  chamber  of  the  Ammonite,  and 
occasionally  to  be  seen  lying  in  situ,  as  an  operculum  which  partially 
closes  the  mouth  of  the  shell ;   this  structure  is  known  to  exist  even 

*  Cf.  Owen,  "These  shells  [Nautilus  and  Ammonites]  are  revolutely  spiral  or 
coiled  over  the  back  of  the  animal,  not  involute  like  Spirula^':  Palaeontology, 
1861,  p.  97;   cf.  Memoir  cm  the  Pearly  Nautilus,  4832;   also  P.Z.S.  1878,  p.  95.5. 


XI]  THE  SHELLS  OF  CEPHALOPODS  839 

in  connection  with  the  early  embryonic  shell.  In  form  the  Anap- 
tychus,  or  the  pair  of  conjoined  Aptychi,  shew  an  upper  and  a  lower 
border,  the  latter  strongly  convex,  the  former  sometimes  shghtly 
concave,  sometimes  shghtly  convex,  and  usually  shewing  a  median 
projection  or  slightly  developed  rostrum.  From  this  rostral 
border  the  curves  of  growth  start,  and  course  round  parallel  to, 
finally  constituting,  the  convex  border.  It  is  this  convex  border 
which  fits  into  the  free  margin  of  the  mouth  of  the  Ammonite's 
shell,  while  the  other  is  appHed  to  and  overlaps  the  preceding  whorl 
of  the  spire.  Now  this  relationship  is  precisely  what  we  should 
expect,  were  we  to  imagine  as  our  starting-point  a  shell  similar  to 
that  of  Hyalaea :  in  which  however  the  dorsal  part  of  the  split  cone 
had  become  separate  from  the  ventral  half,  had  remained  fiat,  and 
had  grown  comparatively  slowly,  while  at  the  same  time  it  kept 
slipping  forward  over  the  growing  and  coiUng  spire  into  which  the 
ventral  half  of  the  original  shell  develops*.  In  short,  I  think  there 
is  reason  to  believe,  or  at  least  to  suspect,  that  we  have  in  the  shell 
and  Aptychus  of  the  Ammonites,  two  portions  of  a  once  united 
structure;  of  which  other  Cephalopods  retain  not  both  parts  but 
only  one  or  other,  one  as  the  ventrally  situated  shell  of  Nautilus, 
the  other  as  the  dorsally  plated  shell  for  example  of  Sepia  or  of 
Spirula. 

In  the  case  of  the  bivalv-e  shells  of  the  Lamellibranchs  or  of  the 
Brachiopods,  we  have  to  deal  w^ith  a  phenomenon  precisely  analogous 
to  the  split  and  flattened  cone  of  our  Pteropods,  save  only  that  the 
primitive  cone  has  been  split  into  two  portions,  not  incompletely, 
as  in  the  Pteropod  {Hyalaea),  but  completely,  so  as  to  forin  two 
separate  valves.  Though  somewhat  greater  freedom  is  given  to 
growth  now  that  the  two  valves  are  separate  and  hinged,  yet  still 
the  two  valves  oppose  and  hamper  one  another,  so  that  in  the 
longitudinal  direction  each  is  capable  of  only  a  moderate  curvature. 
This  curvature,  as  we  have  seen,  is  recognisable  as  an  equiangular 
spiral,  but  only  now  and  then  does  the  growth  of  the  spiral  continue 
so  far  as  to  develop  successive  coils :  as  it  does  in  a  few  symmetrical 
forms  such  as  Isocardia  cor ;  and  as  it  does  still  more  conspicuously 
in  a  few  others,  such  as  Gryphaea  and  Caprinella,  where  one  of  the 

*  The  case  of  Terebratula  or  of  Gryphaea  would  be  closely  analogous,  if  the  smaller 
valve  were  less  closely  connected  and  co- articulated  with  the  larger. 


840  THE  EQUIANGULAR  SPIRAL  [ch. 

two  valves  is  stunted,  and  the  growth  of  the  other  is  (relatively 
speaking)  unopposed. 

Of  septa 

Before  we  leave  the  subject  of  the  molluscan  shell,  we  have  still 
another  problem  to  deal  with,  in  regard  to  the  form  and  arrangement 
of  the  septa  which  divide  up  the  tubular  shell  into  chambers,  in 
the  Nautilus,  the  Ammonite  and  their  allies. 

The  existence  of  septa  in  a  nautiloid  shell  may  probably  be 
accounted  for  as  follows.  We  have  seen  that  it  is  a  property  of 
a  cone  that,  while  growing  by  increments  at  one  end  only,  it  con- 
serves its  original  shape :  therefore  the  animal  within,  which  (though 
growing  by  a  different  law)  also  conserves  its  shape,  will  continue 
to  fill  the  shell  if  it  actually  fills  it  to  begin  with:  as  does  a  snail 
or  other  Gastropod.  But  suppose  that  our  mollusc  fills  a  part  only 
of  a  conical  shell  (as  it  does  in  the  case  of  Nautilus) ;  then,  unless 
it  alter  its  shape,  it  must  move  upward  as  it  grows  in  the  growing 
con^,  until  it  comes  to  occupy  a  space  similar  in  form  to  that  which 
it  occupied  before:  just,  indeed,  as  a  little  ball  drops  far  down  into 
the  cone,  but  a  big  one  must  stay  farther  up.  Then,  when  the 
animal  after  a  period  of  growth  has  moved  farther  up  in  the  shell,  the 
mantle- surface  continues  or  resumes  its  secretory  activity,  and  that 
portion  which  had  been  in  contact  with'  the  former  septum  secretes 
a  septum  anew.  In  short,  at  any  given  epoch,  the  creature  is  not 
secreting  a  tube  and  a  septum  by  separate  operations,  but  is  secreting 
a  shelly  case  about  its  rounded  body,  of  which  case  one  part  appears 
as  the  continuation  of  the  tube,  and  the  other  part,  merging  with  it 
by  indistinguishable  boundaries,  appears  as  the  septum*. 

The  various- forms  assumed  by  the  septa  in  spiral  shells  |  present 
us  with  a  number  of  problems  of  great  beauty,  simple  in  their 
essence,  but  whose  full  investigation  would  soon  lead  us  into  difficult 
mathematics. 

*  "It  has  been  suggested,  and  I  think  in  some  quarters  adopted  as  a  dogma, 
that  the  formation  of  successive  septa  [in  Nautilus]  is  correlated  with  the  recurrence 
of  reproductive  periods.  This  is  not  the  case,  since,  according  to  my  observations, 
propagation  only  takes  place  after  the  last  septum  is  formed";  Willey,  Zoological 
Results,  1902,  p.  746. 

f  Cf.  Henry  Woodward,  On  the  structure  of  camerated  shells,  Pop.  Sci.  Rev. 
XI,  pp.  113-120,  1872. 


XI]  OF  SEPTA  841 

We  do  not  know  how  these  septa  are  laid  down  in  an  Ammonite, 
but  in  the  Nautilus  the  essential  facts  are  clear  *.  The  septum  begins 
as  a  very  thin  cuticular  membrane  (composed  of  a  substance  called 
conchjolin),  which  is  secreted  by  the  skin,  or  mantle-surface,  of  the 
animal ;  and  upon  this  membrane  nacreous  matter  is  gradually  laid 
down  on  the  mantle-side  (that  is  to  say  between  the  animal's'  body 
and  the  cuticular  membrane  which  has  been  thrown  off  from  it), 
so  that  the  membrane  remains  as  a  thin  pellicle  over  the  hinder 
surface  of  the  septum,  and  so  that,  to  begin  with,  th.e  membranous 
septum  is  moulded  on  the  flexible  and  elastic  surface  of  the  animal, 
within  which  the  fluids  of  the  body  must  exercise  a  uniform,  or 
nearly  uniform  pressure. 

Let  us  think,  then,  of  the  septa  as  they  would  appear  in  their 
uncalcified  condition,  formed  of,  or  at  least  superposed  upon,  an 
elastic  membrane.  They  must  follow  the  general  law,  applicable 
to  all  elastic  membranes  under  uniform  pressure,  that  the  tension 
varies  inversely  as  the  radius  of  curvature ;  and  we  come  back  once 
more  to  our  old  equation  of  Laplace  and  Plateau,  that 


p  =  r 


^r^v] 


Moreover,  since  the  cavity  below  the  septum  is  practically  closed, 
and  is  filled  either  with  air  or  with  water,  P  will  be  constant  over 
the  whole  area  of  the  septum.  And  further,  we  must  assume,  at 
least  to  begin  with,  that  the  membrane  constituting  the  incipient 
septum  is  homogeneous  or  isotropic. 

Let  us  take  first  the  case  of  a  straight  cone,  of  circular  section, 
more  or  less  like  an  Orthoceras ;  and  let  us  suppose  that  the  septum 
is  attached  to  the  shell  in  a  plane  perpendicular  to  its  axis.  The 
septum  itself  must  then  obviously  be  spherical.  Moreover  the  extent 
of  the  spherical  surface  is  constant,  and  easily  determined.  For 
obviously,  in  Fig.  417,  the  angle  LCL'  equals  the  supplement  of 
the  angle  (LOU)  of  the  cone;  that  is  to  say,  the  circle  of  contact 
subtends  an  angle  at  the  centre  of  the  spherical  surface,  which  is 
constant,  and  which  is  equal  to  tt  —  2/?.  The  case  is  not  excluded 
where,  owing  to  an  asymmetry  of  tensions,  the  septum  meets  the 

*  See  Willey,  op.  cit.,  p.  749.  Cf.  also  Bather,  Shell-growth  in  Cephalopoda, 
Ann.  Mag.  N.H.  (6),  i,  pp.  298-310,  1888;  ibid.  pp.  421-427,  and  other  papers  by 
Blake,  Riefstahl,  etc.  quoted  therein. 


842  THE  l^QUIANGULAR  SPIRAL  [ch. 

side  walls  of  the  cone  at  other  than  a  right  angle,  as  in  Fig.  416;  and 
here,  while  the  septa  still  remain  portions  of  spheres,  the  geometrical 
construction  for  the  position  of  their  centres  is  equally  easy. 

If,  on  the  other  hand,  the  attachment  of  the  septum  to  the'  inner 
walls  of  the  cone  be  in  a  plane  oblique  to  the  axis,  then  the  outhne  of 
the  septum  will  be  an  elhpse,  but  its  surface  will  still  be  spheroidal.    If 


Fig.  41fi. 

the  attachment  of  the  septum  be  not  in  one  plane,  but  forms  a  sinuous 
line  of  contact  with  the  cone,  then  the  septum  will  be  a  saddle-shaped 
surface,  of  great  complexity  and  beauty.  In  all  cases,  provided  only 
that  the  membrane  be  isotropic,  the  form  assumed  will  be  precisely 
that  of  a  soap-bubble  under  similar  conditions  of  attachment :  that 
is  to  say,  it  will  be  (with  the  usual  limitations  or  conditions)  a  surface 
of  minimal  area,  and  of  constant  mean  curvature. 

If  our  cone  be  no  longer  straight,  but  curved,  then  the  septa  will 
by  symmetrically  deformed  in  consequence.  A  beautiful  and  in- 
teresting case  is  afforded  us  by  Nautilus  itself.  Here  the  outline 
of  the  septum,  referred  to  a  plane,  is  approximately  bounded  by 
two  elliptic  curves,  similar  and  similarly  situated,  whose  areas  are 
to  one  another  in  a  definite  ratio,  namely  as 

A_^l^'l_        4^C0ta 
^a       ^2^  2 

and  a  similar  ratio  exists  in  Ammonites  and  all  other  close-whorled 
spirals,   in   whi^h   however   we   cannot   always   make   the   simple 


XI]  TRE  SETTA  0¥  NAUTILUS  843 

assumption  of  elliptical  form.  In  a  median  section  of  Nautilus,  we 
see  each  septum  forming  a  tangent  to  the  inner  and  to  the  outer 
wall,  just  as  it  did  in  a  section  of  the  strsiight  Orthoceras;    but  the 


Fig.  418.     Section  of  Xaufilus,  shewing  the  contour  of  the  septa 
in  the  median  plane. 

curvatures  in  the  neighbourhood  of  these  two  points  of  contact  are 
not  identical,  for  they  now  vary  inversely  as  the  radii,  drawn  from 
the  pole  of  the  spiral  shell.  The  contour  of  the  septum  in  this 
median  plane  is  a  spiral  curve — the  conformal  spiral  transformation 
of  the  spherical  septum  of  the  rectihnear  Orthoceratite. 


844  THE  EQUIANGULAR  SPIRAL  [ch. 

But  while  the  outline  of  the  septum  in  median  section  is  simple 
and  easy  to  determine,  the  curved  surface  of  the  septum  in  its 
entirety  is  a  very  complicated  matter,  even  in  Nautilus  which  is 
one  of  the  simplest  of  actual  cases.  For,  in  the  first  place,  since 
the  form  of  the  septum,  as  seen  in  median  section,  is  that  of  a 
logarithmic  spiral,   and  as  therefore  its  curvature  is  constantly 


Fig.  419.     C&,Bi  oi  the  mteviov  oi Nautilus-,  to  shew  the  contours  of 
the  septa  at  their  junction  with  the  shell-waU. 

altering,  it  follows  that,  in  successive  transverse  sections,  the  curva- 
ture is  also  constantly  altering.  But  in  the  case  of  Nautilus,  there 
are  other  aspects  of  the  phenomenon,  which  we  can  illustrate,  but 
only  in  part,  in  the  following  simple  manner.  Let  us  imagine  a  ^ack 
of  cards,  in  which  we  have  cut  out  of  each  card  a  similar  concave 
arc  of  a  logarithmic  spiral,  such  as  we  actually  see  in  the  median 
section  of  the  septum  of  a  Nautilus.  Then,  while  we  hold  the  cards 
together,  foursquare,  in  the  ordinary  position  of  the  pack,  we  have 
a  simple  ''ruled"  surface,  which  in  any  longitudinal  section  has  the 


XI]  THE  SEVTA  0¥  NAUTILUS  845 

form  of  a  logarithmic  spiral  but  in  any  transverse  section  is  a  straight 
horizontal  line.  If  we  shear  or  slide  the  cards  upon  one.  another, 
thrusting  the  middle  cards  of  the  pack  forward  in  advance  of  the 
others,  till  the  one  end  of  the  pack  is  a  convex,  and  the  other  a 
concave,  elUpse,  the  cut  edges  which  combine  to  represent  our 
septum  will  now  form  a  curved  surface  of  much  greater  complexity ; 
and  this  is  part,  but  not  by  any  means  all,  of  the  deformation 
produced  as  a  direct  consequence  of  the  form  in  Nautilus  of  the 
section  of  the  tube  within  which  the  septum  has  to  He.  The 
complex  curvature  of  the  surface  will  be  manifested  in  a  sinuous 
outline  of  the  edge,  or  line  of  attachment  of  the  septum  to  the  tube, 
and  will  vary  according  to  the  configuration  of  the  latter.  In  the 
case  of  Nautilus,  it  is  easy  to  shew  empirically  (though  not  perhaps 
easy  to  demonstrate  mathematically),  that  the  sinuous  or  saddle- 
shaped  contour  of  the  "  suture"  (or  line  of  attachment  of  the  septum 
to  the  tube)  is  such  as  can  be  precisely  accounted  for  in  this  manner ; 
and  we  may  find  other  forms,  such  as  Ceratites,  where  the  septal 
outline  is  only  a  httle  more  sinuous,  and  still  precisely  analogous 
to  that  of  Nautilus.  It  is  also  easy  to  see  that,  when  the  section  of 
the  tube  (or  "generating  curve")  is  more  comphcated  in  form,  when 
it  is  flattened,  grooved,  or  otherwise  ornamented,  the  curvature  of  the 
septum  and  the  outline  of  its  sutural  attachment  will  become  very 
complicated  indeed  * ;  but  it  will  be  comparatively  simple  in  the  "case 
of  the  first  few  sutures  of  the  young  shell,  laid  down  before  any  over- 
lapping of  whorls  has  taken  place,  and  this  comparative  simplicity  of 
the  first-formed  sutures  is  a  marked  feature  among  Ammonites f. 

*  The  "lobes"  and  "saddles"  which  arise  in  this  manner,  and  on  whose  arrange- 
ment the  modern  classification  of  the  nautiloid  and  ammonitoid  shells  largely 
depends,  were  first  recognised  and  named  by  Leopold  von  Buch,  Ann.  Sci.  Nat. 
XXVII,  xxvni,  1829. 

t  Blake  has  remarked  upon  the  fact  (op.  cit.  p.  248)  that  in  some  Cyrtocerata 
we  may  have  a  curved  shell  in  which  the  ornaments  approximately  run  at  a  constant 
angular  distance  from  the  pole,  while  the  septa  approximate  to  a  radial  direction; 
and  that  "thus  one  law  of  growth  is  illustrated  by  the  inside,  and  another  by  the 
outside."  In  this  there  is  nothing  at  which  we  need  wonder.  It  is  merely  a  case 
where  the  generating  curve  is  set  very  obliquely  to  the  axis  of  the  shell ;  but  where 
the  septa,  which  have  no  necessary  relation  to  the  mouth  of  the  shell,  take  their 
places,  as  usual,  at  a  certain  definite'  angle  to  the  walls  of  the  tube.  This  relation 
of  the  septa  to  the  walls  of  the  tube  arises  after  the  tube  itself  is  fully  formed, 
and  the  obliquity  of  growth  of  the  open  end  of  the  tube  has  no  relation  to  the 
matter. 


846  THE  EQUIANGULAR  SPIRAL  [ch. 

We  have  other  sources  of  complication,  besides  those  which  are 
at  once  .introduced  by  the  sectional  form  of  the  tube.  For  instance, 
th'e  siphuncle,  or  little  inner  tube  which  perforates  the  septa,  exercises 
a  certain  amount  of  tension,  sometimes  evidently  considerable,  upon 
the  latter:  which  tension  is  made  manifest  in  Spirula  (and  slightly 
so  even  in  Nautilus)  by  a  dip  in  the  septal  floor  where  it  meets  the 
siphuncle.  We  can  no  longer,  then,  consider  each  septum  as  an 
isotropic  surface  under  uniform  pressure;  and  there  may  be  other 
structural  modifications,   or  inequalities,   in  that  portion  of  the 


Fig.  420.     Ammonites  Sowerbyi.     From  Zittel. 

animal's  body  with  which  the  septum  is  in  contact,  and  by  which 
it  is  conformed.  It  is  hardly  likely,  for  all  these  reasons,  tha»t  we 
shall  ever  attain  to  a  full  and  particular  explanation  of  the  septal 
surfaces  and  their  sutural  outlines  throughout  the  whole  range  of 
Cephalopod  shells;  but  in  general  terms,  the  problem  is  probably 
not  beyond  the  reach  of  mathematical  analysis.  The  problem  might 
be  approached  experimentally,  after  the  manner  of  Plateau's  experi- 
ments, by  bending  a  wire  into  the  complicated^  form  of  the  suture-line, 
and  studying  the  form  of  the  liquid  film  which  constitutes  the 
corresponding  surface  'minimae  areae. 

In  certain  Ammonites  the  septal  outline  is  further  complicatec" 
in  another  way.     Superposed  upon  the  usual  sinuous  outline,  witl 


XI]  THE  SEPTA  OF  AMMONITES  847 

its  "lobes"  and  "saddles,"  we  have  here- a  minutely  ramified,  or 
arborescent  outline,  in  which  all  the  branches  terminate  in  wavy, 
more  or  less  circular  arcs — looking  just  like  the  "landscape  marble" 
from  the  Bristol  Rhaetic.  We  have  no  difficulty  in  recognising  in 
this  a  surface-tension  phenomenon.  The  figures  are  precisely  such 
as  we  can  imitate  (for  instance)  by  pouring  a  few  drops  of  milk 
upon  a  greasy  plate,  or  of  oil  upon  an  alkaUne  solution*;  they  are 
what  Charles  Tomlinson  called  "cohesion  figures." 


Fig.  421.     Suture-line  of  a  Triassic  Ammonite  {Pinacoceras).     From  Zittel. 

We  must  not  forget  that  while  the  nautilus  and  the  ammonite 
resemble  one  another,  and  are  mathematically  identical  in  their 
spiral  curves,  they  are  really  very  different  things.  The  one  is  an 
external,  the  other  an  internal  shell.  The  nautilus  occupies  the 
large  terminal  chamber  of  the  many-chambered  shell,  and  "Still 
as  the  spiral  grew.  He  left  the  past  year's  dwelling  for  the  new." 
But  even  the  largest  ammonites  never  contained  the  body  of  the 
animal,  but  lay  hidden,  as  Spirula  does,  deep  within  the  substance 
of  the  mantle.  How  the  comphcated  septa  and  septal  outHnes  of 
the  ammonites  are  produced  I  do  not  knowt. 

We  have  very  far  from  exhausted,  we  have  perhaps  little  more 
than  begun,  the  study  of  the  logarithmic  spiral  and  the  associated 
curves  which  find  exemplification  in  the  multitudinous  diversities 
of  molluscan  shells.  But,  with  a  closing  word  or  two,  we  must 
now  bring  this  chapter  to  an  end. 

*  "The  Fimbriae,  or  Edges,  appeared  on  the  Surface  like  the  Outlines  of  some 
curious  Foliage.  This,  upon  Examination  of  them,  I  found  to  proceed  from  the 
Fulness  of  the  Edges  of  the  Diaphragms,  whereby  the  Edges  were  waved  or  plaited 
somewhat  in  the  manner  of  a  Ruff"  (R.  Hooke,  op.  cit.). 

t  In  certain  rare  cases  the  complicated  sutural  pattern  of  an  ammonite  is  found 
upside  down,  but  unchanged  otherwise.  Cf.  Otto  Haas,  A  case  of  inversion  of 
suture  hues  in  Hysteroceras,  Amer.  Jl.  of  Sci.  ccxxxix,  p.  661,  1941. 


848  THE  EQUIANGULAK  SPIRAL  [ch. 

In  the  spiral  shell  we  have  a  problem,  or  a  phenomenon,  of  growth, 
immensely  simplified  by  the  fact  that  each  successive  increment  is 
no  sooner  formed  than  it  is  fixed  irrevocably,  instead  of  remaining 
in  a  state  of  flux  and  sharing  in  the  further  changes  which  the 
organism  undergoes.  In  such  a  structure,  then,  we  have  certain 
primary  phenomena  of  growth  manifested  in  their  original  simplicity, 
undisturbed  by  secondary  and  conflicting  phenomena.  What  actually 
grows  is  merely  the  lip  of  an  orifice,  where  there  is  produced  a  ring 
of  solid  material,  whose  form  we  have  discussed  under  the  name  of 
the  generating  curve ;  and  this  generating  curve  grows  in  magnitude 
without  alteration  of  its  form.  Besides  its  increase  in  areal  magnitude, 
the  growing  curve  has  certain  strictly  limited  degrees  of  freedom, 
which  define  its  motions  in  space.  And,  though  we  may  know  nothing 
whatsoever  about  the  actual  velocities  of  any  of  these  motions,  we 
do  know  that  they  are  so  correlated  together  that  their  relative 
velocities  remain  constant,  and  accordingly  the  form  and  symmetry 
of  the  whole  system  remain  in  general  unchanged. 

But  there  is  a  vast  range  of  possibilities  in  regard  to  every  one 
of  these  factors :  the  generating  curve  may  be  of  various  forms,  and 
even  when  of  simple  form,  such  as  an  elhpse,  its  axes  may  be  s4t 
at  various  angles  to  the  system;  the  plane  also  in  which  it  lies 
may  vary,  almost  indefinitely,  in  its  angle  relatively  to  that  of  any 
plane  of  reference  in  the  system;  and  in  the  several  velocities  of 
growth,  of  rotation  and  of  translation,  and  therefore  in  the  ratios 
between  all  these,  we  have  again  a  vast  range  of  possibihties.  We 
have  then  a  certain  definite  type,  or  group  of  forms,  mathematically 
isomorphous,  but  presenting  infinite  diversities  of  outward  appear- 
ance: which  diversities,  as  Swammerdam  said,  ex  sola  nascuntur 
diver sitate  gyrationum ;  and  which  accordingly  are  seen  to  have  their 
origin  in  differences  of  rate,  or  of  magnitude,  and  so  to  be,  essentially, 
neither  more  nor  less  than  differences,  of  degree. 

In  nature,  we  find  these  forms  presenting  themselves  with  but  little 
relation  to  the  character  of  the  creature  by  which  they  are  produced. 
Spiral  forms  of  certain  particular  kinds  are  common  to  Gastropods  and 
to  Cephalopods,  and  to  diverse  famihes  of  each ;  while  outside  the  class 
of  molluscs  altogether,  among  the  Foraminifera  and  among  the  worms 
(as  in  Spirorbis,  Spirographis,  and  in  the  Dentalium -like  shell  of 
Ditrupa),  we  again  meet  with  similar  and  corresponding  spirals. 


XI]  CONCLUSION  849 

Again,  we  find  the  same  forms,  or  forms  which  (save  for  external 
ornament)  are  mathematically  identical,  repeating  themselves  in  all 
periods  of  the  world's  geological  history;  and  we  see  them  mixed 
up,  one  with  another,  irrespective  of  cHmate  or  local  conditions,  in 
the  depths  and  on' the  shores  of  every  sea.  It  is  hard  indeed  (to 
my  mind)  to  see  in  such  a  case  as  this  where  Natural  Selection 
necessarily  enters  in,  or  to  admit  that  it  has  had  any  share  what- 
soever in  the  production  of  these  varied  conformations.  Unless 
indeed  we  use.  .the  term  Natural  Selection  in  a  sense  so  wide  as  to 
deprive  it  of  any  purely  biological  significance;  and  so  recognise 
as  a  sort  of  natural  selection  whatsoever  nexus  of  causes  suffices 
to  differentiate  between  the  likely  and  the  unlikely,  the  scarce  and 
the  frequent,  the  easy  and  the  hard:  and  leads  accordingly,  under 
the  pecuhar  conditions,  limitations  and  restraints  which  we  call 
"ordinary  circumstances,"  one  type  of  crystal,  one  form  of  cloud, 
one  chemical  compound,  to  be  of  frequent  occurrence  and  another 
to  be  rare*. 

*  Cf.  Bacon,  Advancement  of  Learning,  Bk.  ii  (p.  254) :  "  Doth  any  give  the  reason, 
why  some  things  in  nature  are  so  common  and  in  so  great  mass,  and  others  so  rare 
and  in  so  small  quantity?" 


CHAPTER  XII 

THE  SPIRAL  SHELLS  OF  THE  FORAMINIFERA 

We  have  already  dealt  in  a  few  simple  cases  with  the  shells  x)f  the 
Foraminifera  * ;  and  we  have  seen  that  wherever  the  shell  is  but  a 
single  unit  or  single  chamber,  its  form  may  be  explained  in  general 
by  the  laws  of  surface-tension :  the  argument  (or  assumption)  being 
that  the  little  mass  of  protoplasm  which  makes  the  simple  shell 
behaves  as  a  fluid  drop,  the  form  of  which  is  perpetuated  when  the 
protoplasm  acquires  its  solid  covering.  Thus  the  spherical  Orbulinae 
and  the  flask-shaped  Lagenae  represent  drops  in  equilibrium,  under 
various  conditions  of  freedom  or  constraint;  while  the  irregular, 
amoeboid  body  of  Astrorhiza  is  a  manifestation  not  of  equilibrium, 
but  of  a  varying  and  fluctuating  distribution  of  surface  energy. 
When  the  foraminiferal  shell  becomes  multilocular,  the  same  general 
principles  continue  to  hold;  the  growing  protoplasm  increases  drop 
by  drop,  and  each  successive  drop  has  its  particular  phenomena  of 
surface  energy,  manifested  at  its  fluid  surface,  and  tending  to  confer 
upon  it  a  certain  place  in  the  system  and  a  certain  shape  of  its  own. 

It  is  characteristic  and  even  diagnostic  of  this  particular  group 
of  Protozoa  (1)  that  development  proceeds  by  a  well-marked  alterna- 
tion of  rest  and  of  activity — of  activity  during  which  the  protoplasm 
increases,  and  of  rest  during  which  the  shell  is  formed;  (2)  that  the 
shell  is  formed  at  the  outer  surface  of  the  protoplasmic  organism, 
and  tends  to  constitute  a  continuous  or  all  but  continuous  covermg ; 
and  it  follows  (3)  from  these  two  factors  taken  together  that  each 
successive  increment  is  added  on  outside  of  and  distinct  from  its 
predecessors,  that  the  successive  parts  or  chambers  of  the  shell  are 
of  different  and  successive  ages,  so  that  one  part  of  the  shell  is  always 
relatively  new,  and  the  rest  old  in  various  grades  of  seniority. 

The  forms  which  we  set  together  in  the  sister-group  of  Radiolaria 
are  very  differently  characterised.  Here  the  cells  or  vesicles  of 
which  each  little  composite  organism  is  made  up  are  but   httle 

*  Cf.  pp.  420,  702,  etc. 


CH.  XII]  THE  FORAMINIFERA  851 

separated,  and  in  no  way  walled  off,  from  one  another;  the  hard 
skeletal  matter  tends  to  be  deposited  in  the  form  of  isolated  spicules 
or  of  little  connected  rods  or  plates,  at  the  angles,  the  edges  or  the 
interfaces  of  the  vesicles;  the  cells  or  vesicles  form  a  coordinated 
and  cotemporaneous  rather  than  a  successive  series.  In  a  word, 
the  whole  quasi-fluid  protoplasmic  body  may  be  likened  to  a  little 
mass  of  froth  or  foam :  that  is  to  say,  to  an  aggregation  of  simul- 
taneously formed  drops  or  bubbles,  whose-physical  properties  and 
geometrical  relations  are  very  different  from  those  of  a  system  of 


Fig.  422.     Hastigerina  sp.;   to  shew  the  "mouth." 

drops  or  bubbles  which  are  formed  one  after  another,  each  solidifying 
before  the  next  is  formed. 

With  the  actual  origin  or  mode  of  development  of  the  foraminiferal 
shell  we  are  now  but  little  concerned.  The  main  factor  is  the 
adsorption,  and  subsequent  precipitation  at  the  surface  of  the 
organism,  of  calcium  carbonate — the  shell  so  formed  being  interrupted 
by  pores  or  by  some  larger  interspace  or  "mouth"  (Fig.  422),  which 
interruptions  we  may  doubtless  interpret  as  being  due  to  unequal 
distributions  of  surface  energy.  In  many  cases  the  fluid  protoplasm 
"picks  up"  sand-grains  and  other  foreign  particles,  after  a  fashion 
which  we  have  already  described  (p.  702);  and  it  cements  these 
together  with  more  or  less  of  calcareous  material.  The  calcareous 
shell  is  a  crystalline  structure,  and  the  micro-crystals  of  calcium 
carbonate  are  so  set  that  their  little  prisms  radiate  outwards  in  each 
chamber  through  the  thickness  of  the  wall — which  symmetry  is 


852  THE  SPIRAL  SHELLS  [ch. 

subject  to  corresponding  modification  when  the  spherical  chambers 
are  more  or  less  symmetrically  deformed*. 

In  various  ways  the  rounded,  drop-Uke  shells  of  the  Foraminifera, 
both  simple  and  compound,  have  been  artificially  imitated.  Thus, 
if  small  globules  of  mercury  be  immersed  in  water  in  which  a  little 
chromic  acid  is  allowed  to  dissolve,  as  the  little  beads  of  quicksilver 
become  slowly  covered  with  a  crystalline  coat  of  mercuric  chromate 
they  assume  various  forms  reminiscent  of  the  monothalamic  Fora- 
minifera. "the  mercuric  chromate  has  a  higher  atomic  volume  than 
the  mercury  which  it  replaces,  and  therefore  the  fluid  contents  of 
the  drop  are  under  pressure,  which  increases  with  /the  thickness 
of  the  pellicle ;  hence  at  some  weak  spot  in  the  latter  the  contents 
will  presently  burst  forth,  so  forming  a  mouth  to  the  little  shell. 
Sometimes  a  long  thread  is  formed,  just  as  in  Rhabdammina  linearis; 
and  sometimes  unduloid  swelhngs  make  their  appearance  on  such, 
a  thread,  just  as  in  R.  discreta.  And  again,  by  appropriate  modi- 
fications of  the  experimental  conditions,  it  is  possible  (as  Rhumbler 
has  shewn)  to  build  up  a  chambered  shell  f. 

In  a  few  forms,  such  as  Globigerina  and  its  close  allies,  the  shell 
is  beset  during  life  with  excessively  long  and  dehcate  calcareous 
spines  or  needles.  It  is  only  in  oceanic  forms  that  these  are  present, 
because  only  when  poised  in  water  can  such  delicate  structures 
endure;  in  dead  shells,  such  as  we  are  much  more  familiar  with, 
every  trace  of  them  is  broken  and  rubbed  away.  The  growth  of 
these  long  needles  may  be  partly  explained  (as  we  have  already  said 
on  p.  675)  by  the  phenomenon  which  Lehmann  calls  orientirte 
Adsorption — the  tendency  for  a  crystalline  structure  to  grow  by 
accretion,  not  necessarily  in  the  outward  form  of  a  "crystal,"  but 
continuing  in  any  direction  or  orientation  which  has  once  been 
impressed  upon  it:  in  this  case  the  spicular  growth  is  in  direct 
continuation    of   the   'radial    symmetry    of   the    micro-crystalhne 

*  In  a  few  cases,  according  to  Awerinzew  and  Rhumbler,  where  the  chambers  are 
added  on  in  concentric  series,  as  in  Orbitolites,  we  have  the  crystalline  structure 
arranged  radially  in  the  radial  walls  but  tangentially  in  the  concentric  ones: 
whereby  we  tend  to  obtain,  on  a  minute  scale,  a  system  of  orthogonal  trajectories, 
comparable  to  that  which  we  shall  presently  study  in  connection  with  the  structure 
of  bone.  Cf.  S.  Awerinzew,  Kalkschale  der  Rhizopoden,  Z.  f.  w.  Z.  Lxxiv, 
pp.  478-490,  1903. 

t  L.  Rhumbler,  Die  Doppelschalen  von  Orbitolites  und  anderer  Foraminiferen, 
etc.,  Arch.  f.  Protistenkunde,  i,  pp.  193-296,  1902;  and  other  papers.  Also  Die 
Foraminiferen  der  Planktonexpedition,  i,  pp.  50-56,  1911. 


XII]  OF  THE  FORAMINIFERA  853 

elements  of  the  shell-wall.  But  the  calcareous  needles  are  secreted 
in,  or  by,  no  less  long  and  dehcate  pseudopodia  or  "filopodia," 
and  much  has  been  learned  since  this  book  was  written  of  the 
molecular,  or  micellar,  orientation  of  the  protoplasm  in  such 
filamentous  structures;  it  is  known  that  the  long  pseudopodia  of 
the  Foraminifera  are  doubly  refractive,  and  it  follows  that  their 
molecules  are  anisotropically  arranged*.  Whether  the  slender  form 
and  asymmetrical  structure  of  calcareous  rod  and  protoplasmic 
thread  be  independent  phenomena,  or  merely  two  aspects  of  one 
and  the  same  phenomenon,  is  a  hard  question,  and  not  one  for  us 
to  discuss.  Nor  can  we  profitably  discuss  (much  as  we  should  like 
to  know)  how  far  these  patterns  of  molecular  structure  in  threads, 
films  and  surface-pelHcles  aifect  the  "fluidity"  of  the  substance, 
and  conflict  with  the  capillary '  forces  which  influence  its  outward 
form.  But  we  may  safely  say  that  the  effects  of  surface-tension 
on  cell-form  have  been  so  plainly  seen  all  through  this  book  that 
any  counter-effects  due  to  protoplasmic  asymmetry  must  be 
phenomena  of  a  second  order,  and  inconspicuous  on  the  whole. 
Over  the  whole  surface  of  the  shell  of  Globigerina  the  radiating 
spicules  tend  to  occur  in  a  hexagonal  pattern,  symmetrically 
grouped  around  the  pores  which  perforate  the  shell.  Rhumbler 
has  suggested  that  this  arrangement  is  due  to  diffusion-currents, 
forming  little  eddies  about  the  base  of  the  pseudopodia  issuing  from 
the  pores:  the  idea  being  borrowed  from  Benard,  to  whom  is  Hue 
the  discovery  of  this  type  or  order  of  vortices  f.  In  one  of  Benard's 
experiments  a  thin  layer  of  parafiin  is  strewn  with  particles  of 
graphite,  then  warmed  to  melting,  whereupon  each  little  sohd  granule 
becomes  the  centre  of  a  vortex ;  by  the  interaction  of  these  vortices 
the  particles  tend  to  be  repelled  to  equal  distances  from  one  another, 
and  in  the  end  they  are  found  to  be  arranged  in  a  hexagonal  pattern  J. 

*  Cf.  W.  J.  Schmidt,  Die  Bausteine  des  Tierkorpers  in  polarisiertem  Lichte, 
Bonn,  1924;  Ueber  den  Feinbau  der  Filopodien;  insb.  ihre  Doppelbrechung  bei 
Miliola,  Protoplasma,  xxvii,  p.  587,  1937;  also  D.  L.  Mackinnon,  Optical  pro- 
perties of  contractile  organs  in  Heliozoa,  Jl.  Physiol,  xxxviii,  p.  254,  1909; 
R.  0.  Herzog,  Lineare  u.  laminHre  Feinstrukturen,  Kolloidzschr.  lxi,  p.  280,  1932. 
See,  for  discussion  and  bibliography,  L.  E.  Picken,  op.  cit. 

t  H.  Benard,  Les  tourbillons  cellulaires,  Ann.  de  Chimie  (8),  xxiv,  1901.  Cf. 
also  the  pattern  of-  cilia  on  an  Infusorian,  as  figured  by  Biitschli  in  Bronn's 
Protozoa,  iii,  p.  1281,  1887. 

X  A  similar  hexagonal  pattern  is  obtained  by  the  mutual  repulsion  of  floating 
magnets  in  Mr  R.  W.  Wood's  experiments,  Phil.  Mag.  xlvi,  pp.  162-164,  1898. 


854  THE  SPIRAL  SHELLS  [ch. 

The  analogy  is  plain  between  this  experiment  and  those  diffusion 
experiments  by  which  Leduc  produces  his  beautiful  hexagonal 
systems  of  artificial  cells,  with  which  we  have  dealt  in  a  previous 
chapter. 

But  let  us  come  back  to  the  shell  itself,  and  consider  particularly 
its  spiral  form.  That  the  shell  in  the  Foraminifera  should  tend 
towards  a  spiral  form  need  not  surprise  us ;  for  we  have  learned  that 
one  of  the  fundamental  conditions  of  the  production  of  a  concrete 
spiral  is  just  precisely  what  we  have  here,  namely  the  develop- 
ment of  a  structure  by  means  of  successive  graded  increments 
superadded  to  its  exterior,  which  then  form  part,  successively,  of 
a  permanent  and  rigid  structure.  This  condition  is  obviously  forth- 
coming in  the  foraminiferal,  but  not  at  all  in  the  radiolarian,  shell. 
Our  second  fundamental  condition  of  the  production  of  a  logarithmic 
spiral  is  that  each  successive  increment  shall  be  so  posited  and  so 
conformed  that  its  addition  to  the  system  leaves  the  form  of  the 
whole  system  unchanged.  We  have  now  to  enquire  into  this  latter 
condition;  and  to  determine  whether  the  successive  increments,  or 
successive  chambers,  of  the  foraminiferal  shell  actually  constitute 
gnomons  to  the  entire  structure. 

It  is  obvious  enough  that  the  spiral  shells  of  the  Foraminifera 
closely  resemble  true  logarithmic  spirals.  Indeed  so  precisely  do 
the  minute  shells  of  many  Foraminifera  repeat  or  simulate  the  spiral 
shells  of  Nautilus  and  its  allies  that  to  the  naturalists  of  the  early 
nineteenth  century  they  were  known  as  the  Cephalopodes  micro- 
scopiques*,  until  Dujardin  shewed  that  their  little  bodies  comprised 
no  complex  anatomy  of  organs,  but  consisted  merely  of  that  sUme-like 
organic  matter  which  he  taught  us  to  call  "sarcode,"  and  which 
we  learned  afterwards  from  Schwann  to  speak  of  as  "protoplasm." 

One  striking  -difference,  however,  is  apparent  between  the  shell 
of  Nautilus  and  the  little  nautiloid  or  rotaline  shells  of  the  Fora- 
minifera: namely  that  the  septa  in  these  latter,  and  in  all  other 
chambered  Foraminifera,  are  convex  outwards  (Fig.  423),  whereas 
they  are  concave  outwards  in  Nautilus  (Fig.  347)  and  in  the  rest 
of  the  chambered  molluscan  shells.     The  reason  is  perfectly  simple. 

*  Cf.  Ale.  d'Orbigny,  Tableau  methodique  de  la  classe  des  Cephalopodes,  Ann. 
des  Sci.  Nat.  (1),  vii,  pp.  245-315,  1826;  Felix  Dujardin,  Observations  nouvelles 
sur  les  pretendus  Cephalopodes  microscopiques,  ibid.  (2),  iii,  pp.  108,  109,  312-315, 
1835,   Recherches  sur  les  organismes  inferieurs,  ibid,  iv,  pp.  343-377,  1835;   etc. 


XII]  OF  THE  FORAMINIFERA  855 

In  both  cases  the  curvature  of  the  septum  was  determined  before 
it  became  rigid,  and  at  a  time  when  it  had  the  properties  of 
a  fluid  film  or  an  elastic  membrane.  In  both  cases  the  actual 
curvature  is  determined  by  the  tensions  of  the  membrane  and  the 
pressures  to  which  it  was  exposed.  Now  it  is  obvious  that  the 
extrinsic  pressure  which  the  tension  of  the  membrane  has  to  with- 
stand is  on  opposite  sides  in  the  two  cases.  In  Nautilus,  the  pressure 
to"  be  resisted  is  that  produced  by  the  growing  body  of  the  animal, 
lying  to  the  outer  side  of  the  septum,  in  the  outer,  wider  portion  of 
the  tubular  shell.  In  the  Foraminifer  the  septum  at  the  time  of  its 
formation  was  no  septum  at  all ;  it  was  but  a  portion  of  the  convex 


Fig.  423.     Nummulina  antiquior  R.  and  V.     After  V.  von  Moller. 

surface  of  a  drop — that  portion  namely  which  afterwards  became 
overlapped  and  enclosed  by  the  succeeding  drop ;  and  the  curvature 
of  the  septum  is  concave  towards  the  pressure  to  be  resisted,  which 
latter  is  inside  the  septum,  being  simply  the  hydrostatic  pressure 
of  the  fluid  contents  of  the  drop.  The  one  septum  is,  speaking 
generally,  the  reverse  of  the  other;  the  organism,  so  to  speak,  is 
outside  the  one  and  inside  the  other;  and  in  both  cases  alike,  the 
septum  tends  to  assume  the  form  of  a  surface  of  minimal  area,  as 
permitted,  or  as  defined,  by  all  the  circumstances  of  the  case. 

The  logarithmic  spiral  is  easily  recognisable  in  typical  cases*  (and 

*  It  is  obvious  that  the  actual  outline  of  a  foraminiferal,  just  as  of  a  molluscan 
shell,  may  depart  widely  from  a  logarithmic  spiral.  When  we  say  here,  for  short, 
that  the  shell  is  a  logarithmic  spiral,  we  merely  mean  that  it  is  essentially  related 
to  one:  that  it  can  be  inscribed  in  such  a  spiral,  or  that  corresponding  points 
(such,  for  instance,  as  the  centres  of  gravity  of  successive  chambers,  or  the 
extremities  of  successive  septa)  will  be  found  to  lie  upon  such  a  sj)iral. 


856  THE  SPIRAL  SHELLS  [ch. 

especially  where  the  spire  makes  more  than  one  visible  revolution 
about  the  pole),  by  its  fundamental  property  of  continued  similarity: 
that  is  to  say,  by  reason  of  the  fact  that  the  big  many-chambered 
shell  is  of  just  the  same  shape  as  the  smaller  and  younger  shell — 
which  phenomenon  is  as  apparent  and  even  obvious  in  the  nautiloid 
Foraminifera,  as  in  Nautilus  itself:  but  nevertheless  the* nature  of 
the  curve  must  be  verified  by  careful  measurement,  just  as  Moseley 
determined  or  verified  it  in  his  original  study  of  Nautilus  (cf.  p.  770). 
This  has  accordingly  been  done,  by  various  writers :  and  in  the  first 
instance  by  Valerian  von  Moller,  in  an  elaborate  study  of  Fusulina — 
a  palaeozoic  genus  whose  little  shells  have  built  up  vast  tracts  of 
carboniferous  limestone  in  European  Russia*. 

In  this  genus  a  growing  surface  of  protoplasm  may  be  conceived 
as  wrapping  round  and  round  a  small  initial  chamber,  in  such  a  way 
as  to  produce  a  fusiform  or  ellipsoidal  shell — a  transverse  section 
of  which  reveals  the  close- wound  spiral  coil.  The  following  are 
measurements  of  the  successive  whorls  in  a  couple  of  species  of 
this  genus: — 

F.  cylindrica  Fischer  F.  Bocki  v.  Moller 

Breadth  (in  miUimetres) 


Vhofl 

Observed 

Calculated 

Observed 

Calculated 

I 

0-132 

— 

0-079 

— 

II 

0-195 

0198 

0120 

0119 

III 

0-300 

0-297 

0180 

0179 

IV 

0-449 

0-445 

0-264 

0-267 

V 

— 

— 

0-396 

0-401 

In  both  cases  the  successive  whorls  are  very  nearly  in  the  ratio 
of  1  :  1-5;   and  on  this  ratio  the  calculated  values  are  based. 

Here  is  another  of  von  MoUer's  series  of  measurements  of  F. 
cylindrica,  the  measurements  being  those  of  opposite  whorls — that 
is  to  say  of  whorls  180°  apart: 


Breadth  (mm.) 

0096 

0117 

0-144 

0-176 

0-216 

;  0-264 
'  0-422 

0-323 

0-395 

Log.  of  do. 

0-982 

0-068 

0-158 

0-246 

0-334 

0-509 

0-597 

Diff.  of  logs. 

— 

0-086 

0090 

0-088 

0-088 

0-088 

0-087 

0-088 

The  mean  logarithmic  difference  is  here  0-088,  =  log  1-225;  or  the 
mean  difference  of  alternate  logs  (corresponding  to  a  vector  angle 
of  27T,  i.e.  to  consecutive  measurements  along  the  same  radius)  is 
0-176,  =  log  1-5,  the  same  value  as  before.  And  this  ratio  of  1-5 
between  the  breadths  of  successive  whorls  corresponds  (as  we  see 

*  V.  von  Moller,  Die  spiral-gewundenen  Foraminifera  des  riissischen  Kohlen- 
kalks,  Mim.  de  VAcad.  Imp.  Sci.f  St  Petersbourg  (7),  xxv,  1878. 


XIl] 


OF  THE  FORAMINIFERA 


857 


by  our  table  on  p.  791)  to  a  constant  angle  of  about  86°,  or  just 
such  a  spiral  as  we  commonly  meet  with  in  the  Ammonites  (cf. 
p.  796). 

In  Fusulina,  and  in  some  few  other  Foraminifera  (cf.  Fig.  424,  A), 
the  spire  seems  to  wind  evenly  on,  with  little  or  no  external  sign 
of  the  successive  periods  of  growth,  or  successive  chambers  of  the 
shell.  The  septa  which  mark  off  the  chambers,  and  correspond  to 
retardations  or  cessations  in  the  periodicity  of  growth,  are  still  to  be 
found  in  sections  of  the  shell  of  Fusulina,  but  they  are  somewhat 
irregular  and  comjiaratively  inconspicuous;  the  measurements  we 
have  just  spoken  of  are  taken  without  reference  to  the  segments  or 
chambers,  but  only  with  reference  to  the  whorls,  or  in  other  words 
with  direct  reference  to  the  vectorial  angle. 


A  B 

Fig.  424.     A,  Cornuspira  foliacea  Phil.;   B,  Operculinu  complatiata  Defr. 

The  linear  dimensions  of  successive  chambers  have  been  measured 
in  a  number  of  cases.  Van  Iterson*  has  done  so  in  various 
Miliolinidae,  with  such  results  as  the  following: 

Triloculina  rotunda  d'Orb. 

No.  of  chamber 12        3        4        5         6 

Breadth  of  chamber  in  /x     —     34      45      61 


7 
142 


8 
182 


9 
246 


10 
319 


__      __       .        84      114 
Breadth  of  chamber  in  /x, 
calculated         —     34      45      60      79      105 

*  G.  van  Iterson,  Mathem.  u.  mikrosk.-anat.  Studien  iiber  Blattstellungen,  nebst 
Betrachtungen  liber  den  Schalenbau  der  MilioUnen,  331  pp.,  Jena,  1907. 


140      187      243      319 


858  THE  SPIRAL  SHELLS  [ch. 

Here  the  mean  ratio  of  breadth  of  consecutive  chambers  may  be 
taken  as  1-323  (that  is  to  say,  the  eighth  root  of  319/34);  and  the 
calculated  values,  as  given  above,  are  based  on  this  determination. 
Again,  Rhumbler  has  measured  the  linear  dimensions  of  a  number 
of  rotaline  forms,  for  instance  Pulvinulina  menardi  (Fig.  363) :  in 
which  common  species  he  finds  the  mean  linear  ratio  of  consecutive 
chambers  to  be  about  1-187.  In  both  cases,  and  especially  in  the 
latter,  the  ratio  is  not  strictly  constant  from  chamber  to  chamber, 
but  is  subject  to  a  small  secondary  fluctuation*. 

When  the  linear  dimensions  of  successive  chambers  are  in  con- 
tinued proportion,  then,  in  order  that  the  whole  shell  may  constitute 
a  logarithmic  spiral,  it  is  necessary  that  the  several  chambers  should 
subtend  equal  angles  of  revolution  at  the  pole.  In  the  case  of  the 
Miliolidae  this  is  obviously  the  case  (Fig.  425);  for  in  this  family 
the  chambers  lie  in  two  rows  (Biloculina),  or  three  rows  (Triloculina), 
or  in  some  other  small  number  of  series :  so  that  the  angles  subtended 
by  them  are  large,  simple  fractions  of  the  circular  arc,  such  as 
180°  or  120°.  In  many  of  the  nautiloid  forms,  such  as  Cyclammina 
(Fig.  426),  the  angles  subtended,  though  of  less  magnitude,  are  still 
remarkably  constant,  as  we  may  see  by  Fig.  427;  where  the  angle 
subtended  by  each  chamber  is  made  equal  to  20°,  and  this  diagram- 
matic figure  is  not  perceptibly  different  from  the  other.  In  some 
cases  the  subtended  angle  is  less  constant;  and  in  these  it  would 
be  necessary  to  equate  the  several  linear  dimensions  with  the 
corresponding  vector  angles,  according  to  our  equation  r  =  e^cota 
It  is  probable  that,  by  so  taking  account  of  variations  of  6,  such 
variations  of  r  as  (according  to  Rhumbler's  measurements)  Pul- 
vinulina and  other  genera  appear  to  shew,  would  be  found  to 
diminish  or  even  to  disappear. 

The  law  of  increase  by  which  each  chamber  bears  a  constant 
ratio  of  magnitude  to  the  next  may  be  looked  upon  as  a  simple 

*  Hans  Przibram  asserts  that  the  linear  ratio  of  successive  chambers  tends  in 
many  Foraminifera  to  approximate  to  1-26,  which  =  V2;  in  other  words,  that 
the  volumes  of  successive  chambers  tend  to  double.  This  Przibram  would  bring 
into  relation  with  another  law,  viz.  that  insects  and  other  arthropods  tend  to 
moult,  or  to  metamorphose,  just  when  they  double  their  weights,  or  increase  their 
linear  dimensions  in  the  ratio  of  1 :  V  2.  (Die  Kammerprogression  der  Foraminiferen 
als  Parallele  zur  Hautungsprogression  der  Mantiden,  Arch.  f.  Entw.  Meek,  xxxiv, 
p.  680,  1813.)     Neither  rule  seems  to  me  to  be  well  grounded  (see  above,  p.  165). 


XII] 


OF  THE  FORAMINIFERA 


859 


Fig.  425.     1,  2,  Miliolina  pulchella  d'Orb.;  3-5,  M.  liyinaeana  d'Orb, 
After  Brady. 


Fig.  426.     Cyclammina  cancellata  Brady, 


860  THE  SPIRAL  SHELLS  [ch. 

consequence  of  the  structural  uniformity  or  homogeneity  of  the 
orgai^ism ;  we  have  merely  to  suppose  (as  this  uniformity  would 
naturally  lead  us  to  do)  that  the  rate  of  increase  is  at  each  instant 
proportional  to  the  whole  existing  mass.     For  if  Vq,  Vi,  etc.  be 


Fig.  427.     Cydammitia  sp.     (J>iagramniatic.) 

the  volumes  of  the  successive  chambers,  let  Fj  bear  a  constant 
proportion  to  Fq,  so  that  V^  =  qV^,  and  let  Fg  bear  the  same 
proportion  to  the  whole  pre-existing  volume:   then 

n  =  ?  (Fo  +  Fi)  =^  q  (F„  +  ?F„)  =  ?F„  (1  +  q)   and    VJV^  =  1  +  ?. 

This  ratio  of  1/(1  +  q)  is  easily  shewn  to  be  the  constant  ratio 
running  through  the  whole  series,  from  chamber  to  chamber;  and 
if  this  ratio  of  volumes  be  constant,  so  also  are  the  ratios  of  corre- 
sponding surfaces,  and  of  corresponding  linear  dimensions,  provided 
always  that  the  successive  increments,  or  successive  chambers,  are 
similar  in  form. 

We  have  still  to  discuss  the  similarity  of  form  and  the  symmetry 
of  position  which  characterise  the  successive  chambers,  and  which, 
together  with  the  law  of  continued  proportionahty  of  size,  are  the 
distinctive  characters  and  the  indispensable  conditions  of  a  series 
of  "gnomons." 

The  minute  size  of  the  foraminiferal  shell  or  at  least  of  each 


XII]  OF  THE  FORAMINIFERA  861 

successive  increment  thereof,  taken  in  connection  with  the  fluid  or 
semi-fluid  nature  of  the  protoplasmic  substance,  is  enough  to  suggest 
that  the  molecular  forces,  and  especially  the  force  of  surface-tension, 
must  exercise  a  controlling  influence  over  the  form  of  the  whole 
structure;  and  this  suggestion,  or  behef,  is  already  implied  in  our 
statement  that  each  successive  increment  of  growing  protoplasm 
constitutes  a  separate  drop.  These  "drops,"  partially  concealed  by 
their  successors,  but  still  shewing  in  part  their  rounded  outhnes, 
are  easily  recognisable  in  the  various  foraminiferal  shells  which  are 
illustrated  in  this  chapter. 


Fig.  428.     Orbulina  imiversa  d'Orb. 

The  accompanying  figure  represents,  to  begin  with,  the  spherical 
shell  characteristic  of  the  common,  floating,  oceanic  Orbulina.  In 
the  specimen  illustrated,  a  second  chamber,  superadded  to  the  first, 
has  arisen  as  a  drop  of  protoplasm  which  exuded  through  the  pores 
of  the  first  chamber,  accumulated  on  its  surface,  and  spread  over 
the  latter  till  it  came  to  rest  in  a  position  of  equihbrium.  We  may 
take  it  that  this  position  of  equilibrium  is  determined,  at  least  in 
the  first  instance,  by  the  "law  of  the  constant  angle,"  which  holds, 
or  tends  to  hold,  in  all  cases  where  the  free  surface  of  a  given  Uquid 
is  in  contact  with  a  given  sohd,  in  presence  of  another  liquid  or  a  gas. 
The  corresponding  equations  *are  precisely  the  same  as  those  which 
we  have  used  in  discussing  the  form  of  a  drop  (on  p.  466);  though 
some  slight  modification  must  be  made  in  our  definitions,  inasmuch 
as  the  consideration  of  snriace-tension  is  no  longer  appropriate  at 
the  solid  surfaces,  and  the  concept  of  snviace-energy  must  take  its 


862  THE  SPIRAL  SHELLS  [ch. 

place.  Be  that  as  it  may,  it  is  enough  for  us  to  observe  that,  in 
such  a  case  a^  ours,  when  a  given  fluid,  (namely  protoplasm)  is  in 
surface  contact  with  a  solid  (viz.  a  calcareous  shell),  in  presence  of 
another  fluid  (sea- water),  then  the  angle  of  contact,  or  angle  by 
which  the  common  surface  (or  interface)  of  the  two  liquids  abuts 
against  the  solid  wall,  tends  to  be  constant:  and  that  being  so,  the 
drop  will  have  a  certain  definite  form,  depending  {inter  alia)  on  the 
form  of  the  surface  with  which  it  is  in  contact.  After  a  period  of 
rest,  during  which  the  surface  of  our  second  drop  becomes  rigid  by 
calcification,  a  new  period  of  growth  will  recur  and  a  new  drop  of 
protoplasm  be  accumulated.  Circumstances  remaining  the  same, 
this  new  drop  will  meet  the  solid  surface  of  the  shell  at  the  same  angle 
as  did  the  former  one;  and,  the  other  forces  at  work  on  the  system 
remaining  the  same,  the  form  of  the  whole  drop,  or  chamber,  will 
be  the  same  as  before. 

According  to  Rhumbler,  this  "law  of  the  constant  angle"  is 
the  fundamental  principle  in  the  mechanical  conformation  of  the 
foraminiferal  shell,  and  proV-ides  for  the  symmetry  of  form  as  well 
as  of  position  in  each  succeeding  drop  of  protoplasm:  which  form 
and  position,  once  acquired,  become  rigid  and  fixed  with  the  onset 
of  calcification.  But  Rhumbler's  explanation  brings  with  it  its  own 
difficulties.  It  is  by  no  means  easy  of  verification,  for  on  the  very 
complicated  curved  surfaces  of  the  shell  it  seems  to  me  extraordinarily 
difficult  to  measure,  or  even  to  recognise,  the  actual  angle  of  contact : 
of  which  angle  of  contact,  by  the  way,  but  little  is  known,  save  only 
in  the  particular  case  where  one  of  the  three  bodies  is  air,  as  when 
a  surface  of  water  is  exposed  to  air  and  in  contact  with  glass.  It 
is  easy  moreover  to  see  that  in  many  of  our  Foraminifera  the  angle 
of  contact,  though  it  may  be  constant  in  homologous  positions  from 
chamber  to  chamber,  is  by  no  means  constant  at  all  points  along  the 
boundary  of  each  chamber.  In  Cristellaria,  for  instance  (Fig.  429), 
it  would  seem  to  be  (and  Rhumbler  asserts  that  it  actually  is) 
about  90°  on  the  outer  side  and  only  about  50°  on  the  inner  side 
of  each  septal  partition;  in  Pulvinulina  (Fig.  363),  according  to 
Rhumbler,  the  angles  adjacent  to  the  mouth  are  of  90°,  and  the 
opposite  angles  are  of  60°,  in  each  chamber.  For  these  and  other 
similar  discrepancies  Rhumbler  would  account  by  simply  invoking 
the  heterogeneity  of  the  protoplasmic  drop:    that  is  to  say,  by 


XII]  OF  THE  FORAMINIFERA  863 

assuming  that  the  protoplasm  has  a  different  composition  and 
different  properties  (including  a  very  different  distribution  of 
surface-energy),  at  points  near  to  and  remote  from  the  mouth  of 
the  shell.  Whether  the.  differences  in  angle  of  contact  be  as  great 
as  Rhumbler  takes  them  to  be,  whether  marked  heterogeneities  of 
the  protoplasm  occur,  and  whether  these  be  enough  to  account  for 
the  differences  of  angle,  I  cannot  tell.  But  it  seems  to  me  that 
we  had  better  rest  content  with  a  general  statement,  and  that 
Rhumbler  has  taken  too  precise  and  narrow  a  view. 


Fig.  429.     Cristellaria  reniformis  d'Orb 

In  the  molecular  growth  of  a  crystal,  although  we  must  of 
necessity  assume  that  each  molecule  settles  down  in  a  position  of 
minimum  potential  energy,  we  find  it  very  hard  indeed  to  explain 
precisely,  even  in  simple  cases  and  after  all  the  labours  of  modern 
crystallographers,  why  this  or  that  position  is  actually  a  place  of 
minimum  potential.  In  the  case  of  our  httle  Foraminifer  (just  as 
in  the  case  of  the  crystal),  let  us  then  be  content  to  assert  that  each 
drop  or  bead  of  protoplasm  takes  up  a  position  of  minimum  potential 
energy,  in  relation  to  all  the  circumstances  of  the  case ;  and  let  us 
not  attempt,  in  the  present  state  of  our  knowledge,  to  define  that 
position  of  minimum  potential  by  reference  to  angle  of  contact  or 
any  other  particular  condition  of  equilibrium.  In  most  cases  the 
whole  exposed  surface,  on  some  portion  of  which  the  drop  must 
come  to  rest,  is  an  extremely  complicated  one,  and  the  forces  in- 
volved constitute  a  system  which,  in  its  entirety,  is  more  complicated 


864  THE  SPIEAL  SHELLS  [ch. 

still;  but  from  the  symmetry  of  the  case  and  the  continuity  of  the 
whole  phenomenon,  we  are  entitled  to  believe  that  the  conditions 
are  just  the  same,  or  .very  nearly  the  same,  time  after  time,  from  one 
chamber  to  another :  as  the  one  chamber  is  conformed  so  will  the  next 
tend  to  be,  and  as  the  one  is  situated  relatively  to  the  system  so  will 
its  successor  tend  to  be  situated  in  turn.  The  physical  law  of  minimum 
potential  (including  also  the  law  of  minimal  area)  is  all  that  we  need 
in  order  to  explain,  in  general  terms,  the  continued  similarity  of  one 
chamber  to  another;  and  the  physiological  law  of  growth,  by  which 
a  continued  proportionality  of  size  tends  to  run  through  the  series 
of  successive  chambers,  impresses  the  form  of  a  logarithmic  spiral 
upon  this  series  of  similar  increments. 

In  each  particular  case  the  nature  of  the  logarithmic  spiral,  as 
defined  by  its  constant  angle,  will  be  chiefly  determined  by  the  rate 
of  growth ;  that  is  to  say  by  the  particular  ratio  in  which  each  new 
chamber  exceeds  its  predecessor  in  magnitude.  But  shells  having 
the  same  constant  angle  (a)  may  still  differ  from  one  another  in  many 
ways — in  the  general  form  and  relative  position  of  the  chambers, 
in  their  extent  of  overlap,  and  hence  in  the  actual  contour  and 
appearance  of  the  shell;  'and  these  variations  must  correspond  to  par- 
ticular distributions  of  energy  within  the  system,  which  is  governed 
as  a  whole  by  the  law  of  minimum  potential. 

Our  problem,  then,  becomes  reduced  to  that  of  investigating  the 
possible  configurations  which  may  be  derived  from  the  successive 
symmetrical  apposition  of  similar  bodies  whose  magnitudes  are  in 
continued  proportion;  and  it  is  obvious,  mathematically  speaking, 
that  the  various  possible  arrangements  all  come  under  the  head  of 
the  logarithmic  spiral,  together  with  the  limiting  cases  which  it 
includes.  Since  the  difference  between  one  such  form  and  another 
depends  upon  the  numerical  value  of  certain  coefficients  of  mag- 
nitude, it  is  plain  that  any  One  must  tend  to  pass  into  any  other 
by  small  and  continuous  gradations;  in  other  words,  that  a  classi- 
fication of  these  forms  must  (like  any  classification  whatsoever  of 
logarithmic  spirals  or  of  any  other  mathematical  curves)  be  theoretic 
or  "artificial."  But  we  may  easily  make  such  an  artificial  classi- 
fication, and  shall  probably  find  it  to  agree,  more  or  less,  with  the 
usual  methods  of  classification  recognised  by  biological  students  of 
the  Foraminifera. 


XIl] 


OF  THE  FORAMINIFERA 


865 


Firstly  we  have  the  typically  spiral  shells,  which  occur  in  great 
variety,  and  which  (for  our  present  purpose)  we  need  hardly  describe 
further.  We  may  merely  notice  how  in  certain  cases,  for  instance 
Globigerina,  the  individual  chambers  are  little  removed  from  spheres; 
in  other  words,  the  area  of  contact 
between  the  adjacent  chambers  is 
small.  In  such  forms  as  Cyclamynwa 
and  Pulvinulina,  on  the  other  hand , 
each  chamber  is  greatly  overlapped 
by  its  successor,  and  the  spherical 
form  of  each  is  lost  in  a  marked 
asymmetry.  Furthermore,  in  Glo- 
higerina  and  some  others  we  have 
a  tendency  to  the  development  of 
a  gauche  spiral  in  space,  as  in  so 
many  of  our  univalve  molluscan 
shells.  The  mathematical  problem 
of  how  a  shell  should  grow,  under 
the  assumptions  which  we  have 
made,  would  probably  find  its  most 
general  statement  in  such  a  case  as 
that  of  Globigerina,  where  the  whole 
organism  lives  and  grows  freely 
poised  in  a  medium  whose  density 
is  little  different  from  its  ow^n. 

The  majority  of  spiral  forms, 
on  the  other  hand,  are  plane  or 
discoid  spirals,  and  we  may  take  it 
that  in  these  cases  some  force  has 
exercised  a  controlling  influence,  so 
as  to  keep  all  the  chambers  in  a 
plane.  This  is  especially  the  case  in 
forms  like  Rotalia  or  Biscorhiiui 
(Fig.  430),  where  the  organism  lives 
attached  to  a  rock  or  a  frond  of  sea- weed ;  for  here  (just  as  in  the  case 
of  the  coiled  tubes  which  little  worms  such  as  Serpula  and  Spirorbis 
make,  under  similar  conditions)  the  spiral  disc  is  itself  asymmetrical 
its  whorls  being  markedly  flattened  on  their  attached  surfaces. 


m:Wt^- 


"Fig.  430.     Discorbina  bertheloti  d'Orb. 


866  THE  SPIRAL  SHELLS  [ch. 

We  may  also  conceive,  among  other  conditions,  the  very  curious 
case  in  which  the  protoplasm  may  entirely  overspread  the  surface 
of  the  shell  without  reaching  a  position  of  equilibrium;  in  which 
case  a  new  shell  will  be  formed  enclosing  the  old  one,  whether 
the  old  one  be  in  the  form  of  a  single,  solitary  chamber,  or  have 
already  attained  to  the  form  of  a  chambered  or  spiral  shell.  This 
is  precisely  what  often  happens  in  the  case  of  Orbulina,  when  within 
the  spherical  shell  we  find  a  small,  but  perfectly  formed,  spiral 
''Globigerina*.'^ 

The  various  Miliolidae  (Fig.  425)  only  differ  from  the  typical 
spiral,  or  rotaline  forms,  in  the  large  angle  subtended  by  each 
chamber,  and  the  consequent  abruptness  of  their  inclination  to  each 
other.  In  these  cases  the  outward  appearance  of  a  spiral  tends  to 
be  lost;  and  it  behoves  us  to  recollect,  all  the  more,  that  our  spiral 
curve  is  not  necessarily  identical  with  the  outline  of  the  shell,  but 
is  always  a  line  drawn  through  corresponding  points  in  the  successive 
chambers  of  the  latter. 

We  reach  a  limiting  case  of  the  logarithmic  spiral  when  the 
chambers  are  arranged  in  a  straight  Line ;  and  the  eye  will  tend  to 
associate  with  this  limiting  case  the  much  more  numerous  forms  in 
which  the  spiral  angle  is  small,  and  the  shell  only  exhibits  a  gentle 
curve  with  no  succession  of  enveloping  whorls.  This  constitutes 
the  Nodosarian  type  (Fig.  134,  p.  421);  and  here  again,  we  must 
postulate  some  force  which  has  tended  to  keep  the  chambers  in 
a  rectilinear  series :  such  for  instance  as  gravity,  acting  on  a  system 
of  "hanging  drops." 

In  Textularia  and  its  allies  (Fig.  431)  we  have  a  precise  parallel 
to  the  helicoid  cyme  of  the  botanists  (cf.  p.  767):  that  is  to  say  we 
have  a  screw  translation,  perpendicular  to  the  plane  of  the  underlying 
logarithmic  spiral.  In  other  words,  in  tracing  a  genetic  spiral 
through  the  whole  succession  of  chambers,  we  do  so  by  a  continuous 
vector  rotation  through  successive  angles  of  180°  (or  120°  in  some 
cases),  while  the  pole  moves  along  an  axis  perpendicular  to  the 
original  plane  of  the  spiral. 

Another  type  is  furnished  by  the  "cyclic"  shells  of  the  Orbi- 
tolitidae,  where  small  and  numerous  chambers  tend  to  be  added 

*  Cf.  G.  Schacko,  Ueber  Glohigerina-^in^chluBS  bei  Orbulina,  Wiegmann's 
Archiv,  XLix,  p.  428,  1883;   Brady,  Chall.  Rep.  1884,  p.  607. 


XII]'  OF  THE  FORAMINIFERA  867 

on  round  and  round  the  system,  so  building  up  a  circular  flattened 
disc.  This  again  we  perceive  to  be,  mathematically,  a  hmiting  case 
of  the  logarithmic  spiral;  the  spiral  has  become  wellnigh  a  circle  and 
the  constant  angle  is  wellnigh  90°. 

Lastly  there  are  a  certain  number  of  Forarninifera  in  which, 
without  more  ado,  we  may  simply  say  that  the  arrangement  of  the 
chambers  is  irregular,  neither  the  law  of  constant  ratio  of  magnitude 
nor  that  of  constant  form  being  obeyed.  The  chambers  are  heaped 
pell-mell  upon  one  another,  and  such  forms  are  known  to  naturahsts 
as  the  Acervularidae. 


A  B 

Fig.  431.     A,  Textularia  trochus  d'Orb.     B,  T.  concava  Karrer. 

While  in  these  last  we  have  an  extreme  lack  of  regularity,  we 
must  not  exaggerate  the  regularity  or  constancy  which  the  more 
ordinary  forms  display.  We  may  think  it  hard  to  believe  that  the 
simple  causes,  or  simple  laws,  which  we  have  described  should  operate, 
and  operate  again  and  again,  in  millions  of  individuals  to  produce 
the  same  delicate  and  complex  conformations.  But  we  are  taking 
a  good  deal  for  granted  if  we  assert  that  they  do  so,  and  in  particular 
we  are  assuming,  with  very  little  proof,  the  "constancy  of  species" 
in  this  group  of  animals.  Just  as  Verworn  has  shewn  that  the 
typical  Amoeba  proteus,  when  a  trace  of  alkali  is  added  to  the  water 
in  which  it  lives,  tends,  by  alteration  of  surface  tensions,  to  protrude 
the  more  delicate  pseudopodia  characteristic  of  A.  radiosa — and 
again  when  the  water  is  rendered  a  little  more  alkaline,  to  turn 
apparently  into  the  so-called  A.  liTnax — so  it  is  evident  that  a  very 
slight  modification  in  the  surface-energies  concerned  might  tend 


868  THE  SPIRAL  SHELLS  [ch. 

to  turn  one  so-called  species  into  another  among  the  Foraminifera. 
To  what  extent  this  process  actually  occurs,  we  do  not  know. 

But  that  this,  or  something  of  the  kind,  does  actually  occur  we 
can  scarcely  doubt.  For  example  in  the  genus  Peneroplis,  the  first 
portion  of  the  shell  consists  of  a  series  of  chambers  arranged  in 
a  spiral  or  nautiloid  series;  but  as  age  advances  the  spiral  is  apt  to 
be  modified  in  various  ways*.  Sometimes  the  successive  chambers 
grow  rapidly  broader,  the  whole  shell  becoming  fan-shaped.  Some- 
times the  chambers  become  narrower,  till  they  no  longer  enfold  the 
earlier  chambers  but  only  come-  in  contact  each  with  its  immediate 
predecessor:  the  result  being  that  the  shell  straightens  out,  and 
(taking  into  account  the  earlier  spiral  portion)  may  be  described  as 
crozier-shaped.  Between  these  extremes  of  shape,  and  in  regard  to 
other  variations  of  thickness  or  thinness,  roughness  or  smoothness, 
and  so  on,  there  are  innumerable  gradations  passing  one  into  another 
and  intermixed  without  regard  to  geographical  distribution: — 
"  wherever  Peneroplides  abound  this  wide  variation  exists,  and  nothing 
can  be  more  easy  than  to  pick  out  a  number  of  striking  specimens 
and  give  to  each  a  distinctive  name,  but  in  no  other  way  can  they  be 
divided  into  '  species.' "f  Some  writers  have  wondered  at  the 
peculiar  variability  of  this  particular  shellj;  but  for  all  we  know 
of  the  life-history  of  the  Foraminifera,  it  may  well  be  that  a  great 
number  of  the  other  forms  which  we  distinguish  as  separate  species 
and  even  genera  are  no  more  than  temporary  manifestations  of  the 
same  variabiUty§. 

*  Cf.  H.  B.  Brady,  Challenger  Rep.,  Foraminifera,  1884,  p.  203,  pi.  xin. 

t  Brady,  op.  cit.  p.  206;  Batsch,  one  of  the  earliest  writers  on  Foraminifera, 
had  already  noticed  that  this  whole  series  of  ear-shaped  and  crozier-shaped  shells 
was  filled  in  by  gradational  forms;  Conchylien  des  Seesandes,  1791,  p.  4,  pi.  vi, 
fig.  15  a-/.  See  also,  in  particular,  Dreyer,  Peneroplis ;  eine  Studie  zur  biologischen 
Morphologie  und  zur  Speciesfrage,  Leipzig,  1898;  also  Eimer  und  Fickert,  Artbildung 
und  Verwandschaft  bei  den  Foraminiferen,  Tuhinger  zool.  Arbeiten,  ni,  p.  35, 
1899. 

X  Doflein,  Protozoenlcunde,  1911,  p.  263:  "Was  diese  Art  veranlasst  in  dieser 
Weise  gelegentlich  zu  variiren,  ist  vorlaufig  noch  ganz  rathselhaft." 

§  In  the  case  of  Globigerina,  some  fourteen  species  (out  of  a  very  much  larger 
number  of  described  forms)  were  allowed  by  Brady  (in  1884)  to  be  distinct;  and 
this  list  has  been,  I  believe,  rather  added  to  than  diminished.  But  these  so-called 
species  depend  for  the  most  part  on  slight  differences  of  degree,  differences  in  the 
angle  of  the  spiral,  in  the  ratio  of  magnitude  of  the  segments,  or  in  their  area  of 
contact  one  with  another.     Moreover  with  the  exception  of  one  or  two  "dwarf" 


XII]  OF  THE  FORAMINIFERA  869 

ConclvMon 

If  we  can  comprehend  and  interpret  on  some  such  lines  as  these 
the  form  and  mode  of  growth  of  the  foraminiferal  shell  we  may  also 
begin  to  understand  two  striking  features  of  the  group,  on  the  one 
hand  the  large  number  of  diverse  types  or  famihes  which  exist 
and  the  large  number  of  species  and  varieties  within  each,  and 
on  the  other  the  persistence  of  forms  which  in  many  cases  seem  to 
have  undergone  little  change  or  none  at  all  from  the  Cretaceous  or 
even  frorri  earlier  periods  to  the  present  day.  In  few  other  groups, 
perhaps  only  among  the  Radiolaria,  do  we  seem  to  possess  so  nearly 
complete  a  picture  of  all  possible  transitions  between  form  and 
form,  and  of  the  whole  branching  system  of  the  evolutionary  tree : 
as  though  little  or  nothing  of  it  had  ever  perished,  and  the  whole 
web  of  life,  past  and  present,  were  as  complete  as  ever.  It  leads 
one  to  imagine  that  these  shells  have  grown  according  to  laws  so 
simple,  so  much  in  harmony  with  their  material,  with  their  environ- 
ment, and  with  all  the  forces  internal  and  external  to  which  they 
are  exposed,  that  none  is  better  than  another  and  none  fitter  or  less 
fit  to  survive.  It  invites  one  also  to  contemplate. the  possibility  of 
the  lines  of  possible  variation  being  here  so  narrow  and  determinate 
that  identical  forms  may  have  come  independently  into  being  again 
and  again. 

While  we  can  trace  in  the  most  complete  and  beautiful  manner 
the  passage  of  one  form  into  another  among  these  httle  shells,  and 
ascribe  them  all  at  last  (if  we  please)  to  a  series  which  starts  with  the 
simple  sphere  of  Orhulina  or  with  the  amoeboid  body  of  Astrorhiza, 
the  question  stares  us  in  the  face  whether  this  be  an  "evolution" 
which  we  have  any  right  to  correlate  with  historic  time.  The 
mathematician  can  trace  one  conic  section  into  another,  and  "  evolve  " 
for  example,  through  innumerable  graded  ellipses,  the  circle  from 
the  straight  line:  which  tracing  of  continuous  steps  is  a  true 
"evolution,"  though  time  has  no  part  therein.  It  was  after  this 
fashion  that  Hegel,  and  for  that  matter  Aristotle  himself,  was  an 

forms,  said  to  be  limited  to  Arctic  and  Antarctic  waters,  there  is  no  principle  of 
geographical  distribution  to  be  discerned  amongst  them.  A  species  found  fossil 
in  New  Britain  turns  up  in  the  North  Atlantic;  a  species  described  from  the  West 
Indies  is  rediscovered  at  the  ice- barrier  of  the  Antarctic. 


870  THE  SPIRAL  SHELLS  [ch. 

evolutionist — to  whom  evolution  was  a  mental  concept,  involving 
order  and  continuity  in  thought  but  not  an  actual  sequence  of  events 
in  time.  Such  a  conception  of  evolution  is  not  easy  for  the  modern 
biologist  to  grasp,  and  is  harder  still  to  appreciate.  And  so  it  is 
that  even  those  who,  like  Dreyer*  and  like  Rhumbler,  study  the 
foraminiferal  shell  as  a  physical  system,  who  recognise  that  its 
whole  plan  and  mode  of  growth  is  closely  akin  to  the  phenomena 
exhibited  by  fluid  drops  under  particular  conditions,  and  who 
explain  the  conformation  of  the  shell  by  help  of  the  same  physical 
principles  and  mathematical  laws^yet  all  the  while  abate  no  jot 
or  tittle  of  the  ordinary  postulates  of  modern  biology,  nor  doubt 
the  validity  and  universal  applicability.of  the  concepts  of  Darwinian 
evolution.  For  these  writers  the  biogenetisches  Grundgesetz  remains 
impregnable.  The  Foraminifera  remain  for  them  a  great  family 
tree,  whose  actual  pedigree  is  traceable  to  the  remotest  ages;  in 
which  historical  evolution  has  coincided  with  progressive  change; 
and  in  which  structural  fitness  for  a  particular  function  (or  functions) 
has  exercised  its  selective  action  and  ensured  "the  survival  of  the 
fittest."  By  successive  stages  of  historic  evolution  we  are  supposed 
to  pass  from  the  irregular  Astrorhiza  to  a  Rhahdammina  with  its 
more  concentrated  disc;  to  the  forms  of  the  same  genus  which 
consist  of  but  a  single  tube  with  central  chamber;  to  those  where 
this  chamber  is  more  and  more  distinctly  segmented;  so  to  the 
typical  many-chambered  Nodosariae;  and  from  these,  by  another 
definite  advance  and  later  evolution  to  the  spiral  Trochamminae. 
After  this  fashion,  throughout  the  whole  varied  series  of  the  Fora- 
minifera,  Dreyer  and  Rhumbler  (following  Neumayr)  recognise  so 
many  successions  of  related  forms,  one  passing  into  another  and 
standing  towards  it  in  a  definite  relationship  of  ancestry  or  descent. 
Each  evolution  of  form,  from  simpler  to  more  complex,  is  deemed 
to  have  been  attended  by  an  advantage  to  the  organism,  an 
enhancement  of  its  chances  of  survival  or  perpetuation ;  hence  the 
historically  older  forms  are  on  the  whole  structurally  the  simpler; 
or  conversely,  the  simpler  forms,  such  as  the  simple  sphere,  were 
the  first  to  come  into  being  in  primeval  seas;  and  finally,  the 
gradual  development  and  increasing  comphcation  of  the  individual 

*  F.  Dreyer,  Prinzipien  der  Gerustbildung  bei  Rhizopoden,  etc.,  Jen.  Zeitschr. 
XXVI,  pp.  204-468,  1892. 


XII]  or  THE  FORAMINIFERA  871 

within  its  own  lifetime  is  hfeld  to  be  at  least  a  partial  recapitulation 
of  the  unknown  history  of  its  race  and  dynasty* 

We  encounter  many  difficulties  when  we  try  to  extend  such 
concepts  as  these  to  the  Foraminifera.  We  are  led  for  instance  to 
assert,  as  Rhumbler  does,  that  the  increasing  complexity  of  the 
shell,  and  of  the  manner  in  which  one  chamber  is  fitted  on  another, 
makes  for  advantage;  and  the  particular  advantage  on  which 
Rhumbler  rests  his  argument  is  strength.  Increase  of  strength,  die 
Festigkeitssteigerung,  is  according  to  him  the  guiding  principle  in 
foraminiferal  evolution,  and  marks  the  historic  stages  of  their 
development  in  geologic  time.  But  in  days  gone  by  I  used  to  see 
the  beach  of  a  little  Connemara  bay  bestrewn  with  millions  upon 
millions  of  foraminiferal  shells,  simple  Lagenae,  less  simple  Nodosariae, 
more  complex  Rotaliae :  all  drifted  by  wave  and  gentle  current 
from  their  sea-cradle  to  their  sandy  grave:  all  lying  bleached  and 
dead:  one  more  delicate  than  another,  but  all  (or  vast  multitudes 
of  them)  perfect  and  unbroken.  And  so  I  am  not  incUned  to  beheve 
that  niceties  of  form  affect  the  case  very  much :  nor  in  general  that 
foraminiferal  life  involves  a  struggle  for  existence  wherein  breakage 
is  a  danger  to  be  averted,  and  strength  an  advantage  to  be 
ensured  f. 

In  the  course  of  the  same  argument  Rhumbler  remarks  that 
Foraminifera  are  absent  from  the  coarse  sands  and  gravels  J,  as 
Williamson  indeed  had  observed  many  years  ago:  so  averting,  or 
at  least  escaping,  the  dangers  of  concussion.     But  this  is  after  all 

*  A  difficulty  arises  in  the  case  of  forms  (like  Peneroplis)  where  the  young  shell 
appears  to  be  more  complex  than  the  old,  the  jfirst-formed  portion  being  closely 
coiled  while  the  later  additions  become  straight  and  simple:  "die  biformen  Arten 
verhalten  sich,  kurz  gesagt,  gerade  umgekehrt  als  man  nach  dem  biogenetischen 
Grundgesetz  erwarten  sollte,"  Rhumbler,  op.  cit.  p.  33,  etc. 

f  "Das  Festigkeitsprinzip  als  Movens  der  Weiterentwicklung  ist  zu  interessant 
und  fiir  die  Aufstellung  meiries  Systems  zu  wichtig  um  die  Frage  unerortert  zu 
lassen,  warum  diese  Bevorziigung  der  Festigkeit  stattgefunden  hat.  Meiner 
Ansicht  nach  lautet  die  Antwort  auf  diese  Frage  einfach,  well  die  Foraminiferen 
meistens  unter  Verhaltnissen  leben,  die  ihre  Schalen  in  hohem  Grade  der  Gefahr 
des  Zerbrechens  aussetzen;  es  muss  also  eine  fortwahrende  Auslese  des  Festeren 
stattfinden,"  Rhumbler,  op.  cit.  p.  22. 

X  "  Die  Foraminiferen  kiesige  oder  grobsandige  Gebiete  des  Meeresbodens  nicht 
lieben,  u.s.w.":  where  the  last  two  words  have  no  particular  meaning,  save  only 
that  (as  M.  Aurelius  says)  "of  things  that  use  to  be,  we  say  commonly  that  they 
love  to  be." 


872  THE  SPIRAL  SHELLS  [ch. 

a  very  simple  matter  of  mechanical  analysis.  The  coarseness  or 
fineness  of  the  sediment  on  the  sea-bottom  is  a  measure  of  the 
current:  where  the  current  is  strong  the  larger  stones  are  washed 
clean,  where  there  is  perfect  stillness  the  finest  mud  settles  down; 
and  the  light,  fragile  shells  of  the  Foraminifera  find  their  appropriate 
place,  like  every  other  graded  sediment  in  this  spontaneous  order 
of  levigation. 

The  theorem  of  Organic  Evolution  is  one  thing;  the  problem  of 
deciphering  the  lines  of  evolution,  the  order  of  phylogeny,  the  degrees 
of  relationship  and  consanguinity,  is  quite  another.  Among  the 
higher  organisms  we  arrive  at  conclusions  regarding  these  things 
by  weighing  much  circumstantial  evidence,  by  dealing  with  the 
resultant  of  many  variations,  and  by  considering  the  probability 
or  improbability  of  many  coincidences  of  cause  and  effect;  but 
even  then  our  conclusions  are  at  best  uncertain,  our  judgments  are 
continually  open  to  revision  and  subject  to  appeal,  and  all  the  proof 
and  confirmation  we  can  ever  have  is  that  which  comes  from  the 
direct,  but  fragmentary  evidence  of  palaeontology*. 

But  in  so  far  as  forms  can  be  shewn  to  depend  on  the  play  of 
physical  forces,  and  the  variations  of  form  to  be  directly  due  to 
simple  quantitative  variations  in  these,  just  so  far  are  we  thrown 
back  on  our  guard  before  the  biological  conception  of  consanguinity, 
and  compelled  to  revise  the  vague  canons  which  connect  classification 
with  phylogeny. 

The  physicist  explains  in  terms  of  the  properties  of  matter,  and 
classifies  according  to  a  mathematical  analysis,  all  the  drops  and 
forms  of  drops  and  associations  of  drops,  all  the  kinds  of  froth  and 
foam,  which  he  may  discover  among  inanimate  things;  and  his 
task  ends  there.  But  when  such  forms,  such  conformations  and 
configurations,  occur  among  living  things,  then  at  once  the  biologist 
introduces  his  concepts  of  heredity,  of  historical  evolution,  of  suc- 
cession in  time,  of  recapitulation  of  remote  ancestry  in  individual 
growlh,  of  common  origin  (unless  contradicted  by  direct  evidence) 
of  similar  forms  remotely  separated  by  geographic  space  or  geologic 
time,  of  fitness  for  a  function,  of  adaptation  to  an  environment,  of 
higher  and  lower,  of  "  better  "  and  "  worse."    This  is  the  fundamental 

*  In  regard  to  the  Foraminifera,  "die  Palaeontologie  lasst  uns  leider  an  Anfang 
der  Stammesgeschichte  fast  ganzlich  im  Stiche,"  Rhumbler,  op.  cit.  p.  14. 


XII]  OF  THE  FORAMINIFERA  873 

difference  between  the  "explanations"  of  the  physicist  and  those 
of  the  biologist. 

In  the  order  of  physical  and  mathematical  complexity  there  is 
no  question  of  the  sequence  of  historic  time.  The  forces  that  bring 
about  the  sphere,  the  cylinder  or  the  ellipsoid  are  the  same  yesterday 
and  to-morrow.  A  snow-crystal  is  the  same  to-day  as  when  the  first 
snows  fell.  The  physical  forces  which  mould  the  forms  of  Orbulina, 
of  Astrorhiza,  of  Lagena  or  of  Nodosaria  to-day  were  still  the  same, 
and  for  aught  we  have  reason  to  believe  the  physical  conditions 
under  which  they  worked  were  not  appreciably  different,  in  that 
yesterday  which  we  call  the  Cretaceous  epoch;  or,  for  aught  we 
know,  throughout  all  that  duration  of  time  which  is  marked,  but 
not  measured,  by  the  geological  record. 

In  a  word,  the  minuteness  of  our  organism  brings  its  conformation 
as  a  whole  within  the  range  of  the  molecular  forces;  the  laws  of 
its  growth  and  form  appear  to  lie  on  simple  lines;  what  Bergson 
calls*  the  "ideal  kinship"  is  plain  and  certain,  but  the  "material 
afiihation"  is  problematic  and  obscure;  and,  in  the  end  and  upshot, 
it  seems  to  me  by  no  means  certain  that  the  biologist's  usual  mode 
of  reasoning  is  appropriate  to  the  case,  or  that  the  concept  of  con- 
tinuous historical  evolution  must  necessarily,  or  may  safely  and 
legitimately,  be  employed. 

That  things  not  only  alter  but  improve  is  an  article  of  faith,  and 
the  boldest  of  evolutionary  conceptions.  How  far  it  be  true  -were 
very  hard  to  say;  but  I  for  one  imagine  that  a  pterodactyl  flew  no 
less  w^ell  than  does  an  albatross,  and  that  Old  Red  Sandstone  fishes 
swam  as  well  and  easily  as  the  fishes  of  our  own  seas. 

*  The  evolutionist  theory,  as  Bergson  puts  it,  "consists  above  all  in  establishing 
relations  of  ideal  kinship,  and  in  maintaining  that  wherever  there  is  this  relation  of, 
so  to  speak,  logical  affiliation  between  forms,  there  is  also  a  relation  of  chronological 
succession  between  the  species  in  which  these  forms  are  materialised''^  {Creative 
Evolution,  1911,  p.  26).     Cf.  supra,  p.  412. 


CHAPTEPv  XIII 

THE  SHAPES  OF  HORNS,  AND  OF  TEETH  OR  TUSKS: 
WITH  A  NOTE  ON  TORSION 

We  have  had  so  much  to  say  on' the  subject  of  shell-spirals  that  we 
must  deal  briefly  with  the  analogous  problems  which  are  presented  by 
the  horns  of  sheep,  goats,  antelopes  and  other  horned  quadrupeds; 
and  all  the  more,  because  these  horn-spirals  are  on  the  whole  less 
symmetrical,  less  easy  of  measurement  than  those  of  the  shell,  and 
in  other  ways  also  are  less  easy  of  investigation.  Let  us  dispense 
altogether  in  this  case  with  mathematics;  and  be  content  with  a 
very  simple  account  of  the  configuration  of  a  horn. 

There  are  three  types  of  horn  which  deserve  separate  consideration : 
firstly,  the  horn  of  the  rhinoceros;  secondly,  the  horns  of  the  sheep, 
the  goat,  the  ox  or  the  antelope,  that  is  to  say,  of  the  so-called 
hollow-horned  ruminants;  and  thirdly,  the  solid  bony  horns,  or 
"antlers,"  which  are  characteristic  of  the  deer. 

The  horn  of  the  rhinoceros  presents  no  difficulty.  It  is  physio- 
logically equivalent  to  a  mass  of  consolidated  hairs,  and,  like  ordinary 
hair,  it  consists  of  non-living  or  "formed"  material,  continually 
added  to  by  the  living  tissues  at  its  base.  In  section  the  horn  is 
elliptical,  with  the  long  axis  fore-and-aft,  or  in  some  species  nearly 
circular.  Its  longitudinal  growth  proceeds  with  a  maximum  velocity 
anteriorly,  and  a  minimum  posteriorly;  and  the  ratio  of  these 
velocities  being  constant,  the  horn  curves  into  the  form  of  a  loga- 
rithmic spiral  in  the  manner  that  we  have  already  studied.  The 
spiral  is  of  small  angle,  but  in  the  longer-horned  species,  such  as 
the  great  white  rhinoceros  (Ceratorhinus),  the  spiral  curvature  is 
distinctly  recognised.  As  the  horn  occupies  a  median  position  on 
the  head — a  position,  that  is  to  say,  of  symmetry  in  respect  to  the 
field  of  force  on  either  side — there  is  no  tendency  towards  a  lateral 
twist,  and  the  horn  accordingly  develops  as  a  plane  logarithmic 
spiral.  When  two  median  horns  coexist,  the  hinder  one  is  much 
the  smaller  of  the  two:   which  is  as  much  as  to  say  that  the  force, 


CH.  XIII]  THE  HORNS  OF  SHEEP  AND  GOATS  875 

or  rate,  of  growth  diminishes  as  we  pass  backwards,  just  as  it  does 
within  the  limits  of  the  single  horn.  And  accordingly,  while  both 
horns  have  essentially  the  same  shape,  the  spiral  curvature  is  less 
manifest  in  the  second  one,  by  the  mere  reason  of  its  shortness. 

The  paired  horns  of  the  ordinary  hollow-horned  ruminants,  such 
as  the  -sheep  or  the  goat,  grow  under  conditions  which  are  in  some 
respects  similar,  but  which  differ  in  other  and  important  respects 
from  the  conditions  under  which  the  horn  grows  in  the  rhinoceros. 
As  regards  its  structure,  the  entire  horn  now  consists  of  a  bony  core 
with  a  covering  of  skin ;  the  inner,  or  dermal,  layer  of  the  latter  is 
richly  supplied  with  nutrient  blood-vessels,  while  the  outer  layer, 
or  epidermis,  develops  the  fibrous  or  chitinous  material,  chemically 
and  morphologically  akin  to  a  mass  of  cemented  or  consolidated 
hairs,  which  constitutes  the  "  sheath"  of  the  horn.  A  zone  of  active 
growth  at  the  base  of  the  horn  keeps  adding  to  this  sheath,  ring 
by  ring,  and  the  specific  form  of  this  annular  zone  may  be  taken 
as  the  "generating  curve"  of  the  horn*.  Each  horn  no  longer 
lies,  as  it  does  in  the  rhinoceros,  in  the  plane  of  symmetry  of  the 
animal  of  which  it  forms  a  part;  and  the  limited  field  of  force  con- 
cerned in  the  genesis  and  growth  of  the  horn  is  bound,  accordingly, 
to  be  more  or  less  laterally  asymmetrical.  But  the  two  horns  are 
in  symmetry  one  with  another;  they  form  "conjugate"  spirals,  one 
being  the  "mirror-image"  of  the  other.  Just  as  in  the  hairy  coat 
of  the  animal  each  hair,  on  either  side  of  the  median  "parting," 
tends  to  have  a  certain  definite  direction  of  its  owji,  inclined  away 
from  the  median  axial  plane  of  the  whole  system,  so  is  it  both  with 
the  bony  core  of  the  horn  and  with  the  consolidated  mass  of  hairs 
or  hair-like  substance  which  constitutes  its  sheath;  the  primary  axis 
of  the  horn  is  more  or  less  inclined  to,  and  may  even  be  nearly 
perpendicular  to,  the  axial  plane  of  the  animal. 

The  growth  of  the  horny  sheath  is  not  continuous,  but  more  or 
less  definitely  periodic :  sometimes,  as  in  the  sheep,  this  periodicity 
is  particularly  well-marked,  and  causes  the  horny  sheath  to  be  com- 

*  In  this  chapter  we  keep  to  Moseley's  way  of  regarding  the  equiangular  spiral 
in  space,  of  shell  or  horn,  as  generated  by  a  certain  figure  which  (a)  grows,  (6)  revolves 
about  an  axis,  and  (c)  is  translated  along  or  parallel  to  the  said  axis,  all  at  certain 
appropriate  and  specific  velocities.  This  method  is  simple,  and  even  adequate, 
from  the  naturalist's  point  of  view;  but  not  so,  or  much  less  so,  from  the  mathe- 
matician's, as  we  have  found  in  the  last  chapter  (p.  782). 


876  THE  SHAPES  OF  HORNS  [ch. 

posed  of  a  series  of  all  but  separate  rings,  which  are  supposed  to  be 
formed  year  by  year,  and  so  to  record  the  age  of  the  animal*. 

Just  as  Moseley  sought  for  the  true  generating  curve  in  the  orifice, 
or  "lip,"  of  the  moUuscan  shell,  so  we  begin  by  assuming  that 
in  the  spiral  horn  the  generating  curve  corresponds  to  the  lip  or 
margin  of  one  of  the  horny  rings  or  annuli.  This  annular  margin, 
or  boundary  of  the  ring,  is  usually  a  sinuous  curve,  not  lying  in 
a  plane,  but  such  as  would  form  the  boundary  of  an  anticlastic 
surface  of  great  complexity:  to  the  meaning  and  origin  of  which 
phenomenon  we  shall  return  presently.  But,  as  we  have  already 
seen  in  the  case  of  the  molluscan  shell,  the  complexities  of  the  Up 
itself,  or  of  the  corresponding  lines  of  growth  upon  the  shell,  need 


Fig.  432.     The  Argali  sheep;  Ovis  Ammon.     From  Cook's 
Spirals  in  Nature  and  Art. 

not  concern  us  in  our  study  of  the  development  of  the  spiral: 
inasmuch  as  we  may  substitute  for  these  actual  boundary  lines, 
their  "trace,"  or  projection  on  a  plane  perpendicular  to  the  axis — in 
other  words  the  simple  outline  of  a  transverse  section  of  the  whorl. 
In  the  horn,  this  transverse  section  is  often  circular  or  nearly  so, 
as  in  the  oxen  and  many  antelopes:    it  now  and  then  becomes  of 

*  Cf.  R.  S.  Hindekoper,  On  the  Age  of  the  Domestic  Animals,  Philadelphia  and 
London,  1891,  p.  173.  In  the  case  of  the  ram's  horn,  the  assumption  that  the  rings 
are  annual  is  probably  justified.  In  cattle  they  are  much  less  conspicuous,  but 
are  sometimes  well-marked  in  the  cow;  and  in  Sweden  they  are  then  called 
"calf-rings,"  from  a  belief  that  they  record  the  number  of  offspring.  That  is 
to  say,  the  growth  of  the  horn  is  supposed  to  be  retarded  during  gestation,  and  to 
be  accelerated  after  parturition,  when  superfluous  nourishment  seeks  a  new  outlet. 
(Cf.  Lonnberg,  P.Z.S.  1900,  p.  689.) 


XIII]  OF  SHEEP  AND  GOATS  877 

somewhat  complicated  polygonal  outline,  as  in  a  highland  ram ;  but 
in  many  antelopes,  and  in  most  of  the  sheep,  the  outline  is  that  of 
an  isosceles  or  sometimes  nearly  equilateral  triangle,  a  form  which 
is  typically  displayed,  for  instance,  in  Ovis  Ammon.  The  horn  in 
this  latter  case  is  a  trihedral  prism,  whose  three  faces  are  (1)  an 
upper,  or  frontal  face,  in  continuation  of  the  plane  of  the  frontal 
bone;  (2)  an  outer,  or  orbital,  starting  from  the  upper  margin  of 
the  orbit;  and  (3)  an  inner,  or  nuchal,  abutting  on  the  parietal 
bone*.  Along  these  three  faces,  and  their  corresponding  angles  or 
edges,  we  can  trace  in  the  fibrous  substance  of  the  horn  a  series  of 
homologous  spirals,  such  as  w^e  have  called  in  a  preceding  chapter 
the  ''ensemble  of  generating  spirals"  which  define  or  constitute  the 
surface. 

The  case  of  the  horn  differs  in  ways  of  its  own  from  that  of 
the  molluscan  shell.  For  one  thing,  the  horn  is  always  tubular — 
its  generating  curve  is  actually,  as  well  as  theoretically,  a  closed 
curve;  there  is  no  such  thing  as  "involution,"  or  the  wrapping  of 
one  whorl  within  another,  or  successive  intersection  of  the  generating 
curve.  Again,  while  the  calcareous  substance  of  the  shell  is  laid 
down  once  for  all,  fixed  and  immovable,  there  is  reason  to  believe 
that  the  young  horn  has,  to  begin  with,  a  certain  measure  of  flexi- 
bility, a  certain  freedom,  even  though  it  be  slight,  to  bend  or  fold 
or  wrinkle.  And  this  being  so,  while  it  is  no  harder  in  the  horn 
than  in  the  shell  to  recognise  the  general  field  of  force  or  general 
direction  of  growth,  the  actual  conditions  are  somewhat  more 
complex. 

In  some  few  cases,  of  which  the  male  musk  ox  is-  one  of  the  most 
notable,  the  horn  is  not  developed  in  a  continuous  spiral  curve.  It 
changes  its  shape  as  growth  proceeds;  and  this,  as  we  have  seen, 
is  enough  to  show  that  it  does  not  constitute  a  logarithmic  spiral. 
The  reason  is  that  the  bony  exostoses,  or  horn-cores,  about  which 
the  horny  sheath  is  shaped  and  moulded,  neither  grow  continuously 
nor  even  remain  of  constant  size  after  attaining  their  full  growth. 
But  as  the  horns  grow  heavy  the  bony  core  is  bent  downwards  by 
their  weight,  and  so  guides  the  growth  of  the  horn  in  a  new  direction. 
Moreover  as  age  advances,  the  core  is  further  weakened  and  to 
a  great  extent  absorbed:    and  the  horny  sheath  or  horn  proper, 

*  Cf.  Sir  V.  Brooke,  On  the  large  sheep  of  the  Thian  Shan,  P.Z.S.  1875,  p.  511. 


878 


THE  SHAPES  OF  HORNS 


[CH. 


deprived  of  its  support,  continues -to  grow,  but  in  a  flattened  curve 
very  different  from  its  original  spiral*.  The  chamois  is  a  somewhat 
analogous  case.  Here  the  terminal,  or  oldest,  part  of  the  horn  is 
curved;  it  tends  to  assume  a  spiral  form,  though  from  its  com- 
parative shortness  it  seems  merely  to  be  bent  into  a  hook.  But 
later  on  the  bony  core  within,  as  it  grows  and  strengthens,  stiffens 
the  horn  and  guides  it  into  a  straighter  course  or  form.  The  same 
phenomenon  of  change  of  curvature,  manifesting  itself  at  the  time 
when,  or  the  place  where,  the  horn  is  freed  from  the  support  of  the 
internal  core,  is  seen  in  a  good  many  other  antelopes  (such-  as  the 


Fig.  4.33.     Diagram  of  ram's  horns,     a,  frontal;   h,  orbital;   c,  nuchal  surface. 
After  Sir  Vincent  Brooke,  from  P.Z.S. 

hartebeest)  and  in  many  buffaloes;  and  the  cases  where  it  is  most 
manifest  appear  to  be  those  where  the  bony  core  is  relatively  short, 
or  relatively  weak.  All  these  illustrate  the  cardinal  difference 
between  the  growth  of  the  horn  and  that  of  the  bone  below:  the 
one  dead,  the  other  alive;  the  one  adding  and  retaining  its  successive 
increments,  the  other  mobile,  plastic,  and  in  eontinual  flux  through- 
out. 

But  in  the  great  majority  of  horns  we  have  no  difficulty  in 
recognising  a  continuous  logarithmic  spiral,  nor  in  correlating  it 
with  an  unequal  rate  of  growth  (parallel  to  the  axis)  on  two 
opposite  sides  of  the  horn,  the  inequality  maintaining  a  constant 
ratio  as  long  as  growth  proceeds.  In  certain  antelopes,  such  as  the 
gemsbok,  the  spiral  angle  is  very  smafl,  or  in  other  words  the  horn 

*  Cf.  E.  Lonnberg,  On  the  structure  of  the  musk  ox,  P.Z.S.  1900,  pp.  686-718. 


XIII]  OF  SHEEP  AND  GOATS  879 

is  very  nearly  straight;  in  other  species  of  the  same  genus  OryXy 
such  as  the  Beisa  antelope  and  the  Leucoryx,  a  gentle  curve  (not 
unlike  though  generally  less  than  that  of  a  Dentalium  shell)  is 
evident;  and  the  spiral  angle,  according  to  the  few  measurements 
I  have  made,  is  found  to  measure  from  about  >20°  to  nearly  40°. 
In  some  of  the  large  wild  goats,  such  as  the  Scinde  wild  goat,  we  have 


Fig.  484.     Head  of  Arabian  wild  goat,  Capra  sinaitica. 
After  Sclater,  from  P.Z.S. 

a  beautiful  logarithmic  spiral,  with  a  constant  angle  of  rather  less 
than  70°;  and  we  may  easily  arrange  a  series  of  forms,  such  for 
example  as  the  Siberian  ibex,  the  moufflon,  Ovis  Ammon,  etc.,  and 
ending  with  the  long-horned  Highland  ram:  in  which,  as  we  pass 
from  one  to  anothei',  we  recognise  precisely  homologous  spirals  with 
an  increasing  angular  constant^  the  spiral  angle  being,  for  instance, 
about  75°  or  rather  less  in  Ovis  Ammon,  and  in  the  Highland  ram 
a  very  little  more.  We  have  already  seen  that  in  the  neighbourhood 
of  70°  or  80°  a  small  change  of  angle  makes  a  marked  difference  in 


880  THE  SHAPES  OF  HORNS  [ch. 

the  appearance  of  the  spire ;  and  we  know  also  that  the  actual  length 
of  the  horn  makes  a  very  striking  difference,  for  the  spiral  becomes 
especially  conspicuous  to  the  eye  when  horn  or  shell  is  long  enough 
to  shew  several  whorls,  or  at  least  a  considerable  part  of  one  entire 
convolution. 

Even  in  the  simplest  cases,  such  as  the  wild  goats,  the  spiral  is 
never  a  plane  but  always  a  gauche  spiral :  in  greater  or  less  degree 
there  is  always  superposed  upon  the  plane  logarithmic  spiral  a  helical 
spiral  in  space.  Sometimes  the  latter  is  scarcely  apparent,  for  the 
horn  (though  long,  as  in  the  said  wild  goats)  is  not  nearly  long 
enough  to  shew  a  complete  convolution :  at  other  times,  as  in  the 
ram,  and  still  better  in  many  antelopes  such  as  the  koodoo,  the 
corkscrew  curve  of  the  horn  becomes  its  most  characteristic  feature. 
So  we  may  study,  as  in  the  molluscan  shell,  the  helicoid  component 
of  the  spire — in  other  words  the  variation  in  what  we  have  called 
(on  p.  816)  the  angle  j^.  This  factor  it  is  which,  more  than  the 
constant  angle  of  the  logarithmic  spiral,  imparts  a  characteristic 
appearance  to  the  various  species  of  sheep,  for  instance  to  the  various 
closely  allied  species  of  Asiatic  wild  sheep,  or  Argali.  In  all  of'these 
the  constant  angle  of  the  logarithmic  spiral  is  very  much  the  same, 
but  the  enveloping  angle  of  the  cone  differs  greatly.  Thus  the  long 
drawn  out  horns  of  Ovis  Poli,  four  feet  or  more  from  tip  to  tip, 
diifer  conspicuously  from  those  of  Ovis  Ammon  or  0.  hodgsoni,  in 
which  a  very  similar  logarithmic  spiral  is  wound  (as  it  were)  round 
a  much  blunter  cone. 

Let  us  continue  to  dispense  with  mathematics,  for  the  mathe- 
matical treatment  of  a  gauche  spiral  is  never  very  simple,  and  let 
us  deal  with  the  matter  by  experiment.  We  have  seen  that  the 
generating  curve,  or  transverse  section,  of  a  typical  ram's  horn  is 
triangular  in  form.  Measuring  (along  the  curve  of  the  horn)  the 
length  of  the  three  edges  of  the  trihedral  structure  in  a  specimen  of 
Ovis  Ammon,  and  calling  them  respectively  the  outer,  inner,  and 
hinder  edges  (from  their  position  at  the  base  of  the  horn,  relatively 
to  the  skull),  I  find  the  outer  edge  to  measure  80  cm.,  the  inner 
74  cm.,  and  the  posterior  45  cm. ;  let  us  say  that,  roughly,  they  are 
in  the  ratio  of  9:8:5.  Then,  if  we  make  a  number  of  little 
cardboard  triangles,  equip  each  with  three  httle  legs  (I  make  them 
of  cork),  whose  relative  lengths  are  as  9:8:5,  and  pile  them  up 


XIII]  OF  SHEEP  AND  GOATS  881 

and  stick  them  all  together,  we  straightway  build  up  a  curve  ol 
double  curvature  precisely  analogous  to  the  ram's  horn:  except 
only  that,  in  this  first  approximation,  we  have  not  allowed  for  the 
gradual  increment  (or  decrement)  of  the  triangular  surfaces,  that  is 
to  say,  for  the  tapering  of  the  horn  due  to  the  magnification  of  the 
generating  curve. 

In  this  case  then,  and  in  most  other  trihedral  or  three-sided  horns, 
one  of  the  three  components,  or  three  unequal  velocities  of  growth, 
is  of  relatively  small  magnitude,  but  the  other  two  are  nearly  equal 
one  to  the  other;  it  would  involve  but  little  change  for  these  latter 
to  become  precisely  equal;  and  again  but  little  to  turn  the  balance 
of  inequality  the  other  way.  But  the  immediate  consequence  of 
this  altered  ratio  of  growth  would  be  that  the  horn  would  appear  to 
wind  the  other  way,  as  it  does  in  the  antelopes,  and  also  in  certain 
goats,  e.g.  the  markhor,  Capra  falconeri. 

For  these  two'  opposite  directions  of  twist  Dr  Wherry  has  suggested  a 
convenient  nomenclature.  When  the  horn  winds  so  that  we  follow  it  from 
base  to  apex  in  the  direction  of  the  hands  of  a  watch,  it  is  customary  to  call 
it  a  "left-handed"  spiral.  Such  a  spiral  we  have  in  the  horn  on  the  left-hand 
side  of  a  ram's  head.  Accordingly,  Dr  Wherry  calls  the  condition  homonymous, 
where,  as  in  the  sheep,  a  right-handed  spiral  is  on  the  right  side  of  the  head, 
and  a  left-handed  spiral  on  the  left  sid,e ;  while  he  calls  the  opposite  condition 
heteronymmis,  as  we  have  it  in  the  antelopes,  where  the  right-handed  twist 
is  on  the  left  side  of  the  head,  and  the  left-handed  twist  on  the  right-hand  side. 
Among  the  goats,  we  may  have  either  condition.  Thus  the  domestic  and 
most  of  the  wild  goats  agree  with  the  sheep;  but  in  the  markhor  the  twisted 
horns  are  heteronymous,  as  in  the  antelopes.  The  difference,  as  we  have 
seen,  is  easily  explained;  and  (very  much  as  in  the  case  of  our  opposite  spirals 
in  the  apple-snail,  referred  to  on  p.  820)  it  has  no  very  deep  importance 

Summarised  then  in  a  very  few  words,  the  argument  by  which 
we  account  for  the  spiral  conformation  of  the  horn  is  as  follows : 
The  horn  elongates  by  dint  of  continual  growth  within  a  narrow 
zone,  or  annulus,  at  its  base.  If  the  rate  of  growth  be  identical  on 
all  sides  of  this  zone,  the  horn  will  "grow  straight;  if  it  be  greater 
on  one  side  than  on  the  other,  the  horn  will  become  curved;  and 
it  probably  will  be  greater  on  one  side  than  on  the  other,  because 
each  single  horn  occupies  an  unsymmetrical  field  with  reference  to 
the  plane  of  symmetry  of  the  animal.  If  the  maximal  and  minimal 
velocities  of  growth  be  precisely  at  opposite  sides  of  the  zone  of 


882  THE  SHAPES  OF  HORNS  [ch. 

growth,  the  resultant  spiral  will  be  a  plane  spiral ;  but  if  they  be  not 
precisely  or  diametrically  opposite,  then  the  spiral  will  be  a  gauche 
spiral  in  space ;  and  it  is  by  no  means  likely  that  the  maximum  and 
minimum  will  occur  at  precisely  opposite  ends  of  a  diameter,  for  no 
such  plane  of  symmetry  is  manifested  in  the  field  of  force  to  which 
the  growing  annulus  corresponds  or  appertains. 

Now  we  must  carefully  remember  that  the  rates  of  growth  of  which 
we  are  here  speaking  are  the  net  rates  of  longitudinal  increment,  in 
which  increment  the  activity  of  the  living  cells  in  the  zone  of  growth 
at  the  base  of  the  horn  is  only  one  (though  it  is  the  fundamental) 
factor.  In  other  words,  if  the  horny  sheath  were  continually  being 
added  to  with  equal  rapidity  all  round  its  zone  of  active  growth. 


Fig.  4,*i5.     Marco  Polo's  sheep:  Ovis  FoU.     From  Cook. 

but  at  the  same  time  had  its  elongation  more  retarded  on  one  side 
than  the  other  (prior  to  its  complete  solidification)  by  varying  degrees 
of  adhesion  or  membranous  attachment  to  bhe  bony  core  within, 
then  the  net  result  would  be  a  spiral  curve  precisely  such  as  would 
have  arisen  from  initial  inequalities  in  the  rate  of  growth  itself.  It 
seems  probable  that  this  is  an  important  factor,  and  sometimes  even 
the  chief  factor  in  the  case.  The  same  phenomenon  of  attachment 
to  the  bony  core,  and  the  consequent  friction  or  retardation  with 
which  the  sheath  slides  over  its  surface,  will  lead  to  various  subsidiary 
phenomena:  among  others  to  the  presence  of  transverse  folds  or 
corrugations  upon  the  horn,  and  to  their  unequal  distribution  upon  its 
several  faces  or  edges.  And  while  it  is  perfectly  true  that  nearly  all 
the  characters  of  the  hotn  ca  n  be  accounted  for  by  unequal  velocities 


XIII]  OF  SHEEP  AND  GOATS  883 

of  longitudinal  growth  upon  its  different  sides,  it  is  also  plain  that  the 
actual  field  of  force  is  a  very  complicated  one  indeed.  For  example, 
we  can  easily  see  (at  least  in  the  great  majority  of  cases)  that  the 
direction  of  growth  of  the  horny  fibres  of  the  sheath  is  by  no  means 
parallel  to  the  axis  of  the  core  within;  accordingly  these  fibres  will 
tend  to  wind  in  a  system  of  helicoid  curves  around  the  core,  and  not 
only  this  hehcoid  twist  but  any  other  tendency  to  spiral  curvature 
on  the  part  of  the  sheath  will  tend  to  be  opposed  or  modified  by  the 
resistance  of  the  core  within.  On  the  other  hand  living  bone  is  a 
very  plastic  structure,  and  yields  easily  though  slowly  to  any  forces 
tending  to  its  deformation;  and  so,  to  a  considerable  extent,  the 
bony  core  itself  will  tend  to  be  modelled  by  the  curvature  which  the 


Fig.  43(5.     Head  of  Ovis  Ammon,  shewing  St  Venant's  curves. 

growing  sheath  assumes,  and  the  final  result  will  be  determined  by 
an  equilibrium  between  these  two  systems. 

While  it  is  not  very  safe,  perhaps,  to  lay  down  any  general  rule 
as  to  what  horns  are  more  and  what  are  less  spirally  curved,  I  think 
it  may  be  said  that,  on  the  whole,  the  thicker  the  horn  the  greater 
is  its  spiral  curvature.  It  is  the  slender  horns,  of  such  forms  as 
the  Beisa  antelope,  which  are  gently  curved,  and  it  is  the  robust 
horns  of  goats  or  of  sheep  in  which  the  curvature  is  more  pronounced. 
Other  things  being  the  same,  this  is  what  we  should  expect  to  find ; 
for  it  is  where  the  transverse  section  of  the  horn  is  large  that  we  may 
expect  to  find  the  more  marked  differences  in  the  intensity  of  the 
field  of  force,  whether  of  active  growth  or  of  retardation,  on  opposite 
sides  or  in  different  sectors  thereof. 


884  THE  SHAPES  OF  HORNS  [ch. 

But  there  is  yet  another  and  a  very  remarkable  phenomenon 
which  we  may  discern  in  the  growth  of  a  horn  when  it  takes  the 
form  of  a  curve  of  double  curvature^  namely,  an  effect  of  torsional 
strain;  and  this  it  is  which  gives  rise  to  the  sinuous  "  lines  of  growth,"' 
or  sinuous  boundaries  of  the  separate  horny  rings,  of  which  we  have 
already  spoken.  It  is  not  at  first  sight  obvious  that  a  mechanical 
strain  of  torsion  is  necessarily  involved  in  the  growth  of  the  horn. 
In  our  experimental  illustration  (p.  880),  we  built  up  a  twisted  coil  of 
separate  elements,  and  no  torsional  strain  attended  the  development 
of  the  system.  So  would  it  be  if  the  horny  sheath  grew  by  successive 
annular  increments,  free  save  for  their  relation  to  one  another  and 
having  no  attachment  to  the  'Solid  core  within.  But  as  a  matter 
of  fact  there  is  such  an  attachment,  by  subcutaneous  connective 
tissue,  to  the  bony  core;  and  accordingly  a  torsional  strain  will  be 
set  up  in  the  growing  horny  sheath,  again  provided  that  the  forces 
of  growth  therein  be  directed  more  or  less  obliquely  to  the  axis  of 
the  core;  for  a  -'couple"  is  thus  introduced,  giving  rise  to  a  strain 
which  the  sheath  would  not  experience  were  it  free  (so  to  speak) 
to  shp  along,  impelled  only  by  the  pressure  of  its  own  growth  from 
below.  And  furthermore,  the  successive  small  increments  of  the 
growing  horn  (that  is  to  say,  of  the  horny  sheath)  are  not  instan- 
taneously converted  from  living  to  solid  and  rigid  substance;  Init 
there  is  an  intermediate  stage,  probably  long-continued,  during 
which  the  new-formed  horny  substance  in  the  neighbourhood  of  the 
zone  of  active  growth  is  still  plastic  and  capable  of  deformation. 

Now  we  know,  from  the  celebrated  experiments  of  St  Venant*, 
that  in  the  torsion  of  an  elastic  body,  other  than  a  cylinder  of 
circular  section,  a  very  remarkable  state  of  strain  is  introduced.  If 
the  body  be  thus  cylindrical  (whether  solid  or  hollow),  then  a  twist 
leaves  each  circular  section  unchanged,  in  dimensions  and  in  figure. 
But  in  all  other  cases,  such  as  an  elliptic  rod  or  a  prism  of  any 
particular  sectional  form,  forces  are  introduced  which  act  parallel 
to  the  axis  of  the  structure,  and  which  warp  each  section  into  a 
complex  "  anticlastic  "  surface.     Thus^  in  the  case  of  a  triangular  and 

♦  St  Venant,  De  la  torsion  ties  prismes,  avec  des  considerations  sur  leur  flexion, 
etc.,  Mem.  des  Savants  J'jtrangers,  Paris,  xiv,  pp.  233-560,  1856.  Karl  Pearson 
dedicated  part  of  his  History  of  the  Theory  of  Elasticity  to  the  memory  of  this 
ingenious  and  original  man.  For  a  modern  account  of  the  subject  see  Love's 
Elasticity  (2nd  ed.),  chaj),  xiv. 


XIII 


ST  VENANT'S  EXPERIMENTS 


885 


e(|uilateral  prism,  such  as  is  shewn  in  section  in  Fig.  437  A,  if  the  part 
of  the  rod  represented  in  the  section  be  twisted  by  a  force  acting 
in  the  direction  of  the  arrow,  then  the  originally  plane  section  will 
be  warped  as  indicated  in  the  diagram — where  the  full  contour-lines 
represent  elevation  above,  and  the  dotted  lines  represent  depression 
below,  the  original  level.  On  the  external  surface  of  the  prism, 
then,"  contour-lines  which  were  originally  parallel  and  horizontal 
will  be  found  warped  into  sinuous  curves,  such  that,  on  each  of  the 
three  faces,  the  curve  will  be  convex  upwards  on  one  half,  and 
concave  upwards  on  the  other  half  of  the  face.  The  ram's  horn, 
and  still  better  that  of  Ovis  Ammon,  is  comparable  to  such  a  prism, 


A  Fig.  437. 

save  that  in  section  it  is  not  quite  equilateral,  and  that  its  three 
faces  are  not  plane.  The  warping  is  therefore  not  precisely  identical 
on  the  three  faces  of  the  horn;  but,  in  the  general  distribution  of 
the  curves,  it  is  in  complete  accordance  with  theory*.  Similar 
anticlastic  curves  ?ire  well  seen  in  many,  antelopes;  but  they  are 
conspicuous  by  their  absence  in  the  cylindrical  horns  of  oxen. 

The  better  to  illustrate  this  phenomenon,  the  nature  of  which  is 
indeed  obvious  enough  from  a  superficial  examination  of  the  horn, 
I  made  a  plaster  cast  of  one  of  the  horny  rings  in  a  horn  of  Ovis 
Arnmon,  so  as  to  get  an  accurate  pattern  of  its  sinuous  edge :  and 
then,  filling  the  mould  up  -with  wet  clay,  I  modelled  an  anticlastic 

*  The  case  of  a  thin  conical  shell  under  torsion  is  more  complicated  than  either 
that  of  the  cylinder  or  of  a  prismatic  rod ;  and  the  more  tapering,  horns  doubtless 
deserve  further  study  from  this  point  of  view.  Cf.  R.  V.  Southwell,  On  the  torsion 
of  conical  shells,  Proc.  R.S.  (A),  clxiii,  pp.  337-355,  1937. 


886  THE  SHAPES  OF  HORNS  [ch. 

surface,  such  as  to  correspond  as  nearly  as  possible  with  the  sinuous 
outhne*.  Finally,  after  making  a  plaster  oast  of  this  sectional 
surface,  I  drew  its  contour-lines  (as  shewn  in  Fig.  437  B)  with  the 
help  of  a  simple  form  of  spherometer.  It  will  be  seen  that  in  great 
part  this  diagram  is  precisely  similar  to  St  Venant's  diagram  of  the 
•cross-section  of  a  twisted  triangular  prism;  and  this  is  especially 
the  case  in  the  neighbourhood  of  the  sharp  angle  of  our  prismatic 
section.  That  in  parts  the  diagram  is  somewhat  asymmetrical  is 
not  to  be  wondered  at:  and  (apart  from  inaccuracies  due  to  the 
somewhat  rough  means  by  which  it  was  made)  this  asymmetry  can 
be  sufficiently  accounted  for  by  anisotropy  of  the  material,  by 
inequalities  in  thickness  of  different  parts  of  the  horny  sheath,  and 
especially  (I  think)  by  unequal  distributions  of  rigidity  due  to  the 
presence  of  the  smaller  corrugations  of  the  horn.  It  is  on  account 
of  these  minor  corrugations  that  in  such  horns  as  the  Highland 
ram's,  where  they  are  strongly  marked,  the  main  St  Venant  effect 
is  not  nearly  so  well  shewn  as  in  smoother  horns,  such  as  those  of 
Ovis  Ammon  and  its  congeners  f. 

The  distribution  of  forces  which  manifest  themselves  in  the 
growth  and  configuration  of  a  horn  is  no  simple  nor  merely  super- 
ficial matter.  One  thing  is  coordinated  with  another;  the  direction 
of  the  axis  of  the  horn,  the  form  of  its  sectional  boundary,  the 
specific  rates  of  growth  in  the  mean  spiral  and  at  various  parts 
of  its  periphery — all'  these  play  their  parts,  controlled  in  turn  by 
the  supply  of  nutriment  which  the  character  of  the  adjacent  tissues 
and  the  distribution  of  the  blood-vessels  combine  to  determine. 
To  suppose. that  this  or  that  size  or  shape  of  horn  has  been  pro- 
duced or  altered,  acquired  or  lost,  by  Natural  Selection,  whensoever 
one  type  rather  than  another  proved  serviceable  for  defence  or 
attack  or  any  other  purpose,  is  an  hypothesis  harder  to  define  and 
to  substantiate  than  some  imagine  it  to  be. 

There  are  still  one  or  two  small  matters  to  speak  of  before  we  leave 
these  spiral  horns.  It  is  the  way  of  sportsmen  to  keep  record  of  big 
game  by  measuring  the  length  along  the  curve  of  the  horn  and 
the  span  from  tip  to  tip.     Now  if  we  study  such  measurements  (as 

*  This  is  not  difficult  to  do,  with  considerable  accuracy,  if  the  clay  be  kept 
well  wetted  or  semi-fluid,  and  the  smoothing  be  done  with  a  large  wet  brush. 

t  The  curves  are  well  shewn  in  most  of  Sir  V.  Brooke's  figures  of  the  various 
species  of  Argali,  in  the  paper  quoted  above,  on  p.  877. 


xm]  OF  SHEEP  AND  GOATS  887 

they  may  be  found  in  Mr  Rowland  Ward's  book*),  we  shall  soon 
see  that  the  two  measurements  do  not  tally  one  with  the  other: 
but  that  a  pair  of  horns,  the  longer  when  measured  along  the  curve, 
may  be  the  shorter  from  tip  to  tip,  and  vice  versa.  We  might  set 
this  down  to  mere  variability  of  form,  but  the  true  reason  is  simpler 
still.  If  the  axes  of  the  two  horns  stood  straight  out,  at  right  angles 
to  the  median  plane,  then  growth  in  length  and  in  width  of  span 
would  go  on  together.  But  if  the  two  horns  diverge  at  any  lesser 
angle,  then  as  the  horns  grow  their  spiral  curvature  will  tend  to 
bring  their  tips  nearer  and  farther  apart  alternately. 

There  is  one  last,  but  not  least  curious  property  to  be  seen  in 
a  ram's  horns.  However  large  and  heavy  the  horns  may  be — and 
in  Ovis  Poll  50  or  60  lb.  is  no  unusual  weight  for  the  pair  to  grow  to — 
the  ram  carries  them  with  grace  and  ease,  and  they  neither  endanger 
his  poise  nor  encumber  his  movements.  The  reason  is  that  head 
and  horns  are  very  perfectly  balanced,  in  such  a  way  that  no  bending 
moment  tends  to  turn  the  head  up  or  down  about  its  fulcrum  in 
the  atlas  vertebra ;  if  one  puts  two  fingers  into  the  foramen  magnum 
one  may  lift  up  the  heavy  skull,  and  find  it  hang  in  perfect  equili- 
brium. Moreover,  the  horns  go  on  growing,  but  this  equipoise  is 
never  lost  nor  changed ;  for  the  centre  of  gravity  of  the  logarithmic 
spiral  remains  constant.  There  are  other  cases  where  heavy  horns, 
well  balanced  as  they  doubtless  are,  yet  visibly  affect  the  set  and 
balance  of  the  head.  The  stag  carries  his  head  higher  than  a  horse, 
and  an  Indian  buffalo  tilts  his  muzzle  higher  than  a  cow. 

A  further  note  upon  torsion 

The  phenomenon  of  torsion,  to  which  we  have  been  thus  intro- 
duced, opens  up  many  wide  questions  in  connection  with  form.  Some 
of  the  associated  phenomena  are  admirably  illustrated  in  the  case 
of  climbing  plants ;  but  we  can  only  deal  with  these  still  more  briefly 
and  parenthetically.  The  subject  has  been  elaborately  dealt  with 
not  only  in  Darwin's  books  f,  but  also  by  a  great  number  of 
earlier  and  later  writers.  In  "twining"  plants,  which  constitute 
the  greater  number  of  "climbers,"  the  essential  phenomenon  is  a 
tendency  of  the  growing  shoot  to  revolve  about  a  vertical  axis — 

*  Records  of  Big  Game,  9th  edition,  1928. 

t  Climbing  Plants,  1865  (2nd  ed.  1875);   Power  of  Movement  in  Plants,  1880. 


888  THE  SHAPES  OF  HORNS  [ch. 

a  tendency  long  ago  discussed  by  de  Candolle  and  investigated  by 
Palm,  H.  von  Mohl  and  Dutrochet*.  This  tendency  to  revolution — 
circumvolution,  as  Darwin  calls  it,  revolving  nutation,  as  Sachs  puts 
it— is  very  closely  comparable  to  the  process  by  which  an  antelope's 
horn  (such  as  the  koodoo's)  acquires  its  spiral  twist,  and  is  due,  in  like 
manner,  to  inequahties  in  the  rate  of  growth  of  the  growing  stem : 
with  this  difference  between  the  two,  that  in  the  antelope's  horn 
the  zone  of  active  growth  is  confined  to  the  base  of  the  horn,  while 
in  the  climbing  stem  the  same  phenomenon  is  at  work  throughout 
the  whole  length  of  the  growing  structure.  This  growth  is  in  the 
main  due  to  "turgescence,"  that  is  to  the  extension,  or  elongation, 
of  ready-formed  cells  through  the  imbibition  of  water ;  it  is  a  phe- 
nomenon due  to  osmotic  pressure.  The  particular  stimulus  to  which 
these  movements  (that  is  to  say,  these  inequalities  of  growth)  have 
been  ascribed  can  hardly  be  discussed  here;  but  it  was  hotly 
debated  fifty  years  ago  and  for  many  years  thereafter,  the  point 
at  issue  being  no  other  than  whether  direct  physical  causation,  or 
the  Darwinian  concept  of  fitness  or  adaptation,  should  be  invoked 
as  an  "explanation"  of  biological  phenomena.  The  old  Natur- 
philosophie  had  been  inclined  to  look  for  spirals  everywhere,  and  to 
attribute  them  to  very  simple  causes:  "Man  wird  nicht  gross  irren" 
(said  Okenf)  "wenn  man  sagt,  alle  Pflanzen  entstehen  als  Spirale, 
und  zwar  weil  sie  feststehen  und  ein  End  gegen  die  Sonne  kehren, 
die  taglich  einen  Spiralgang  um  sie  ihacht,  u.s.w."  When  de 
Candolle  saw  a  shoot  curve  under  the  influence  of  light  (by  helio- 
tropism,  as  we  are  told  to  call  it),  he  was  content  to  regard  the 
curvature  as  the  result  of  difi'erent  rates  of  growth  on  one  side 
or  other  of  the  shoot,  and  these  in  turn  as  the  direct  result  of 
differences  of  illumination.  But  by  the  Darwins,  father  and  son, 
and  by  Sachs  and  by  the  Wiirzburg  school,  the  curvature  was 
ascribed  to  "irritability,"  a  "stimulus"  on  one  side  of  the  shoot 
being  followed  by  a  "motor-reaction"  on  the  other.  The  curvature 
was  thus  taken  to  be  a  "response"  to  external  stimuli  (such  as  light 
and  gravity);  and  stimulus  and  response  were  supposed  to  have 

*  Palm,  Ueber  das  Winden  der  Pflanzen,  1827;  H.  von  Mohl,  Bau  und  Windert 
der  Ranken,  etc.,  1827;  R.  H.  J.  Dutrochet,  Sur  la  volubilite  des  tiges  de  certains 
vegetaux,  et  sur  la  cause  de  ce  phenomene,  Ann.  Sc.  Nat.  {Bot.),  ii,  pp.  156-167, 
1844,  and  other  papers, 

t  I  sis,  I,  p.  222,  1817. 


XIII]  A  NOTE  UPON  TORSION  889 

evolved  together  in  the  course  of  ages,  to  bring  about  something 
more  and  more  fitted  for  survival  in  the  struggle  for  existence. 
They  were,  in  short,  of  the  nature  of  acquired  habits,  rather  than 
physical  phenomena.  But  there  was  no  gainsaying  the  fact  that  the 
immediate  cause  of  curvature  was  inequahty  of  growth  on  opposite 
sides*. 

A  simple  stem  growing  upright  in  the  dark,  or  in  uniformly  diffused 
light,  would  be  in  a  position  of  equilibrium  to  a  field  of  force  radially 
symmetrical  about  its  vertical  axis.  But  this  complete  radial  sym- 
metry will  not  often  occur;  and  the  radial  anomalies  may  be  such 
as  arise  intrinsically  from  structural  pecuHarities  in  "the  stem  itself, 
or  externally  to  it  by  reason  of  unequal  illumination  or  through 
various  other  locahsed  forces.  The  essential  fact,  so  far  as  we  are 
concerned,  is  that  in  twining  plants  we  have  a  very  marked  tendency 
to  inequahties  in  longitudinal  growth  on  different  aspects  of  the 
stem^a  tendency  which  is  but  an  exaggerated  manifestation  of  one 
which  is  more  or  less  present,  under  certain  conditions,  in  all  plants 
whatsoever.  Just  as  in  the  case  of  the  ruminants'  horns  so  we  find 
here  that  this  inequality  may  be,  so  to  speak,  positive  or  negative, 
the  maximum  lying  to  the  one  side  or  the  other  of  the  twining  stem ; 
and  so  it  coiftes  to  pass  that  some  climbers  twine  to  the  one  side 
and  some  to  the  other :  the  hop  and  the  honeysuckle  following  the 
sun,  and  the  field-convolvulus  twining  in  the  reverse  direction ;  there 
are  also  some,  like  the  woody  nightshade  (/So/ai<Mm  Dulcamara),  which 
twine  indifferently  either  way. 

Together  with  this  circumnutatory  movement,  there  is  very 
generally  to  be  seen  an  actual  torsioyi  of  the  twining  stem — a  twist, 
that  is  to  say,  about  its  own  axis;  and  Mohl  made  the  curious 
observation,  confirmed  by  Darwin,  that  when  a  stem  twines  around 
a  smooth  cylindrical  stick  the  torsion  does  not  take  place,  save 
■'only  in  that  degree  which  follow^  as  a  mechanical  necessity  from 
the  spiral  winding":  but  that  stems  which  had  cHmbed  around 
a  rough  stick  were  all  more  or  less;  and  generally  much,  twisted. 
Here  Darwin  did  not  refrain  from  introducing  that  teleological  argu- 
ment which  pervades  his  whole  train  of  reasoning:  "The  stem," 
he  says,  "probably  gains  rigidity  by  being  twisted  (on  the  same 

*  On  the  whole  controversy,  see  F.  P.  Blackman's  obituary  notice  of  Francis 
Darwin  in  Proc.  R.8.  (B),  ex,  1932. 


890  THE  SHAPES  OF  HORNS  [ch. 

principle  that  a  much  twisted  rope  is  stiffer  than  a  slackly  twisted 
one),  and  is  thus  indirectly  benefited  so  as  to  be  able  to  pass  over 
inequalities  in  its  spiral  ascent,  and  to  carry  its  own  weight  when 
allowed  to  revolve  freely."  The  mechanical  explanation  would 
appear  to  be  very  simple,  and  such  as  to  render  the  teleological 
hypothesis  unnecessary.  In  the  case  of  the  roughened  support, 
there  is  a  temporary  adhesion  or  "clinging"  between  it  and  the 
growing  stem  which  twines  around  it;  and  a  system  of  forces  is 
thus  set  up,  producing  a  "couple,"  just  as  it  was  in  the  case  of  the 
ram's  or  antelope's  horn  through  direct  adhesion  of  the  bony  core 
to  the  surrounding  sheath.  The  twist  is  the  direct  result  of  this 
couple,  and  it  disappears  when  the  support  is  so  smooth  that  no 
such  force  comes  to  be  exerted. 

Another  important  class  of  climbers  includes  the  so-called  "leaf- 
climbers."  In  these,  some  portion  of  the  leaf,  generally  the  petiole, 
sometimes  (as'  in  the  fumitory)  the  elongated  midrib,  curls  round 
a  support;  and  a  phenomenon  of  like  nature  occurs  in  many,  though 
not  all,  of  the  so-called  "tendril-bearers."  Except  that  a  different 
part  of  the  plant,  leaf  or  tendril  instead  of  stem,  is  concerned  in  the 
twining  process,  the  phenomenon  here  is  strictly  analogous  to  our 
former  case ;  but  in  the  resulting  helix  there  is,  as  a  rul^,  this  obvious 
difference,  that,  while  the  twining  stem,  for  instance  of  the  hop, 
makes  a  slow  revolution  about  its  support,  the  typical  leaf-climber 
makes  a  close,  firm  coil:  the  axis  of  the  latter  is  nearly  perpendicular 
and  parallel  to  the  axis  of  its  support,  while  in  the  twining  stem  the 
angle  between  the  two  axes  is  comparatively  small.  Mathematically 
speaking,  the  difference  merely  amounts  to  this,  that  the  component 
in  the  direction  of  the  vertical  axis  is  large  in  the  one  case,  and  the 
corresponding  component  is  small,  if  not  absent,  in  the  other;  in 
other  words,  we  have  in  the  climbing  stem  a  considerable  vertical 
component,  due  to  its  own  tendency  to  grow  in  height,  while  this 
longitudinal  or  vertical  extension  of  the  whole  system  is  not  apparent, 
or  little  apparent,  in  the  other  cases.  But  from  the  fact  that  the 
twining  stem  tends  to  run  obliquely  to  its  support,  and  the  coiling 
petiole  of  the  leaf-climber  tends  to  run  transversely  to  the  axis  of 
its  support,  there  immediately  follows  this  marked  difference,  that 
the  phenomenon  of  torsion,  so  manifest  in  the  former  case,  will  be 
absent  in  the  latter. 


xiii]  A  NOTE  UPON  TORSION  891 

There  is  one  other  phenomenon  which  meets  us  in  the  twining 
and  twisted  stem,  and  which  is  doubtless  illustrated  also,  though 
not  so  well,  in  the  antelope's  horn;  it  is  a  phenomenon  which  forms 
the  subject  of  a  second  chapter  of  St  Venant's  researches  on  the 
effects  of  torsional  strain  in  elastic  bodies.  We  have  already  seen 
how  one  effect  of  torsion,  in  for  instance  a  prism,  is  to  produce 
strains  parallel  to  the  axis,  elevating  parts  and  depressing  other 
parts  of  each  transverse  section.  But  m  addition  to  this,  the  same 
torsion  has  the  effect  of  materially  altering  the  form  of  the  section 
itself,  as  we  may  easily  see  by  twisting  a  square  or  oblong  piece  of 
india-rubber.  If  we  start  with  a  cylinder,  such  as  a  round  piece 
of  catapult  india-rubber,  and  twist  it  on  its  own  long  axis,  we  have 
already  seen  that  it  suffers  no  other  distortion;  it  still  remains 
a  cyhnder,  that  is  to  say,  it  is  still  in  section  everywhere  circular. 
But  if.it  be  of  any  other  shape  than  cylindrical  the  case  is  different, 
for  now  the  sectional  shape  tends  to  alter  under  the  strain  of  torsion. 
Thus,  if  our  rod  be  eUiptical  m  section  to  begin  with,  it  will,  under 
torsion,  become  a  more  elongated  ellipse ;  if  it  be  square,  its  angles 
will  become  more  prominent  and  its  sides  will  curve  inwards,  till  at 
length  the  square  assumes  the  appearance  of  a  four-pomted  star 
with  rounded  angles.  Furthermore,  looking  at  the  results  of  this 
process  of  modification,  we  find  experimentally  that  the  resultant 
figures  are  more  easily  twisted,  less  resistant  to  torsion,  than  were 
those  from  which  we  evolved  them;  and  this  is  a  very  curious 
physical  or  mathematical  fact.  So  a  cylinder,  which  is  especially 
resistant  to  torsion,  is  very  easily  bent  or  flexed;  while  projecting 
ribs  or  angles,  such  as  an  engineer  makes  in  a  bar  or  pillar  of  iron 
for  the  purpose  of  increasing  its  resistance  to  bending,  actually  make 
it  much  weaker  than  before  (for  the  same  amount  of  metal  per  unit 
length)  in  the  way  of  resistance  to  torsion. 

In  the  hop  itself,  and  in  a  very  considerable  number  of  other 
twining  and  twisting  stems,  the  ribbed  or  channelled  form  of  the 
stem  is  a  conspicuous  feature.  We  may  safely  take  it,  (1)  that  such 
stems  are  especially  susceptible  of  torsion;  and  (2)  that  the  effect 
of  torsion  will  be  to  intensify  any  such  peculiarities  of  sectional 
outline  which  they  may  possess,  though  not  to  initiate  them  in  an 
originally  cylindrical  structure.  In  the  leaf-climbers  the  case  does 
not  present  itself,  for  there,  as  we  have  seen,  torsion  itself  is  not, 


892  THE  SHAPES  OF  HORNS  [ch. 

or  is  very  slightly,  manifested.  There  are  very  distinct  traces  of 
the  phenomenon  in  the  horns  of  certain  antelopes,  but  the  reason 
why  it  is  not  a  more  conspicuous  feature  of  the  antelope's  horn  or 
of  the  ram's  is  apparently  a  very  simple  one:  namely,  that  the 
presence  of  the  bony  core  within  tends  to  check  that  deformation 
which  is  perpendicular,  while  it  pfermits  that  which  is  parallel,  to 
the  axis  of  the  horn. 

Uf  deer's  antlers 

But  let  us  return  to  our  subject  of  the  shapes  of  horns,  and  con- 
sider briefly  our  last  class  of  these  structures,  namely  the  bony 
antlers  of  the  elk  and  deer*.  The  problems  which  these  present  to 
us  are  very  different  from  those  which  we  have  had  to  do  with  in 
the  antelope  or  the  sheep. 

With  regard  to  its  structure,  it  is  plain  that  the  bony  antler  corre- 
sponds, upon  the  whole,  to  the  bony  core  of  the  antelope's  horn; 
while  in  place  of  the  hard  horny  sheath  of  the  latter,  we  have  the 
soft  "velvet,"  which  every  season  covers  the  new  growing  antler, 
and  protects  the  large  nutrient  blood-vessels  by  help  of  which  the 
antler  grows -f.  The  main  difference  hes  in  the  fact  that  in  the 
one  case  the  bony  coTe,  imprisoned  within  its  sheath,  is.  rendered 
incapable  of  branching  and  incapable  also  of  lateral  expansion,  and 
the  whole  horn  is  only  permitted  to  grow  in  length  while  retaining 
a  sectional  contour  that  is  identical  with  (or  but  little  altered  from) 
that  which  it  possesses  at  its  growing  base :  but  in  the  antler  on  the 
other  hand  no  such  restraint  is  imposed,  and  the  living,  growing 
fabric  of  bone  is  free  to  expand  into  a  broad  flat  plate  over  which 
the  blood-vessels  run.  In  the  immediate  neighbourhood  of  the  main 
blood-vessels  growth  will  be  most  active,  in  the  interspaces  between 
it  may  wholly  fail:  with  the  result  that  we  may  have  great  notches 
cut  out  of  the  flattened  plate,  or  may  at  length  find  it  reduced  to  the 

*  For  an  elaborate  study  of  antlers,  see  A.  Rorig,  Arch.  f.  Entw.  Mech.  x, 
pp.  525-644,  1900;  xi,  pp.  65-148,  225-309,  1901;  C.  Hoffmann,  Zur  Morpholdgie 
der  rezenten  Hirsche,  75  pp.,  23  pis.,  1901;  also  Sir  V^ictor  Brooke,  On  the 
classification  of  the  Cervidae,  P.Z.S.  1878,  pp.  883-928.  For  a  discussion  of  the 
development  of  horns  and  antlers,  see  H.  Gadow,  P.Z.S.  1902,  pp.  206-222,  and 
works  quoted  therein. 

t  Cf.  L.  Rhumbler,  Ueber  die  Abhangigkeit  des  Geweihwachsturas  der  Hirsche, 
speziell  des  Edelhirsches,  vom  Verlauf  der  Blutgefasse  im  Kalbengeweih,  Zeitsckr. 
f.  Forst.  und  Jagdwesen,  1911,  pp.  295-314, 


XIII]  OF  THE  ANTLERS  OF  DEER  893 

form  of  a  simple  branching  structure.  The  main  point  is  that  the 
"horn"  is  essentially  an  axial  rod^  while  the  "antler"  is  essentially 
an  outspread  sicrface*.  In  other  words,  the  whole  configuration 
of  an  antler  is  more  easily  understood  by  conceiving  it  as  a  plate 
or  a  surface,  more  and  more  notched  and  scolloped  till  but  a  slender 
skeleton  remains,  than  to  look  upon  it  the  other  way,  namely  as  an 
axial  stem  (or  beam)  giving  off  branches  (or  tines),  the  interspaces 
between  which  latter  may  sometimes  fill  up  to  form  a  continuous 
surface. 


Fig.  438.     Antlers  of  Swedish  elk..    After  Lonnberg,  from  P.Z.S. 

In  a  sense  it  matters  very  little  whether  we  regard  the  broad 
plate-like  antlers  of  the  elk  or  the  slender  branching  antlers  of  the 
stag  as  the  more  primitive  type;,  for  we  are  not  concerned  here 
with  questions  of  hypothetical  phylogeny,  and  even  from  the 
mathematical  point  of  view  it  makes  little  or  no  difference  whether 
we  describe  the  plate  as  constituted  by  the  interconnection  of 
branches,  or  the  branches  as  derived  by  the  notching  or  incision 

*  The  fact  that  in  one  very  small  deer,  the  little  South  American  Coassus,  the 
antler  is  reduced  to  a  simple  short  spike,  does  not  preclude  the  general  distinction 
which  I  have  drawn.  In  Coassus  we  have  the  beginnings  of  an  antler,  which  has 
not  yet  manifested  its  tendency  to  expand;  and  in  the  many  allied  species  of  the 
American  genus  Cariacus,  we  find  the  expansion  manifested  in  various  sim,ple 
modes  of  ramification  or  bifurcation. 


894  THE  SHAPES  OF  HORNS  [ch. 

of  a  plate.  The  important  point  for  us  is  to  recognise  that 
(save  for  occasional  slight  irregularities)  the  branching  system  in 
the  one  conforms  essentially  to  the  curved  plate  or  surface  which  we 
see  plainly  in  the  other.  In  short  the  arrangement  of  the  branches 
is  more  or  less  comparable  to  that  of  the  veins  in  a  leaf,  or  to  that  of 
the  blood-vessels  as  they  course  over  the  curved  surface  of  an  organ. 
It  is  a  process  of  ramification,  not,  like  that  of  a  tree,  in  various 
planes,  but  strictly  limited  to  a  single  surface.  And  just  as  the 
veins  within  a  leaf  are  not  necessarily  confined  (as  they  happen  to 


Fig.  439.     Head  and  antlers  of  the  Indian  swamp-deer  (Cervus  Duvaureli). 
After  Lydekker,  from  P.Z.S. 

be  in  most  ordinary  leaves)  to  a  plane  surface,  but,  as  in  the  petal 
of  a  tulip  or  the  capsule  of  a  poppy,  may  have  to  run  their  course 
within  a  curved  surface,  so  does  the  analogy  of  the  leaf  lead  us 
directly  to  the  mode  of  branching  which  is  characteristic  of  the  antler. 
The  surface  to  which  the  branches  of  the  antler  tend  to  be  confined 
is  a  more  or  less  spheroidal,  or  occasionally  an  ellipsoidal  one;  and 
furthermore,  when  we  inspect  any  well-developed  pair  of  antlers, 
such  as  those  of  a  red  deer,  a  sambur  or  a  wapiti,  we  have  no  difficulty 
in  seeing  that  the  two  antlers  make  up  between  tliem  a  single  surface, 


XIII]  OF  THE  ANTLERS  OF  DEER  895 

and  constitute  a  symmetrical  figure,  each  half  being  the  mirror-image 
of  the  other.     It  is  what  the  ghillies  call  the  "cup  of  the  antler". 

To  put  the  case  in  another  way,  a  pair  of  antlers  (apart  from 
occasional  slight  irregularities)  tends  to  constitute  a  figure  such  that 
we  could  conceive  an  elastic  sheet  stretched  over  or  round  the  entire 
system,  and  to  form  one  continuous  and  even  surface;  and  not 
only  would  the  surface  curvature  be  on  the  whole  smooth  and  even, 
but  the  boundary  of  the  surface  would  also  tend  to  be  an  even  curve : 
that  is  to  say  the  tips  of  all  the  tines  would  approximately  have 
their  locus  in  a  continuous  curve. 

It  follows  from  this  that  if  we  want  to  make  a  simple  model  of 
a  set  of  antlers,  we  shall  be  very  greatly  helped  by  taking  some 
appropriate  spheroidal  surface  as  our  groundwork  or  scaffolding. 
The  best  form  of  surface  is  a  matter  for  trial  and  investigation  in 
each  particular  case ;  but  even  in  a  sphere,  by  selecting  appropriate 
areas  thereof,  we  can  obtain  sufficient  varieties  of  surface  to  meet  all 
ordinary  cases.  With  merely  a  bit  of  sculptor's  clay  or  plasticine, 
we  should  be  put  hard  to  it  to  model  the  horns  of  a  wapiti  or  a 
reindeer*  but  if  we  start  with  an  orange  (or  a  round  florence  flask) 
and  lay  our  little  tapered  rolls  of  plasticine  upon  it,  in  simple  natural 
curves,  it  is  surprising  to  see  how  quickly  and  successfully  we  can 
imitate  one  type  of  antler  after  another.  In  either  case,  we. shall  be 
struck  by  the  fact  that  our  model  may  vary  in  its  mode  of  branching 
within  very  considerable  limits,  and  yet  look  perfectly  natural;  for 
the  same  wide  range  of  variation  is  characteristic  of  the  natural 
antlers  themselves.  As  Sir  V.  Brooke  says  (op,  cit.  p.  892),  "No 
two  antlers  are  ever  exactly  alike;  and  the  variation  to  which  the 
antlers  are  subject  is  so  great  that  in"  the  absence  of  a  large  series 
they  would  be  held  to  be  indicative  of  several  distinct  species*." 
But  all  these  many  variations  lie  within  a  limited  range,  for  they  are 
all  subject  to  our  general  rule  that  the  entire  structure  is  essentially 
confined  to  a  single  curved  surface.  A  sheet  of  stiff  paper  makes 
an  even  simpler  model.  Fold  it  in  two;  cut  a  deer's  head  out  of 
the  double  sheet,  and  leave  a  large  oval  where  the  antlers  are  to  be; 
cut  a  few  notches  in  this  oval  leaf,  for  the  spaces  between  the  tines 
(Fig.  440).     The  likeness  to  a  pair  of  antlers  seems  remote  to  begin 

*  Cf.  also  the  immense  range  of  variation  in  elks'  horns,  a,s  described  by 
Lonnberg,  P.Z..S'.  ir,  pp.  352-360,  1902. 


896 


THE  SHAPES  OF  TEETH 


[CH. 


with ;  but  it  is  wonderfully  improved  as  we  separate  the  two  antlers 
and  give  a  twist  to  each,  turning  antler,  tines  and  all,  into  the 
appropriate  curved  or  twisted  surface. 

It  is  probable  that  in  the  curvatures  both  of  the  beam  and  of  its 
tines,  in  the  angles  by  which  these  latter  meet  the  beam,  and  in 
the  contours  of  the  entire  system,  there  are  involved  many  elegant 
mathematical  problems  with  which  we  cannot  attempt  to  deal. 
Nor  must  we  attempt  meanwhile  to  enquire  into  the  physical 
meaning  or  origin  of  these  phenomena,  for  as  yet  the  clue  seems  to 
be  lacking  and  we  should  only  heap  one  hypothesis  upon  another. 
That  there  is  a  complete  contrast  of  mathematical  properties  be- 
tween the  horn  and  the  antler  is  the  main  lesson  with  which,  in  the 
meantime,  we  must  rest  content. 


A  ---^  B 

Fig.  44(T.     Diagrams  of  antlers,  before  twisting  into  shape. 
A,  Red-deer;   B,  Swamp-deer. 


Of  teeth,  and  of  beak  mid  clciw 

In  a  fashion  similar  to  that  manifested  in  the  shell  or  the  horn, 
we  find  the  equiangular  spiral  to  be  implicit  in  a  great  many  other 
organic  structures  where  the  phenomena  of  growth  proceed  in  a 
similar  way:  that  is  to  say,  where  about  an  axis  there  is  some 
asymmetry  leading  to  unequal  rates  of  longitudinal  growth,  and 
where  the  structure  is  of  such  a  kind  that  each  new  increment  is 
added  on  as  a  permanent  and  unchanging  part  of  the  entire  con- 


XIII]  AND  OF  BEAK  AND  CLAW  897 

formation.  Nail  and  claw,  beak  and  tooth,  all  come  under  this 
category.  The  logarithmic  spiral  always  tends  to  manifest  itself  in 
such  structures  as  these,  though  it  usually  only  attracts  our  attention 
in  elongated  structures,  where  (that  is  to  say)  the  radius  vector  has 
described  a  considerable  angle.  When  the  canary-bird's  claws  grow 
long  from  lack  of  use,  or  when  the  incisor  tooth  of  a  rabbit  or  a  rat 
grows  long  by  reason  of  disease  or  of  injury  of  the  opponent  tooth 
against  which  it  was  wont  to  bite*,  we  know  that  the  tooth  or  claw 
tends  to  grow  into  a  spiral  curve,  and  we  speak  of  it  as  a  mal- 
formation!.  But  there  has  been  no  fundamental  change  of  form, 
only  an  abnormal  increase  in  length;  the  elongated  tooth  or 
claw  has  the  selfsame  curvature  which  it  had  when  it  was  short, 
but  the  spiral  becomes  more  and  more  manifest  the  longer  it  grows. 
It  is  only  natural,  but  nevertheless  it  is  curious  to  see,  how 
closely  a  rabbit's  abnormally  overgrown  teeth  come  to  resemble 
the  tusks  of  swine  or  elephants,  of  which  the  normal  state  is  one 
of  hypertrophy.  A  curiously  analogous  case  is  that  of  the  New 
Zealand  Huia  bird,  in  which  the  beak  of  the  male  is  comparatively 
short  and  straight,  while  that  of  the  female  is  long  and  curved; 
it  is  easy  to  see  that  there  is  a  shght  but  identical  curve  also  in  the 
beak  of  the  male,  and  that  the  beak  of  the  female  shews  nothing 
but  an  extension  or  prolongation  of  the  same.  In  the  case  of  the 
more  curved  beaks,  such  as  those  of  an  eagle  or  a  parrot,  we  may, 
if  we  please,  determine  the  constant  angle  of  the  logarithmic  spiral, 
just  as  we  have  done  in  the  case  of  the  Nautilus  shell;  and  here 
again,  as  the  bird  grows  older  or  the  beak  longer,  the  spiral  nature 
of  the  curve  becomes  more  and  more  apparent,  as  in  the  hooked 
beak  of  an  old  eagle,  or  in  the  great  beak  of  a  hyacinthine  macaw. 
Let  us  glance  at  one  or  two  instances  to  illustrate  the  spiral 
curvature  of  teeth. 

*  Cf.  John  Hunter,  Natural  History  of  the  Human  Teeth  (3rd  ed.),  1808,  p.  110: 
"Where  a  tooth  has  lost  its  opposite,  it  will  in  time  become  really  so  much  longer 
than  the  rest  as  the  others  grow  shorter  by  abrasion".  Cf.  James  Murie,  Notes 
on  some  diseased  dental  conditions  in  animals,  Tr.  Odontol.  Soc.  1867-8,  pp.  37-69, 
257-298.  We  now  know  that  a  Coenurus-cjat  in  a  rabbit's  masseter  muscle  may 
twist  the  jaw  sideways,  so  that  the  incisors  fail  to  meet,  and  grow  accordingly: 
H.  A.  Baylis,  Trans.  B.  Soc.  Trop.  Medicine,  xxxiii,  p.  4,  1939. 

t  See  Professor  W.  C.  Mcintosh's  paper  on  "Abnormal  teeth  in  certain  mammals, 
especially  in  the  rabbit,"  Trans.  B.S.E.  lvi,  pp.  333-407,  for  a  large  collection  of 
instances  admirably  illustrated. 


898  THE  SHAPES  OF  TEETH  [ch. 

A  dentist  knows  that  every  tooth  has  a  curvature  of  its  own, 
and  that  in  pulHng  the  tooth  he  must  follow  the  direction  of  the 
curve;  but  in  an  ordinary  tooth  this  curvature  is  scarcely  visible, 
and  is  least  so  when  the  diameter  of  the  tooth  is  large  compared 
with  its  length.  In  simple,  more  or  less  conical  teeth,  such  as  those 
of  the  dolphin,  and  in  the  more  or  less  similarly  shaped  canines 
and  incisors  of  mammals  in  general,  the  curvature  of  the  tooth  is 
particularly  well  seen.  We  see  it  in  the  little  teeth  of  a  hedgehog, 
and  in  the  canines  of  a  dog  or  a  cat  it  is  very  obvious  indeed.  When 
the  great  canine  of  the  carnivore  becomes  still  further  enlarged 
or  elongated,  as  in  Machairodus,  it  grows  into  the  strongly  curved 
sabre-tooth  of  that  extinct  tiger;  and- the  boar's  canine  grows  into 
the  spiral  tusk  of  wart-hog  or  babirussa.  In  rodents,  it  is  the  incisors 
which  undergo  elongation ;  their  rate  of  growth  differs,  though  but 
slightly,  on  the  two  sides  of  the  axis,  and  by  summation  of  these 
slight  differences  in  the  rapid  growth  of  the  tooth  an  unmistakable 
logarithmic  spiral  is  gradually  built  up;  we  see  it  admirably  in 
the  beaver,  or  in  the  great  ground-rat  Geornys.  The  elephant  is  a 
similar  case,  save  that  the  tooth  or  tusk  remains,  owing  to  com- 
parative lack  of  wear,  in  a  more  perfect  condition.  In  the  rodent 
(save  only  in  those  abnormal  cases  mentioned  on  the  last  page) 
the  tip,  or  first-formed  part  of  the  tooth  wears  away  as  fast  as  it  is 
added  to  from  behind;  and  in  the  grown  animal,  all  those  portions 
of  the  tooth  near  to  the  pole  of  the  logarithmic  spiral  have  long 
disappeared.  In  the  elephant,  on  the  other  hand,  we  see,  practically 
speaking,  the  whole  unworn  tooth,  from  point  to  root; .  and  its 
actual  tip  nearly  coincides  with  the  pole  of  the  spiral.  If  we 
assume  (as  with  no  great  inaccuracy  we  may  do)  that  the  tip 
actually  coincides  with  the  pole,  then  we  may  very  easily  con- 
struct the  continuous  spiral  of  which  the  existing  tusk  constitutes 
a  part;  and  by  so  doing,  we  see  the  short,  gently  curved  tusk 
of  our  ordinary  elephant  growing  gradually  into  the  spiral  tusk 
of  the  mammoth.  No  doubt,  just  as  in  the  case  of  our  molluscan 
shells,  we  have  a  tendency  to  variation,  both  individual  and  specific, 
in  the  constant  angle  of  the  spiral ;  some  elephants,  and  some  species 
of  elephant,  undoubtedly  have  a  higher  spiral  angle  than  others. 
But  in  most  cases,  the  angle  would  seem  to  be  such  that  a  spiral 
configuration  would  become  very  manifest  indeed  if  only  the  tusk 


XIII]  OF  DOLPHINS'  TEETH  899 

pursued  its  steady  growth,  uiiclianged  otherwise  in  form,  till  it 
attained  the  dimensions  which  we  meet  with  in  the  mammoth.  In  a 
species  such  as  Mastodon  angustidens,  or  M.  arvernensis,  the  specific 
angle  is  low  and  the  tusk  comparatively  straight ;  but  the  American 
mastodons  and  the  existing  species  of  elephant  have  tusks  which  do 
not  differ  appreciably,  except  in  size,  from  the  great  spiral  tusks  of 
the  mammoth,  though  from  their  comparative  shortness  the  spiral 
is  httle  developed  and  only  appears  to  the  eye  as  a  gentle  curve. 
Wherever  the  tooth  is  very  long  indeed,  as  in  the  mammoth  or  the 
beaver,  the  effect  of  some  slight  and  all  but  inevitable  lateral 
asymmetry  in  the  rate  of  growth  begins  to  shew  itself:  in  other 
words,  the  spiral  is  seen  to  lie  not  absolutely  in  a  plane,  but  to  be 
a  gauche  curve,  like  a  twisted  horn.  We  see  this  condition  very 
well  in  the  huge  canine  tusks  of  the  babirussa;  it  is  a  conspicuous 
feature  in  the  mammoth,  and  it  is  more  or  less  perceptible  in  any 
large  tusk  of  the  ordinary  elephants. 

The  simplest  of  mammaUan  teeth  are,  Uke  those  of  reptiles,  conical 
bads  which  spring  by  single  roots  from  a  common  origin:  much  as 
the  pinnules  of  a  compound  leaf  spring  from  a  common  petiole. 
A  dolphin's  teeth  are  typical  of  what  .Cope*  called  a  haplodont 
dentition;  a  sloth's  (whether  degenerate  or  no)  are  no  further 
advanced ;  canines  remain  unaltered  throughout  the  mammaha,  and 
incisors  vary  little  save  for  some  flattening  due  to  crowding  in  a 
foreshortened  jaw.  Like  the  leaf  and  its  pinnules,  the  tooth-germ 
buds  and  branches  in  endless  ways;  and  we  have  no  criterion  of 
comparison  (nor  any  right  to  expect  it)  between  the  individual 
cusps  of  a  dog's,  an  elephant's  and  a  horse's  teeth,  any  more  than 
between  the  several  pinnules,  cusps  or  leaflets  of  a  rose,  a  maple 
and  a  horse-chestnut.  The  tooth-buds  remain  apart  or  coalesce  in 
various  numbers  and  degrees ;  and  croWding,  abrasion  and  mechanical 
pressure  play  a  large  part  in  the  final  arrangement  and  conformation  f. 

The  dolphin's  teeth,  used  only  for  prehension,  do  not  impinge 
on  one  another,  and  stay  sharp  accordingly ;  those  of  the  carnivores 

*  E.  D.  Cope,  On  the  homologies  and  origin  of  the  types  of  molar  teeth  in  mam- 
malian dentition,  Pr.  Ac.  N.  S.  Philad.  xxv,  p.  371,  1873;  Journ.  Ac.  N.  S.  Philad. 
(c),  vm,  pp.  71-89,  1874. 

t  Cf.  J.  A.  Ryder,  Mechanical  genesis  of  tooth  forms,  Pr.  Ac.  N.  S.  Philad. 
1878,  pp.  45-80. 


900  THE  SHAPES  OF  TEETH  [ch. 

interlock,  rather  than  meet  and  oppose* ;  in  herbivorous  animals  the 
molars  grind  one  against  another,  and  wear  their  crowns  away. 
The  teeth  of  ungulates  have  been  studied  with  especial  care  by  the 
palaeontologists  on  the  basis  of  Cope's  well-known  tritubercular 
theory,  and  one  is  greatly  daring  who  ventures  to  deal  with  them 
in  a  different  wayj.  The  case  is  neither  plain  nor  easy.  We  are 
accustomed  to  speak  of  a  "tooth"  as  a  single  unit,  however  com- 
plicated it  may  be ;  but  we  may  err  in  doing  so,  and  we  encounter 
other  difficulties  in  studying  teeth  whose  crowns  are  worn  away, 
and  in  interpreting  the  "patterns"  which  successive  stages  of  wear 
and  tear  expose. 

The  elephant's  molar  is  manifestly  composite.  We  see  on  its 
worn  surface  a  long  succession  of  "enamel  ridges,"  each  marking 
a  narrow  ring  or  island,  lying  transversely,  filled  with  dentine,  sur- 
rounded by  interstitial  cement,  and  with  a  root  or  roots  of  its  own. 
The  molars  develop  one  after  another  during  the  animal's  lifetime; 
and  each  consists,  to  begin  with,  of  so  many  separate  island-elements, 
not  yet  cemented  together  nor  worn  down,  each  with  its  own  roots, 
its  own  covering  of  enamel  and  its  own  transversely  cuspidate  crown. 
These  are  true  dental  units,  the  primitive  individual  "teeth",  corre- 
sponding to  the  still  simpler  teeth  of  the  dolphin ;  and  they  illustrate, 
and  go  far  to  confirm  the  view  that  the  molar  tooth  is  formed,  both 
here  and  elsewhere,  by  "concrescence "J.  These  rudimenta  dentium, 
as  old  Patrick  Blair  called  them,  or  denticules  as  Owen  did,  soon 
fuse  together,  and  begin  to  wear  down  as  soon  as  the  great  composite 
tooth  rolls  forward  and  emerges  from  the  gum.  As  each  denticule 
begins  to  wear  away,  it  first  appears  as  a  transverse  row  of  separate 
rings,  the  so-called  column^,  which  represent  the  cusps  of  the  original 
crown  and  vary  in  size,  number  and  proportion  with  the  species. 

*  This  is  precisely  what  Aristotle  means  when  he  describes  the  dog's  teeth  as 
car  char  odont,  br  sharklike,  i.e.  interlocking — Kapxcpobovra  yap  €<ttlu  6aa  eTraXXdrrei 
TOLS  686uTai  Ta<r  o^eTs,  H.A.,  ii,  501  a  18. 

t  See  E.  D.  Cope,  loc.  cit.  and  H.  F.  Osborn,  passim.  Cf.  also  W.  K.  Gregory, 
A  half  century  of  trituberculy,  Proc.  Am.  Phil.  Soc.  lxxiii,  pp.  169-317,  1934,  who 
says  that  "even  the  most  complex  molar  patterns  of  the  Ungulates  are  referable 
to  the  trituberculate  type,  in  strict  accord  with  the  steps  postulated  by  Cope  and 
Osborn." 

J  A  view  held  by  Gaudry,  Giebel,  Kiikental  and  others,  but  stoutly  opposed  by 
Cope  and  Osborn,  who  see  in  the  molar  tooth  a  single  unit,  complicated  by  "dif- 
ferentiation". 


XIII 


OF  ELEPHANTS'  TEETH 


901 

These  columns  soon  fuse  and  vanish  as  the  cusps  wear  down;   and 
each  denticule  now  appears  as  a  continuous  ring  of  enamel,  within 


Fig.  441, 


b  c 

Abnormal  incisor  teeth:  a,  b,  of  rabbit;  c,  of  beaver. 
After  Mcintosh. 


which  the  dentine  is  exposed  and  around  which  the  cement  accumu- 
lates*.    A  single  great  molar  is  made  up  of  nearly  a  dozen  of  these 


Fig.  442.     A  dental  unit,  or  element  of  the  composite  molar  tooth,  of  an  Indian 
elephant.     It  consists  of  five  "columns",  terminating  in  yet  unworn  "cusps". 

*  See  Blair's  Osteographia  elephantma,  1713,  Tab.  ni,  19;  also  the  figures  in 
F.  van  Gaver's  :6tude  de  la  tete  d'un  jeune  Elephant  d'Asie,  Ann.  Mus.  Marseille, 
XX,  1925,  Cf.  also  L.  Bolk,  Zur  Ontogenie  des  Elefantengebisses,  Odontologische 
Studien,  ni,  Leipzig,  1919. 


902 


THE  SHAPES  OF  TEETH 


[CH. 


S  :5i 


P^JS 


.^     S3 

II 


XIII]  OF  ELEPHANTS'  TEETH  903 

elements  in  the  African,  and  of  twice  as  many  (twice  as  much 
flattened  or  compressed)  in  the  Indian  elephant;  in  the  mastodon 
they  are  much  fewer  and  much  larger,  and  their  great  tuberculated 
crowns  never  wholly  wear  away.  In  an  old  but  admirable  paper 
on  the  Indian  elephant*,  Mr  John  Corse  says:  "The  number  of 
teeth  of  w^hich  a  grinder  is  composed  varies  from  four  to  twenty- 
three,  according  as  the  elephant  advances  in  years;  so  that  a 
grinder,  or  case  of  teeth,  in  full-grown  elephants,  is  more  than 
sufficient -to  fill  one  side  of  the  mouth....  The  same  number  of 
laminae  generally  fills  the  jaw  of  a  young  or  of  an  old  elephant; 
and  from  three  till  fifty  years  there  are  from  ten  to  twelve  teeth  or 
laminae  in  use,  in  each  side  of  either  jaw,  for  the  mastication  of 
the  food." 

The  molar  teeth  of  a  mouse,  a  hare  or  a  capybara  are  Hkewise 
composite  structures;  they  shew,  precisely  after  the  fashion  of  the 
elephant,  successive  narrow  annular  islands  of  enamel,  with  dentine 
within  and  cement  between,  all  in  varying  degrees  of  independence 
or  coalescence  I . 

The  molars  of  a  hippopotamus  are  composite  but  to  a  less  degree ; 
his  upper  molars  have  each  two  pair  of  roots,  the  last  molar  one 
root  more.  A  block  of  dentine  lying  transversely  to  the  jaw,  with 
a  pair  of  roots  below  and  a  pair  of  enamel-covered  cusps  above,  is  the 
unit  of  dentition,  and  is  analogous  to  the  young  toothlet  of  the 
elephant. 

In  the  horse  and  its  kind  the  teeth  are  long  and  deeply  sunk  in 
the  jaw,  very  much  as  in  a  rabbit  or  hare.  Their  length  is  made 
up  not  of  root  but  of  elongated  crown,  in  which  the  deep  valleys 
between  the  once  high  cusps  are  filled  or  flooded  with  cement;  and 
these  long  crowns  are  soon  worn  down  to  an  all  but  level  surface, 

*  J.  Corse,  Observations  on  the  different  species'  of  Asiatic  elephant,  and  their 
mode  of  dentition,  Phil.  Trans.  1799,  pp.  205-236;  and  cf.  Owen's  Comp.  Anat. 
m,  p.  361. 

t  The  elephant  (in  my  opinion)  shews  its  likeness  or  affinity  to  the  rodents 
throughout  its  whole  anatomy,  the  metaoromial  process  of  its  scapula  being  one 
conspicuous  indication.  Hyrax  and  Elephas  are  two  isolated  forms  lying  near  the 
common  origin  of  ungulates  and  rodents;  the  one  lying  rather  to  the  ungulate 
side,  the  other  to  the  rodent  side,  of  the  vague  and  indefinable  border-line.  On 
the  relation  of  the  rodent's  dentition  to  the  elephant's  (a  view  strongly  opposed 
by  Dr  \V.  K.  Gregory),  see  M.  Friant,  Contribution  a  Tetude.  .  .des  dents  jugales 
chez  les  Mammiferes,  Bull.  Mus.  Hist.  Nat.  i,  pp.  1-132,  1933. 


904  THE  SHAPES  OF  TEETH  [ch. 

in  which  the  enamel-layer  which  covered  the  hills  and  lined  the 
valleys  is  seen  in  sectional  contour.  A  horse's  incisor  is  the  simplest 
case.  On  its  worn  surface  we  see  an  inner  ring  of  enamel  concentric 
with  the  enamel  of  the  outer  edge  or  surface  of  the  tooth;  cement 
fills  up  the  inner  ring,  and  dentine  the  space  between.  The  tip  of 
the  tooth  has  sunk  down,  or  been  tucked  in,  till  it  forms  a  cement- 
filled  lake  on  the  top  of  the  hill;  the  lake  narrows  in,  and  at  last 
vanishes  as  the  horse  grows  old  and  the  tooth  wears  down;  in  the 
"aged"  horse  we  see  the  "mark"  no  more.  To  recognise  this  lake 
or  pit  in  the  simple  contours  of  the  young  incisor  is  an  easy  matter ; 
but  in  the  abraded  molar  the  enamel-layer  which  once  covered  all 
its  ups  and  downs  forms  a  contour-line,  or  "curve  of  level,"  of  great 
complexity.  This  contour-line  alters  as  the  levels  change,  and  varies 
from  one  tooth  to  the  next  and  from  one  year  to  another,  so  long  as 
wear  and  tear  continue.  The  geographer  reads  the  lie  of  the  land, 
with  all  its  ups  and  downs,  from  a  many-contoured  map*,  but  the 
worn  tooth  shews  us  only  one  level  and  one  contour  at  a  time ;  we 
must  eke  out  its  scanty  evidence  by  older  and  younger  teeth  in  other 
phases  or  degrees  of  wear.  The  "  pattern  "  of  a  horse's  molar  tooth  is 
indeed  so  closely  akin  to  a  map-maker's  contours  that  some  of  the 
terms  he  uses  may  be  useful  to  us.  He  speaks,  for  instance,  of  ridge- 
lines  and  course-lines,  lignes  defaite  and  lignes  de  thalweg;  of  a  gap, 
or  lowland  way  between  two  hills,  in  contrast  to  a  col  or  saddle  at 
the  summit  of  a  mountain-pass;  or  of  a  gorge,  which  is  a  narrow 
steep-sided  valley;  or  a  scarp,  which  is  a  long  steep-faced  hillside. 
We  must  take  care  all  the  while  to  see  which  side  of  our  contour-line 
is  positive  or  negative — on  which  side  the  ground  slopes  up  and  on 
which  down.  In  our  tooth  we  find  that  every  enamel-contour  has 
dentine  on  the  one  side  and  more  or  less  cement  on  the  other;  the 
dentine  belongs  to  the  closed  interior  of  the  tooth  itself,  and  on  the 
other  side  of  the  enamel-line  are  spaces  open  to  the  world. 

In  a  horse's  molar  we  see  the  sinuous  contours  of  two  small  lakes, 
remains  of  the  two  valleys  which  lay  between  the  three  transverse 
ridges  of  the  compound  tooth;    and  outside  the  enamel-edges  of 

*  Contour-lines  or  horizontals,  as  some  geographers  prefer  to  call  them,  were 
invented  by  Buache,  in  1752.  These  are  discussed  by  Cayley,  On  contours  and 
slope-lines,  Phil.  Mag.  xvni,  pp.  294-8,  1859;  and  by  Clerk  Maxwell,  On  hills  and 
dales,  ibid,  xl,  pp.  421-7,  1870. 


xni]  OF  CONTOUR  LINES  905 

these  contoured  lakes  is  the  dentinal  substance  of  the  tooth,  sur- 
rounded again  by  the  outer  covering  of  enamel.  The  space  between 
the  outer  and  the  inner  contours  is  narrowed  in  each  case  at  a 
certain  point,  suggesting  a  "col";  while  it  broadens  out  at  other 
places,  suggesting  the  former  sites  of  cusps  or  hills.  In  neighs 
bouring  sections  (B,  C)  we  rise  above  the  level  of  the  cols,  find  a 
way  open  to  the  valleys,  and  see  the  separate  transverse  mountain- 
ranges  (or  lophs)  of  which  the  tooth  is  composed.     The  general  plan 


Fig.  444.     Upper  molar  teeth  of  elephants.     A,  E.  meridionalis  (Pliocene),  largest 
of  elephants.     B,  Indian,  and  C,  African  elephants. 


of  this  tooth  is  characteristic  of  the  Perissodactyles ;  but  the  varying 
steepness  of  the  hills,  the  depth  of  the  valleys,  and  the  amount  of 
abrasion  or  erosion,  lead  to  an  infinite  variety  of  patterns,  varying 
with  the  species,  with  the  age  of  the  animal,  and  with  the  order 
of  succession  of  the  particular  tooth ;  and  so  rendering  it  (as  Osborn 
says)  "one  of  the  most  difficult  objects  to  define  and  describe  in 
the  whole  field  of  vertebrate  palaeontology*." 

In  Elasmotherium  the  hillsides  are  ridged  and  channelled,  and 
their  contours  folded  or  sinuous  accordingly.  In  Rhinoceros  broad 
gaps  replace  the  narrow  cols,  and  certain  jutting  crags  figure  on 

*  H.  F.  Osborn,  Equidae  of  the  Oligocene,  etc.,  Mem.  Amer.  Mus.  of  N.H.  (n.s.) 
n,  p.  3,  1918. 


906  THE  SHAPES  OF  TEETH  [ch. 

the  contour-lines  as  the  so-called  crochet  and  anticrochet.  In  Anchi- 
therium,  erosion  goes  no  farther  than  the  summits  of  the  several 
cusps  or  hill-tops. 

These,  to  my  thinking,  are  the  few  and  simple  lines  on  which  we 
may  study  the  architecture  of  the  Perissodactyle  tooth.  But  to  say 
how  far  we  may  rely  on  the  innumerable  minor  differences  of  pattern 


Fig.  445  A.  Third  upper  molar  of  a  horse,  a,  the  ecioloph,  with  its  three  styles,, 
separated  by  two  indents  (Owen);  b,  three  transverse  ridges,  the  protoloph,  mesoloph 
and  metaloph  (Osborn);  c,  c',  two  lakes,  valleys  or  fossettes ;  d,  d',  what  Owen  calls 
the  entries  of  the  valleys;  x,  x' ,  cols,  where  a  less  worn  tooth  would  shew  open  roads 
or  passes;  'o,  o,  cusps  or  conules,  the  sites  of  worn-down  hills  or  hillocks. 


B  C 

Fig.  445  B  and  C.     The  same  tooth,  but  younger  and  less  worn 
down  than  A.     Diagrammatic. 

as  evidence  of  blood-relationship  and  evolutionary  descent  is  quite 
another  story,  and  deserves  much  more  anxious  consideration*. 

*  In  the  vast  literature  of  mammalian  dentition  the  foUowing  are  conspicuous: 
R.  Owen,  Odontography,  1845;  L.  Riitimeyer,  Zur  Kenntniss  der  fossilen  Pferde, 
und  zu  einer  vergl.  Odontographie  der  Hufthiere,  Verh.  Naturf.  Ges.  Basel,  iii, 
1963;  W.  Leche,  Zur  Entwicklungsgeschichte  des  Zahnsystems  der  Saugetiere, 
Bibl.  Zool.  1894-5,  160  pp.;  E.  D.  Cope,  On  the  trituberculate  type  of  molar  tooth 
in  the  Mammalia,  Proc.  Amer.  Philos.  Sac.  xxi,  pp.  324-326,  1885;  W.  K.  Gregory, 
A  half-century  of  trituberculy,  ibid.  Lxxiii,  pp.  161-317,  1934. 


XIII]  OF  HORSES'  TEETH  907 

The  "horn"  or  tusk  of  the  narwhal  is  a  very  remarkable  and 
a  very  anomalous  thing.  It  is  the  only  tooth  in  the  creature's  head 
to  come  to  maturity;    it  grows  to  an  immense  and  apparently 


Kg.  446.     Enamel  patterns  (diagrammafic)  of  certain  fossil  Equidae. 
A,  Protohipptcs ;   B,  Hyohippus;   C,  Neohipptis. 

unwieldy  size,  say  to  eight  or  even  nine  feet  long;  it  never  curves 
nor  bends,  but  grows  as  straight  as  straight  can  be — a  very  singular 
and  exceptional  thing;  it  looks  as  though  it  were  twisted,  but  really 


Fig.  447.  Enamel  pattern  (diagrammatic)  of  the  upper  molar  teeth  of  Rhinoceros. 
The  back-tooth  (to  the  right-hand  side)  is  the  least  worn,  and  its  contour-line  lies 
at  the  highest  level. 

carries  on  its  straight  axis  a  screw  of  several  contiguous  low-pitched 
threads;  and  (last  and  most  anomalous  thing  of  all)  when,  as 
happens  now  and  then,  two  tusks  are  developed  instead  of  one,  one 
on  either  side,  these  two  do  not  form  a  conjugate  or  symmetrical 
pair,  they  are  not  mirror-images  of  one  another,  but  are  identical 
screws,  with  both  threhds  running  the  same  way*. 

*  The  male  narwhal  carries  the  horn,  the  female  being  tuskless;  but  the  whalers 
say  that  the  rare  two-horned  specimens  are  ull  females.  A  famous  two-horned 
skull  in  the  Hamburg  Museum  is  known  to  have  belonged  to  a  pregnant  female. 
It  was  brought  home  in  1684,  and  is  one  of  the  oldest  museum  specimens  in  the 
world;  the  tusks  measure  242  and  236  cm.  During  my  thirty  years'  close 
acquaintance  with  the  Dundee  whalers,  only  four  two-horned  narwhals  passed 
through  my  hands.  Bateson  {Problems  of  Genetics,  1913,  p.  44)  makes  the  curious 
remark  that  "the  Narwhal's  tusks,  in  being  both  twisted  in  the  same  direction, 
are  highly  anomalous,  and  are  comparable  with  pairs  of  twins." 


908  THE  SHAPES  OF  TEETH  [ch. 

All  ordinary  teeth,  as  we  have  seen,  have  their  own  natural 
curvature,  less  or  more,  which  becomes  more  manifest  and  con- 
spicuous the  longer,  they  grow.  We  cannot  suppose  that  the  field 
of  force  (internal  and  external)  in  which  the  narwhal's  tusk  develops 
is  so  simple  and  uniform  as  to  allow  it  to  grow  in  perfect  symmetry, 
year  after  year,  without  the  least  bias  or  intention  toward  either 
side ;  we  must  rather  suppose  that  the  resistances  which  the  growing 
tusk  encounters  average  out  and  cancel  one  another,  and  leave  no 
one-sided  resultant.  The  long,  straight,  tapering  tooth  is  commonly 
said  to  have  a  "spiral  twist,"  but  there  is  no  twist  at  all;  the  ivory 
is  straight-grained  and  uniform,  through  and  through.  The  tusk, 
in  short,  is  a  straight,  right-handed,  low-pitched  screw  or  helix,  with 
several  threads;  which  threads,  in  the  form  of  alternate  grooves 
and  ridges,  wind  evenly  and  continuously  from  one  end  of  the  tusk 
to  the  other,  even  extending  to  its  root,  deep-set  in  the  socket  or 
alveolus  of  the  upper  jaw. 

How  this  composite  spiral  thread  is  formed  is  quite  unknown.^ 
We  have  just  seen  that  it  is  not  due  to  any  twisting  of  the  dentinal 
axis  of  the  tooth.  That  it  is  uniform  and  unbroken  from  end  to 
end  shews  that  the  tooth  somehow  fashions  it  as  a  whole ;  and  that 
it  extends  deep  down  within  the  alveolus  is  enough  to  shew  that  it 
is  not  impressed  or  graven  on  the  tooth  by  any  external  agency. 
We  note,  as  a  minor  feature,  that  the  several  grooves  or  ridges  which 
constitute  the  composite  thread  have  their  individual  or  accidental 
differences;  a  broader  or  a  narrower  groove  continues  unchanged 
and  recognisable  from  one  end  of  the  tooth  to  the  other;  in  other 
words,  whatever  makes  each  ridge  or  groove  goes  on  acting  in  the 
selfsame  way,  as  long  as  growth  goes  on.  A  screw  is  made,  in 
general,  by  compounding  a  translatory  with  a  rotatory  motion,  and 
by  bringing  the  latter  into  relation  with  the  mould  or  matrix  by 
which  the  thread  is  fashioned  or  imposed;  and  I  cannot  see  how 
to  avoid  believing  that  the  narwhal's  tooth  must  revolve  in  like 
manner,  very  slowly  on  its  longitudinal  axis,  all  the  while  it  grows — 
however  strange,  anomalous  and  hard  to  imagine  such  a  mode  of 
growth  may  be.  We  know  that  the  tooth  grows  throughout  life  in 
its  longitudinal  direction,  the  open  root  and  "permanent  pulp" 
accounting  for  this ;  and  only  by  a  simultaneous  and  equally  con- 
tinuous rotation  (so  far  as  I  can  see)  can  we  account  for  the  perfect 


XIII]  OF  THE  NARWHAL'S  HORN  909 

straightness  of  the  tusk,  for  the  grooving  or  "rifling"  of  the  surface 
accompanied  by  no  internal  twist,  for  the  extension  of  that  rifling 
to  the  alveolar  portion  of  the  tusk  within  the  jaw,  and  for  the  fact 
that  the  several  associate  grooves  and  ridges  preserve  their  individual 
character  as  they  pass  along  and  wind  their  way  around.  A  very 
slow  rotation  is  all  we  need  demand — say  four  or  five  complete 
revolutions  of  the  tusk  in  the  whole  course  of  a  lifetime. 

The  progress  of  a  whale  or  dolphin  through  the  water  may  be 
explained  as  the  reaction  to  a  wave  which  is  caused  to  run  from 
head  to  tail,  the  creature  moving '  through  the  water  somewhat 
slower  than  the  wave  travels.  The  same  is  true,  so  far,  of  a  fish; 
but  the  w^ave  tends  to  be  in  one  plane  in  the  fish,  the  dorsal  and 
ventral  fins  helping  to  keep  it  so;  while  in  the  dolphin  it  may  be 
said  to  be  "circularly  polarised,"  or  resoluble  into  two  oscillations  in 
planes  normal  to  one  another,  and  caused  by  tail  and  tail-end  swishing 
around  in  circular  orbits  which  alter  in  phase  from  one  transverse 
section  to  another.  Just  as  in  the  case  of  a  screw-propeller,  or  as 
in  a  torpedo  (where  it  is  specially  corrected  or  compensated),  this 
mode  of  action  entails  a  certain  waste  of  energy;  it  comes  of  the 
development  of  a  "harmful  moment,"  which  tends  to  rotate  the 
body  about  its  axis,  and  to  screw  the  animal  along  its  course.  A 
slight  left-handed  curvature  of  the  dolphin's  tail  goes  some  little 
way  towards  correcting  this  tendency.  M.  Shuleikin's  study  of  the 
kinematics  of  the  dolphin  * — a  fine  piece  of  work  both  on  its  experi- 
mental and  its  theoretical  side — shows  the  dolphin  to  be  a  better 
swimmer  than  the  fish,  inasmuch  as  its  speed  of  progression  comes 
nearer  to  the  velocity  of  the  wave  which  is  propagated  along  its 
body;  the  so-called  "step,"  or  fraction  of  the  body-length  travelled 
in  a  single  period,  is  found  to  be  about  0-7  in  the  dolphin,  against 
0-57  in  a  fast-swimming  fish  (tunny  or  mackerel). 

Shuleikin  makes  the  curious  remark  that  the  asymmetry  of  the 
skull  (discernible  in  all  Cetacea),  which  in  the  dolphin  shews  a  screw- 
twist  with  a  pitch  about  equal  to  the  length  of  the  body,  acts  as 
a  compensatory  check  to  the  screw-component  in  the  creature's 
movement  of  progression,  and  that  "the  till  now  obscure  purpose 

*  Wassilev  Shuleikin,  Kinematics  of  a  dolphin  (Russian),  Bull.  Acad.  Sci.  U.R.S.S. 
{CI.  sci.  math,  et  phys.),  1935,  pp.  651-671;  also,  Dynamics,  external  and  internal, 
of  a  fish,  ibid.  1934,  pp.  1151-1186.  On  the  latter  subject  see  James  Gray, 
Crobnian  Lecture,  1940,  and  other  papers. 


910  THE  SHAPES  OF  TEETH  [ch. 

of  the  skull's  asymmetry"  is  accordingly  explained.  I  should  put 
this  differently,  and  suggest  that  this  counter-spirality  of  the  skull, 
is  the  direct  result  of  the  spiral  component  in  locomotion.  It  implies, 
I  take  it,  a  lagging  and  incomplete  response  in  the  fore-part  of  the 
body  to  the  rotatory  impulse  of  the  pai  ts  behind :  or,  in  the  plain 
words  of  the  engineer,  a  torque  of  inertia.    • 

This  tendency,  dimly  seen  in  the  dolphin's  skullj  is  clearly  demon- 
strated in  the  narwhal's  "horn,"  and  gives  a  complete  explanation 
of  its  many  singularities-.  The  narwhal  and  its  horn  are  joined 
together,  and  move  together  as  one  piece — nearly,  but  not  quite! 
Stiif,  straight  and  heavy,  the  great  tusk  has  its  centre  of  inertia 
well  ahead  of  the  animal,  and  far  from  the  driving  impulse  of  its 
tail.  At  each  powerful  stroke  of  the  tail  the  creature  not  only  darts 
forward,  but  twists  or  slews  all  of  a  sudden  to  one  side;  and  the 
heavy  horn,  held  only  by  its  root,  responds  (so  to  speak)  with 
difficulty.  For  at  its  slender  base  the  "couple",  by  which  it  has 
to  follow  the  twisting  of  the  body,  works  at  no  small  disadvantage. 
A  "torque  of  inertia"  is  bound  to  manifest  itself.  The  horn  does 
not  twist  round  in  perfect  synchronism  with  the  animal;  but  the 
animal  (so  to  speak)  goes  slowly,  slowly,  little  by  little,  round  its 
own  horn!  The  play  of  motion,  the  lag,  between  head  and  horn 
is  sUght  indeed;  but  it  is  repeated  with  every  stroke  of  the  tail. 
It  is  felt  just  at  the  growing  root,  the  permanent  pulp,  of  the  tooth; 
and  it  puts  a  strain,  or  exercises  a  torque,  at  the  very  seat,  and  during 
the  very  process,  of  calcification. 

Suppose  that  at  every  sweep  of  the  tail  there  be  a  lag  of  no  more 
than  a  fifth  part  of  a  second  of  arc*  between  the  rotation  of  the 
tusk  and  of  the  body,  that  small  amount  would  amply  suffice  to 
account,  on  a  rough  estimate  of  the  age  and  of  the  activity  of  the 
animal,  for  as  many  turns  of  the  screw  as  a  fair-sized  tusk  is  found 
to  exhibit. 

According  to  this  explanation,  or  hypothesis,  the  slow  rotation 
of  the  tusk  corrects  all  tendency  to  flexure  or  curvature  in  one 
direction  or  another ;  the  grooves  and  ridges  which  constitute  th6 
"thread"  of  the  screw  are  the  result  of  irregularities  or  inequalities 
within  the  alveolus,  which  "rifle"  the  tusk  as  it  grows;    and  the 

*  Or  say  a  hundred-thousandth  part  of  the  angle  subtended  by  a  minute  on  the 
clock. 


XIII]  THE  TIGHT  FIT  OF  THE  TEETH  911 

identity  of  direction  in  the  two  horns  of  a  pair  is  at  once  accounted 
for. 

Beautiful  as  the  spiral  pattern  of  the  tusk  is,  it  obviously  falls 
short,  in  regularity  and  elegance,  of  what  we  find,  for  instance,  in 
a  long  tapering  Terebra  or  Turritella,  or  any  other  spiral  gasteropod 
shell.  In  the  narwhal  we  have,  as  we  suppose,  only  a  general  and 
never  a  precise  agreement  between  rate  of  torsion  and  rate  of  growth ; 
for  these  two  velocities — of  translation  and  rotation — are  separate 
and  independent,  and  their  resultant  keeps  fairly  steady  but  no 
more.  In  the  snail-shell,  on  the  other  hand,  actual  tissue-growth 
is  the  common  cause  of  both  longitudinal  and  torsional  displace- 
ments, and  the  resultant  spiral  is  very  perfect  and  regular. 

Before  we  leave  the  teeth,  let  us  note  that  their  extreme  tightness 
in  their  sockets  is  a  remarkable  thing.  A  thin  "periodontal  mem- 
brane," less  than  0-25  mm.  thick,  fills  up  the  space  between  tooth  and 
socket ;  and  this  membrane,  elastic,  homogeneous  and  incompressible, 
is  analogous  to  the  thin  layer  of  viscous  liquid  dealt  with  in  modern 
theories  of  lubrication.  The  equilibrium  of  the  system,  the  tightness 
of  the  fit,  the  displacement  of  the  tooth  under  given  forces,  and 
the  conditions  of  stress  and  strain  in  the  membrane,  are  all  open  to 
mathematical  treatment ;  distributions  of  pressure  can  be  assigned 
to  the  tooth,  a  centre  of  rotation  can  be  found,  a  critical  load  can 
be  approximately  determined,  and  the  pressures  calculated  at  various 
points.  If  the  membrane  thickens,  the  tooth  loosens;  its  freedom 
of  movement  or  range  of  displacement  varies  with  the  cube  of  the 
thickness  of  the  membrane,  and  is  at  most  exceedingly  small*. 

*  J.  L.  Synge,  The  tightness  of  the  teeth,  etc.,  Phil.  Trans.  (A),  ccxxxi,  pp.  435- 
477,  1933. 


CHAPTER  XIV 

ON  LEAF-ARRANGEMENT,  OR  PHYLLOTAXIS 

The  beautiful  configurations  produced  by  the  orderly  arrangement 
of  leaves  or  florets  on  a  stem  have  long  been  an  object  of  admiration 
and  curiosity;  and  not  the  least  curious  feature  of  the  case  is  the 
limited,  even  the  small  number  of  possible  arrangements  which  we 
observe  and  recognise.  Leonardo  da  Vinci  would  seem,  as  Sir  Theodore 
Cook  tells  us,  to  have  been  the  first  to  record  his  thoughts  upon  this 
subject;  but  the  old  Greek  and  Egyptian  geometers  are  not  likely 
to  have  left  unstudied  or  unobserved  the  spiral  traces  of  the  leaves 
upon  a  palm-stem,  or  the  spiral  order  of  the  petals  of  a  lotus  or  the 
florets  in  a  sunflower.  For  so,  as  old  Nehemiah  Grew  says,  "from 
the  contemplation  of  Plants,  men  might  first  be  invited  to  Mathe- 
matical Enquirys*." 

The  spiral  leaf-order  has  been  regarded  by  many  learned  botanists 
as  involving  a  fundamental  law  of  growth,  of  the  deepest  and  most 
far-reaching  importance;  while  others,  such  as  Sachs,  have  looked 
upon  the  whole  doctrine  of  " phyllotaxis "  as  "a  sort  of  geometrical 
or  arithmetical  playing  with  ideas,"  and  "the  spiral  theory  as  a 
mode  of  view  gratuitously  introduced  into  the  plant."  Sachs  even 
went  so  far  as  to  declare  this  doctrine  to  be  "in  direct  opposition 
to  scientific  investigation,  and  based  upon  the  idealism  of  the 
Naturphilosophie " — the  mystical  biology  of  Oken  and  his  school. 

The  essential  facts  of  the  case  are  not  difficult  to  understand; 
but  the  theories  built  upon  them  are  so  varied,  so  conflicting,  and 
sometimes  so  obscure,  that  we  must  not  attempt  to  submit  them 
to  detailed  analysis  and  criticism.  There  are  said  to  be  two  chief 
ways  by  which  we  may  approach  the  question,  according  to  whether 
we  regard  as  the  more  fundamental  and  typical,  one  or  other  of 
two  chief  modes  in  which  the  phenomenon  presents  itself.  That  is 
to  say,  we  may  hold  that  the  phenomenon  is  displayed  in  its  essential 

*  N.  Grew,  The  Anatomy  of  Plants,  1682,  p.  152. 


ON  LEAF-ARRANGEMENT 


913 


CH.  XIV] 

simplicity .  by  the  corkscrew  spirals,  or  helices,  which  mark  the 
position  of  the  leaves  on  a  cylindrical  stem  or  tapering  fir-cone ;  or,  on 


Fig.  448.     A  giant  sunflower,  Helianthus  maximus.     From  H.  A.  Naber, 
after  M.  Brocard. 

the  other  hand,  we  may  be  more  attracted  by,  and  may  regard  as  of 
greater  importance,  the  spirals  traced  by  the  curving  rows  of  florets 
in  the  discoidal  inflorescence  of  a  sunflower.     Whether  one  way  or 


9U  ON  LEAF-ARRANGEMENT  [ch. 

the  other  be  the  better,  or  even  whether  one  be  not  positively  correct 
and  the  other  radically  wrong,  has  been  vehemently  debated ;  but  as 
a  matter  of  fact  they  are,  both  mathematically  and  biologically, 
inseparable  and  even  identical  phenomena.  For  the  face  of  the 
sunflower   is    but    a    shortened    stem,    and   the    curves  upon   its 


Fig.  449.     A  cauliflower,  its  composite  inflorescence  shewing  spiral  patterns 
of  the  first  and  second  order. 


surfaces  are  but  the  projection  on  a  plane  of  a  more  elongated 
inflorescence. 

We  speak,  as  botanists  are  wont  to  do,  of  these  spirals  of  sunflower, 
cauliflower  and  the  rest  as  logarithmic  spirals,  but  not  without 
hesitation.  They  doubtless  resemble  the  logarithmic  or  equiangular 
spiral,  but  different  spirals  may  look  much  alike;  and  these  are 
ill-suited  to  the  careful  admeasurement  and  rigorous  verification 


XIV]  OR  PHYLLOTAXIS  915 

which  Moseley  gave  to  the  spirals  of  his  moUuscan  shells*.  But  in 
the  sunflower,  to  judge  by  the  eye,  the  spirals  remain  self-similar 
as  they  grow;  each  fresh  increment  forms,  or  seems  to  form,  a 
gnofnon  to  what  went  before ;  each  new  floret  falls  into  hne  as  part 
of  a  continuous  and  self-§imilar  curve :  and  this  goes  a  long  way  to 
justify  our  use  of  the  famihar  term  logarithmic,  or  equiangular  spiral. 
But  the  leaf-arrangement  or  the  inflorescence  are  far  less  simple 
than  the  shell.  The  shell  grew  as  one  continuous  and  indivisible 
whole;  its  tip  is  the  oldest  part,  it  remains  the  smallest  part,  and 
the  spiral  tube  expands  continuously  as  it  goes  on.  But  each  floret 
of  the  sunflower  has  its  own  separate  and  individual  growth;  the 
oldest  is  also  the  largest,  and  the  youngest  is  the  least;  and  as 
younger  and  younger  florets  are  added  on,  the  spiral  advances  in 
the  direction  of  its  own  focus,  or  its  own  httle  end.  And  the  con- 
ditions may  be  less  simple  still  in  other  cases,  as  in  the  fir-cone 
itself. 

The  spiral  tesselation  of  th-e  fir-cone  was  carefully  studied  in  the 
middle  of  the  eighteenth  century  by  the  celebrated  Bonnet,  with  the 
help  of  Calandrini  the  mathematician.  Memoirs  pubHshed  about  1 835, 
by  Schimper  and  Braun,  greatly  amphfied  Bonnet's  investigations, 
and  introduced  a  nomenclature  which  still  holds  its  own  in  botanical 
textbooks.  Naumann  and  the  brothers  Bravais  are  among  those 
who  continued  the  investigation  in  the  years  immediately  following, 
and  Hofmeister,  in  1868,  gave  an  admirable  account  and  summary 
of  the  work  of  these  and  many  other  writers  |. 

*  Thus  Dr  A.  H.  Church,  in  his  Interpretation  of  Phyllotaxis  Phenomena,  1920, 
p.  3,  begins  by  saying  that  "angular  measurements  on  actual  plant-specimens.  . . 
can  never  hope  to  come  within  a  range  of  accuracy  admitting  of  an  error  of  less 
than  half  a  degree,  while  precise  mathematical  theory  soon  begins  to  tabulate 
minutes  and  seconds." 

f  Besides  papers  referred  to  below,  and  many  others  quoted  in  Sachs's  Botany 
and  elsewhere,  the  following  are  important:  Alex.  Braun,  Vergl.  Untersuchung 
liber  die  Ordnung  der  Schuppen  an  den  Tannenzapfen,  etc.,  Nova  Acta  Acad.  Car. 
Leop.  XV,  pp.  199-401,  1831;  C.  F.  Schimper's  Vortrage  iiber  die  Moglichkeit 
eines  wissenschaftlichen  Verstandnisses  der  Blattstellung,  etc.,  Flora,  xvui, 
pp.  145-191,  737-756,  1835;  C.  F.  Schimper,  Geometrische  Anordnung  der  um 
eine  Achse  peripherischen  Blattgebilde,  Verhandl.  Schweiz.  Ges.  1836,  pp.  113-117; 
L.  and  A.  Bravais,  Essai  sur  la  disposition  des  feuiUes  curviseriees,  Ann.  Sci.  Nat. 
(2),  VII,  pp.  42-110,  1837;  Sur  la  disposition  symetrique  des  inflorescences, 
ibid.  pp.  193-221,  291-348,  vni,  pp.  11^2,  1838;  Sur  la  disposition  generale  des 
feuilles  rectiseriees,  ibid,  xn,  pp.  5-41,  65-77,  1839;  Memoire  sur  la  disposition 
geometrique  des  feuilles  et  des  inflorescences,  Paris,  1838;   Zeising,  Normalverhdltnisa 


916  ON  LEAF-ARRANGEMENT  [ch. 

The  surface  of  a  pine-cone  shews  a  crowded  assemblage  of  woody 
scales,  close-packed  and  pressed  together  in  such  a  way  that  each 
has  a  quadrangular,  rhomboidal  form*.  Each  scale  forms  part  of, 
and  marks  the  intersection  of,  two  hnear  series;  these  run  upwards 
in  a  spiral  course,  one  in  one  direction  and  one  in  the  other,  and 
are  called  accordingly  diadromous  si^iials.  In  the  little  cones  of  the 
Scotch  Fir  (Pinus  silvestris),  the  whose  assemblage  of  scales  may  be 
looked  on  as  forming  five  linear  series,  or  spiral  bands,  running  side 
by  side  the  one  way,  or  as  eight  such  series  running  the  other.  But 
these  two  sets  are  far  from  being  all  the  spirals  which  we  can  trace 
upon  the  cone.  Sometimes  the  packing  is  closer  still,  especially  if 
the  cone  be  long  and  slender.  Then  each  scale  tends  to  come  in 
contact  with  six  others,  and  so  to  become  roughly  hexagonal;  we 
recognise  a  third  spiral  series  besides  the  other  two,  and  this  new 
series  is  found  to  consist  of  thirteen  rows.  But  let  us  disregard  for 
the  moment  this  perplexing  phenomenon  of  a  cone  composed  of  so 
many  series  of  scales,  five,  eight  or  thirteen  in  number  as  we  happen 
to  look  at  them ;  and  try  to  find  a  single  series  in  which  every  scale 
takes  part.  We  are  in  no  way  limited  to  the  fir-cone,  which  is  a 
somewhat  special  case;  but  may  consider,  in  a  very  general  way, 
the  case  of  any  leafy  stem. 

Starting  from  some  given  level  and  proceeding  upwards,  let  us 
mark  the  position  of  some  one  leaf  (A)  upon  the  cylindrical  stem. 

der  chemischen  und  morphologischen  Proportionen,  Leipzig,  1856;  C.  F.  Naumann, 
Ueber  den  Quincunx  als  Gesetz  der  Blattstellung  bei  Sigiilaria,  etc.,  Neues  Jahrb. 
f.  Miner.  1842,  pp.  410-417;  T.  Lestiboudois,  Phyllotaxie  anatomique,  Paris,  1848; 
G.  Henslow,  Phyllotaxis,  or  the  arrangement  of  leaves  according  to  mathematical 
laws,  Jl.  Victoria  Inst,  vi,  pp.- 129-140,  1873;  On  the  origin  of  the  prevailing 
systems  of  Phyllotaxis,  Tr.  Linn.  Soc.  (Bot.),  i,  pp.  37-45,  1880.  J.  Wiesner, 
Bemerkungen  xiber  rationale  und  irrationale  Divergenzen,  Flora,  Lvin,  pp.  113-115, 
139-143,  1875;  H.  Airy,  On  leaf  arrangement,  Proc.  B.S.  xxi,  p.  176,  1873; 
S.  Schwendener,  Mechanische  Theorie  der  Blattstellungen,  Leipzig,  1878;  F.  Delpino, 
Causa  mecca'nica  della  filotasse  quincunciale,  Genova,  1880;  Teoria  generate  di 
Filotasse,  ibid.  1883;  S.  Giinther,  Das  mathematische  Grundgesetz  im  Bau  des 
Pflanzenkorpers,  Kosmos,  iv,  pp.  270-284,  1879;  F.  Ludwig,  Wichtige  Abschnitte 
aus  der  mathematischen  Botanik,  Zeitschr.  f.  truithem.  u.  naturw.  Unterricht,  xiv, 
p.  161,  1883;  Weiteres  iiber  Fibonacci-Kurven  und  die  numerische  Variation  der 
gesammten  Bluthenstande  der  Kompositen,  Botan.  Chit,  lxviii,  p.  1,  1896;  Alex. 
Dickson,  Phyllotaxis  of  Lepidodendron  and  Knossia,  Jl.  Bot.  ix,  p.  166,  1871.  For 
a  historical  account  of  the  earlier  literature,  see  Casimir  de  Candolle's  Considerations 
generates  sur  Vetude  de  la  phyllotaxie,  Geneve,  1881. 
*  Cf.  supra,  p.  515. 


XIV]  OR  PHYLLOTAXIS  917 

Another,  and  a  younger  leaf  (B)  will  be  found  standing  at  a  certain 
distance  around  the  stem,  and  a  certain  distance  along  the  stem, 
from  the  first.  The  former  distance  may  be  expressed  as  a  fractional 
"divergence"  (such  as  two-fifths  of  the  circumference  of  the  stem) 
as  the  botanists  describe  it,  or  by  an  "angle  of  azimuth"  (such  as 
(f)  =  144°)  as  the  mathematician  would  be  more  likely  to  state  it. 
The  position  of  B  relatively  to  A  may  be  determined,  not  only  by 
this  angle  (/>,  in  the  horizontal  plane,  but  also  by  an  angle  of  slope  {6), 
or  merely  by  linear  distance  from  its  basal  plane;  for  the  height  of 
B  above  the  level  of  ^,  in  comparison  with  the  diameter  of  the 
cylinder,  will  obviously  make  a  great  difference  in  the  appearance  of 
the  whole  system.  But  this  matter  botanical  students  have  not 
concerned  themselves  with ;  in  other  words,  their  studies  have  been 
limited  (or  mainly  limited)  to  the  relation  of  the  leaves  to  one  ailother 
in  azimuth — in  other  words,  to  the  angle  ^  and  its  multiples. 

Whatever  relation  we  have  found  between  A  and  B,  let  precisely 
the  same  relation  subsist  between  B  and  C:  and  so  on.  Let  the 
growth  of  the  system,  that  is  to  say,  be  continuous  and  uniform; 
it  is  then  evident  that  we  have  the  elementary  conditions  for  the 
development  of  a  simple  cyhndrical  helix;  and  this  "primary  helix" 
or  "genetic  spiral"  we  can  now  trace,  winding  round  and  round  the 
stem,  through  A,  B,C,  etc.  But  if  we  can  trace  such  a  helix  through 
A,  B,  C,  it  follows  from  the  symmetry  of  the  system,  that  we  have 
only  to  join  A  to  some  other  leaf  to  trace  another  spiral  helix,  such, 
for  instance,  as  ^,  C,  E,  etc. ;  parallel  to  which  will  run  another  and 
similar  one,  namely  in  this  case  B,  D,  F,  etc.  And  these  spirals 
will  run  in  the  opposite  direction  to  the  spiral  ABC*. 

In  short,  the  existence  of  one  helical  arrangement  of  points  im- 
plies and  involves  the  existence  of  another  and  then  another  helical 
pattern,  just  as,  in  the  pattern  of  a  wall-paper,  our  eye  travels  from 
one  linear  series  to  another. 

A  modification  of  the  helical  system  will  be  introduced  when, 
instead  of  the  leaves  appearing,  or  standing,  in  singular  succession, 
w^  get  two  or  more  appearing  simultaneously  upon  the  same  level. 
If  there  be  two  such,  then  we  shall  have  two  generating  spirals 

*  For  the  spiral  ACE  to  be  different  from  ABC,  the  angle  of  divergence,  or  angle 
of  azimuth  for  one  step,  must  exceed  90°,  so  that  the  nearer  way  from  ^  to  C  is 
backwards;  otherwise  the  spiral  ACE  is  ABCDE,  or  ABC  over  again. 


918  ON  LEAF-ARRANGEMENT  [ch. 

precisely  equivalent  to  one  another;  and  we  may  call  them  A,  B,  C, 
etc.,  and  A',  B',  C\  and  so  on.  These  are  the  cases  which  we  call 
"whorled"  leaves;  or  in  the  simplest  case,  where  the  whorl  consists 
of  two  opposite  leaves  only,  we  call  them  *'  decussate." 

Among  the  phenomena  of  phyllotaxis,  two  points  in  particular 
have  been  found  difficult  of  explanation,  and  have  aroused  dis- 
cussion. These  are  (1),  the  presence  of  the  logarithmic  spirals  such 
as  we  have  already  spoken  of  in  the  sunflower;  and  (2)  the  fact 
that,  as  regards  the  number  of  the  helical  or  spiral  rows,  certain 
numerical  coincidences  are  apt  to  recur  again  and  again,  to  the 
exclusion  of  others,  and  so  to  become  characteristic  features  of  the 
phenomenon. 

As  to  the  first  of  these,  we  have  seen  that  the  curves  resemble, 
and  sometimes  closely  resemble,  the  logarithmic  spiral;  but  that 
they  are,  strictly  speaking,  logarithmic  is  neither  proved  nor  capable 
of  proof.  That  they  appear  as  spiral  curves  (whether  equable  or 
logarithmic)  is  then  a  mere  matter  of  mathematical  "deformation." 
The  stem  which  we  have  begun  to  speak  of  as  a  cyhnder  is  not 
strictly  so,  inasmuch  as  it  tapers  off  towards  its  summit.  The 
curve  which  winds  evenly  around  this  stem  is,  accordingly,  not  a 
true  helix,  for  that  term  is  confined  to  the  curve  which  winds  evenly 
around  the  cylinder :  it  is  a  curve  in  space  which  (like  the  spiral  curve 
we  have  studied  in  our  turbinate  shells)  partakes  of  the  characters  of 
a  helix  and  of  a  spiral,  and  which  is  in  fact  a  spiral  with  its  pole 
drawn  out  of  its  original  plane  by  a  force  acting  in  the  direction  of 
the  axis.  If  we  imagine  a  tapering  cylinder,  or  cone,  projected  by 
vertical  projection  on  a  plane,  it  becomes  a  circular  disc;  and  a 
helix  described  about  the  cone  becomes  in  the  disc  a  spiral  described 
about  a  pole  which  corresponds  to  the  apex  of  our  cone.  In  like 
manner  we  may  project  an  identical  spiral  in  space  upon  such  surfaces 
as  (for  instance)  a  portion  of  a  sphere  or  of  an  ellipsoid ;  and  in  all 
these  cases  we  preserve  the  spiral  configuration,  which  is  the  more 
clearly  brought  into  view  the  more  we  reduce  the  vertical  component 
by  which  it  was  accompanied.  The  converse  is  equally  true,  and 
equally  obvious,  namely  that  any  spiral  traced  upon  a  circular  disc 
or  spheroidal  surface  will  be  transformed  into  a  corresponding  spiral 
helix  when  the  plane  or  spheroidal  disc  is  extended  into  an  elongated 


XIV]  OR  PHYLLOTAXIS  919 

cone  approximating  to  a  cylinder.  This  mathematical  conception  is 
translated,  in  botany,  into  actual  fact.  The  fir-cone  may  be  looked 
upon  as  a  cylindrical  axis  coiltracted  at  both  ends,  until  it  becomes 
approximately  an  ellipsoidal  solid  of  revolution,  generated  about 
the  long  axis  of  the  ellipse ;  and  the  semi-ellipsoidal  capitulum  of  the 
teasel,  the  more  or  less  hemispherical  one  of  the  thistle,  and  the 
flattened  but  still  convex  one  of  the  sunflower,  are  all  beautiful  and 
successive  deformations  of  what  is  typically  a  long,  conical,  and  all 
but  cylindrical  stem.  On  the  other  hand,  every  stem  as  it  grows 
out  into  its  long  cylindrical  shape  is  but  a  deformation  of  the  little 
spheroidal  or  ellipsoidal  or  conical  surface  which  was  its  forerunner 
in  the  bud. 

This  identity  of  the  helical  spirals  around  the  stem  with  spirals 
projected  on  a  plane  was  clearly  recognised  by  Hofmeister,  who  was 
accustomed  to  represent  his  diagrams  of  leaf-arrangement  either  in 
one  way  or  the  other,  though  not  in  a  strictly  geometrical  projection*. 

According  to  Mr  A.  H.  Church  f,  who  has  dealt  carefully  and 
elaborately  with  the  whole  question  of  phyllotaxis,  the  spirals  such 
as  we  see  in  the  disc  of  the  sunflower  have  a  far  greater  importance 
and  a  far  deeper  meaning  than  this  brief  treatment  of  mine  would 
accord  to  them :  and  Sir  Theodore  Cook,  in  his  book  on  the  Curves 
of  Life,  adopted  and  helped  to  expound  and  popularise  Mr  Church's 
investigations. 

Mr  Church,  regarding  the  problem  as  one  of  "uniform  growth," 
easily  arrives  at  the  conclusion  that,  if  this  growth  can  be  conceived 
as  taking  place  sjnnmetrically  about  a  central  point  or  "pole,"  the 
uniform  growth  would  then  manifest  itself  in  logarithmic  spirals, 
including  df  course  the  limiting  cases  of  the  circle  and  straight  line. 
With  this  statement  I  have  Httle  fault  to  find;  it  is  in  essence 
identical  with  much  that  I  have  said  in  a  previous  chapter.  But 
other  statements  of  Mr  Church's,  and  many  theories  woven  about 
them  by  Sir  T.  Cook  and  himself,  I  am  less  able  to  follow.  Mr 
Church  tells  us  that  the  essential  phenomenon  in  the  sunflower  disc 
is  a  series  of  orthogonally  intersecting  logarithmic  spirals.  Unless 
I  wholly  misapprehend  Mr  Church's  meaning,  I  should  say  that  this 

*  Allgemeine  Morphologie  der  Gewdchse,  1868,  p.  442,  etc. 

t  Relation  of  Phyllotaxis  to  Mechanical  Laws,  Oxford,  1901-1903;  cf.  Ann.  Bot, 
XV,  p.  481,  1901. 


920  ON  LEAF-ARRANGEMENT  [ch. 

is  very  far  from  essential.  The  spirals  intersect  isogonally,  but 
orthogonal  intersection  would  be  only  one  particular  case,  and  in  all 
probability  a  very  infrequent  one,  in  the  intersection  of  logarithmic 
spirals  developed  about  a  common  pole.  Again  on  the  analogy  of 
the  hydrodynamic  lines  of  force  in  certain  vortex  movements,  and  of 
similar  lines  of  force  in  certain  magnetic  phenomena,  Mr  Church 
proceeds  to  argue  that  the  energies  of  life  follow  lines  comparable 
to  those  of  electric  energy,  and  that  the  logarithmic  spirals  of  the 
sunflower  are,  so  to  speak,  lines  of  equipotential*.  And  Sir  T.  Cook 
remarks  that  thife  "theory,  if  correct,  would  be  fundamental  for  all 
forms  of  growth,  though  it  would  be  more  easily  observed  in  plant 
construction  than  in  animals."  But  the  physical  analogies  are  remote, 
and  the  deductions  I  am  not  able  to  follow. 

Mr  Church  sees  in  phyllotaxis  an  organic  mystery,  a  something 
for  which  we  are  unable  to  suggest  any  precise  cause :  a  phenomenon 
which  is  to  be  referred,  somehow,  to  waves  of  growth  emanating 
from  a  centre,  but  on  the  other  hand  not  to  be  explained  by  the 
division  of  an  apical  cell,  or  any  other  histological  factor.  As  Sir 
T.  Cook  puts  it,  "at  the  growing  point  of  a  plant  where  the  new 
members  are  being  formed,  there  is  simply  nothing  to  see.'' 

But  it  is  impossible  to  deal  satisfactorily,  in  brief  space,  either 
with  Mr  Church's  theories,  or  my  own  objections  to  themj*.  Let 
it  suffice  to  say  that  I,  for  my  part,  see  no  subtle  mystery  in  the 
matter,  other  than  what  lies  in  the  steady  production  of  similar 
growing  parts,  similarly  situated,  at  similar  successive  intervals  of 
time.     If  such  be  the  case,  then  we  are  bound  to  have  in  consequence 

*  "The  proposition  is  that  the  genetic  spiral  is  a  logarithmic  spiral,  homologous 
with  the  line  of  current-flow  in  a  spiral  vortex ;  and  that  in  such  a  system  the 
action  of  orthogonal  forces  will  be  mapped  out  by  other  orthogonally  intersecting 
logarithmic  spirals— the  '  parastichies ' " ;   Church,  op.  cit.  i,  p.  42. 

t  Mr  Church's  whole  theory,  if  it  be  not  based  upon,  is  interwoven  with,  Sachs's 
theory  of  the  orthogonal  intersection  of  cell- walls,  and  the  elaborate  theories  of 
the  symmetry  of  a  growing  point  or  apical  cell  which  are  connected  therewith. 
According  to  Mr  Church,  "the  law  of  the  orthogonal  intersection  of  cell- walls  at 
a  growing  apex  may  be  taken  as  generally  accepted"  (p.  32);  but  I  have  taken 
a  very  different  view  of  Sachs's  law,  in  the  eighth  chapter  of  the  present  book. 
With  regard  to  his  own  and  Sachs's  hypotheses,  Mr  Church  .makes  the  following 
curious  remark  (p.  42) :  "Nor  are  the  hypotheses  here  put  forward  more  imaginative 
than  that  of  the  paraboloid  apex  of  Sachs  which  remains  incapable  of  proof,  or  his 
construction  for  the  apical  cell  of  Pteris  which  does  not  satisfy  the  evidence  of  his 
own  drawings." 


XIV]  OR  PHYLLOTAXIS  \  921 

a  series  of  symmetrical  patterns,  whose  nature  will  depend  upon  the 
form  of  the  entire  surface.  If  the  surface  be  that  of  a  cylinder,  we 
shall  have  a  system,  or  systems,  of  spiral  helices:  if  it  be  a  plane 
with  an  infinitely  distant  focus,  such  as  we  obtain  by  "unwrapping" 
our  cylindrical  surface,  we  shall  have  straight  lines;  if  it  be  a  plane 
containing  the  focus  within  itself,  or  if  it  be  any  other  symmetrical 
surface  containing  the  focus,  then  we  shall  have  a  system  of  loga- 
rithmic spirals.  The  appearance  of  these  spirals  is  sometimes  spoken 
of  as  a  "subjective"  phenomenon,  but  the  description  is  inaccurate: 
it  is  a  purely  mathematical  phenomenon,  an  inseparable  secondary 
result  of  other  arrangements  which  we,  for  the  time  being,  regard 
as  primary.  When  the  bricklayer  builds  a  factory  chimney,  he  lays 
his  bricks  in  a  certain  steady,  orderly  way,  with  no  thought  of  the 
spiral  patterns  to  which  this  orderly  sequence  inevitably  leads,  and 
which  spiral  patterns  are  by  no  means  "subjective."  The  designer 
of  a  wall-paper  not  only  has  no  intention  of  producing  a  pattern  of 
criss-cross  lines,  but  on  the  contrary  he  does  his  best  to  avoid  them ; 
nevertheless,  so  long  as  his  design  is  a  symmetrical  one,  the  criss-cross 
intersections  inevitably  come.  And  as  the  train  carries  us  past  an 
orchard  we  see  not  one  single  symmetrical  configuration,  but  a 
multiplicity  of  collineations  among  the  trees. 

Let  us,  however,  lea'C^e  this  discussion,  and  return  to  the  facts  of 
the  case. 

Our  second  question,  which  relates  to  the  numerical  coincidences 
so  familiar  to  all  students  of  phyllotaxis,  is  not  to  be  set  and 
answered  in  a  word. 

Let  us,  for  simplicity's  sake,  avoid  consideration  of  simultaneous 
or  whorled  leaf  origins,  and  consider  only  the  more  frequent  cases 
where  a  single  "genetic  spiral"  can  be  traced  throughout  the 'entire 
system. 

It  is  seldom  that  this  primary,  genetic  spiral  catches  the  eye,  for 
the  leaves  which  immediately  succeed  one  another  in  this  genetic 
order  are  usually  far  apart  on  the  circumference  of  the  stem,  and  it 
is  only  in  close-packed  arrangements  that  the  eye  readily  apprehends 
the  continuous  series.  Accordingly  in  such  a  case  as  a  fir-cone,  for 
instance,  it  is  certain  of  the  secondary  spirals  or  "  parastichies " 
which  catch  the  eye ;  and  among  fir-cones,  we  can  easily  count  these, 


922  ON  LEAF-ARRANGEMENT  [ch. 

and  we  find  them  to  be  on  the  whole  very  constant  in  number, 
according  to  the  species. 

Thus  in  many  cones,  such  as  those  of  the  Norway  spruce,  we  can 
trace  five  rows  of  scales  winding  steeply  up  the  cone  in  one  direction, 
and  three  rows  winding  less  steeply  the  other  way ;  in  certain  other 
species,  such  as  the  common  larch,  the  normal  number  is  eight  rows 
in  the  one  direction  and  five  in  the  other;  while  in  the  American 
larch  we  have  again  three  in  the  one  direction  and  five  in  the  other. 
It  not  seldom  happens  that  two  arrangements  grade  into  one  another 
on  different  parts  of  one  and  the  same  cone.  Among  other  cases 
in  which  such  spiral  series  are  readily  visible  we  have,  for  instance, 
the  crowded  leaves  of  the  stone-crops  and  mesembryanthemums,  and 
(as  we  have  said)  the  crowded  florets  of  the  composites.  Among 
these  we  may  find  plenty  of  examples  in  which  the  numbers  of  the 
serial  rows  are  similar  to  those  of  the  fir-cones ;  but  in  some  cases, 
as  in  the  daisy  and  others  of  the  smaller  composites,  we  shall  be 
able  to  trace  thirteen  rows  in  one  direction  and  twenty-one  in  the 
other,  or  perhaps  twenty-one  and  thirty-four;  while  in  a  great  big 
sunflower  we  may  find  (in  one  and  the  same  species)  thirty-four  and 
fifty-five,  fifty-five  and  eighty-nine,  or  even  as  many  as  eighty-nine 
and  one  hundred  and  forty-four.  On  the  other  hand,  in  an  ordinary 
"  pentamerous "  flower,  such  as  a  ranunculus,  we  may  be  able  to 
trace,  in  the  arrangement  of  its  sepals,  petals  and  stamens,  shorter 
spiral  series,  three  in  one  direction  and  two  in  the  other;  and  the 
scales  on  the  little  cone  of  a  Cypress  shew  the  same  numerical 
simplicity.  It  will  be  at  once  observed  that  these  arrangements 
manifest  themselves  in  connection  with  very  different  things,  in  the 
orderly  interspacing  of  single  leaves  and  of  entire  florets,  and  among 
all  kinds  of  leaf-like  structures,  foliage-leaves,  bracts,  cone-scales, 
and  the  various  parts  or  members  of  the  flower.  Again  we  must 
be  careful  to  note  that,  while  the  above  numerical  characters  are 
by  much  the  most  common,  so  much  so  as  to  be  deemed  "normal," 
many  other  combinations  are  known  to  occur. 

The  arrangement,  as  we  have  seen,  is  apt  to  vary  when  the  entire 
structure  varies  greatly  in  size,  as  in  the  disc  of  the  sunflower.  It 
is  also  subject  to  less  regular  variation  within  one  and  the  same 
species,  as  can  always  be  discovered  when  we  examine  a  sufficiently 
large  sample  of  fir-cones.     For  instance,  out  of  505  cones  of  the 


XIV]  OR  PHYLLOTAXIS  923 

Norway  spruce,  Beal*  found  92  per  cent,  in  which  the  spirals  were 
in  five  and  eight  rows ;  in  6  per  cent,  the  rows  were  four  and  seven, 
and  in  4  per  cent,  they  were  four  and  six.  In  each  case  they  were 
nearly  equally  divided  as  regards  direction ;  for  instance,  of  the  467 
cones  shewing  the  five-eight  arrangement,  the  five-series  ran  in 
right-handed  spirals  in  224  cases,  and  in  left-handed  spirals  in  243. 
Omitting  the  "abnormal"  cases,  such  as  we  have  seen  to  occur 
in  a  small  percentage  of  our  cones  of  the  spruce,  the  arrangements 
which  we  have  just  mentioned  may  be  set  forth  as  follows  (the 
fractional  number  used  being  simply  an  abbreviated  symbol  for  the 
number  of  associated  helices  or  parastichies  which  we  can  count 
running  in  the  opposite  directions):  2/3,  3/5,  5/8,  8/13,  13/21,  21/34, 
34/55,  55/89,  89/144.  Now  these  numbers  form  a  very  interesting 
series,  which  happens  to  have  a  number  of  curious  mathematical 
properties  "f.     We  see,  for  instance,  that  the  denominator  of  each 

*  Amer.  Naturalist,  vn,  p.  449,  1873. 

t  This   celebrated   series   corresponds  to  the  continued  fraction   1+1      etc., 

1  + 
and  converges  to  1-618...,  the  numerical  equivalent  of  the  sectio  divina,  or 
''Golden  Mean."  The  series  of  numbers,  1,  1,  2,  3,  5,  8,  ...,  of  which  each  is  the 
sum  of  the  preceding  two,  was  used  by  Leonardo  of  Pisa  (c.  1170-1250),  nicknamed 
Fi  Bonacci,  or  films  bonassi,  in  his  Liber  Abbaci,  a  work  dedicated  to  magister 
mens,  summus  philosophus,  Michael  Scot  {Scritti,  i,  pp.  283-284,  1857).  This  learned 
man  was  educated  in  Morocco,  where  his  father  was  clerk  or  dragoman  to  Pisan 
merchants;  and  he  is  said  to  have  been  the  first  to  bring  the  Arabic  numerals,  or 
"  novem  figurae  Indorum,''  into  Europe.  The  Fibonacci  numbers  were  first  so-caUed 
by  Eduard  Lucas,  Bollettino  di  Bibliogr.  e  Storia  dei  Sci.  Matem.  e  Fis.  x,  p.  129, 1877. 
The  general  expression  for  the  series 

V^Y    (\-V5\ 


^^{(^T-C~)"l' 


was  known  to  Euler  and  to  Daniel  Bernoulli  {Coram.  Acad.  Sci.  Imp.  Petropol. 
1732,  p.  90),  and  was  rediscovered  by  Binet,  C.E.  xviii,  p.  563,  1843;  xix,  p.  939, 
1844)  and  by  Lame,  ibid,  xix,  p.  867,  after  whom  it  is  sometimes  called  Lame's 
series.  But  the  Greeks  were  familiar  with  the  series  2,  3:  5,  7  :  12,  17,  etc.;  which 
converges  to  V2,  as  the  other  does  to  the  Golden  Mean ;  and  so  closely  related  are 
the  two  series,  that  it  seems  impossible  that  the  Greeks  could  have  known  the  one 
and  remained  ignorant  of  the  other.  (See  a  paper  of  mine,  on  "Excess  and  Defect, 
etc.,"  in  Mind,  xxxviii.  No.  149,  1928.) 

The  Fibonacci  (or  Lame)  series  was  well  known  to  Kepler,  who,  in  his  paper 
De  nive  sexangula  (1611,  cf.  supra,  p.  695),  discussed  it  in  connection  with  the  form 
of  the  dodecahedron  and  icosahedron,  and  with  the  ternary  or  quinary  symmetry  of 
the  flower.  (Cf.  F.  Ludwig,  Kepler  iiber  das  Vorkommen  der  Fibonaccireihe  im 
Pflanzenreich,  Bot.  Centralbl.  lxviii,  p.  7,  1896.)  Professor  William  Allman, 
Professor  of  Botany   in  Dublin   (father   of  the   historian  of  Greek  geometry). 


924  ON  LEAF-ARRANGEMENT  [ch. 

fraction  is  the  numerator  of  the  next;  and  further,  that  each  suc- 
cessive numerator,  or  denominator,  is  the  sum  of  the  preceding  two. 
Our  immediate  problem,  then,  is  to  determine,  if  possible,  how  these 
numerical  coincidences  come  about,  and  why  these  particular  numbers 
should  be  so  commonly  met  with  as  to  be  considered  "normal"  and 
characteristic  features  of  the  general  phenomenon  of  phyllotaxis.  The 
following  account  is  based  on  a  short  paper  by  Professor  P.  Gr.  Tait*. 
Of  the  two  following  diagrams,  Fig.  450  represents  the  general 
case,  and  Fig.  451  a  particular  one,  for  the  sake  of  possibly  greater 
simplicity.  Both  diagrams  represent  a  portion  of  a  branch,  or  fir- 
cone, regarded  as  cylindrical,  and  unwrapped  to  form  a  plane  surface. 
A,  a,  at  the  two  ends  of  the  base-line,  represent  the  same  initial 
leaf  or  scale;   0  is  a  leaf  which  can  be  reached  from  ^  by  m  steps 

O 


Fig.  450. 

in  a  right-hand  spiral  (developed  into  the  straight  line  AO),  and  by 
n  steps  from  a  in  a  left-handed  spiral  aO.  Now  it  is  obvious  in  our 
fir-cone,  that  we  can  include  all  the  scales  upon  the  cone  by  taking 
so  many  spirals  in  the  one  direction,  and  again  include  them  all  by  so 

speculating  on  the  same  facts,  put  forward  the  curious  suggestion  that  the  cellular 
tissue  of  the  dicotyledons,  or  exogens,  would  be  found  to  consist  of  dodecahedra, 
and  that  of  the  monocotyledons  or  endogens  of  icosahedra  {On  the  mathematical 
connection  between  the  parts  of  vegetables:  abstract  of  a  Memoir  read  before  the 
Royal  Society  in  the  year  1811  (privately  printed,  n.d.).  Cf.  De  Candolle,  Organo- 
genie  vegetale,  i,  p.  534.  See  also  C.  E.  Wasteels,  Over  de  Fibonaccigetalen, 
Me  Natuur.  Congres,  Antwerpen,  1899,  pp.  25-37;  R.  C.  Archibald,  in  Jay 
Hambidge's  Dynawfc  Fiymmetry,  1920,  pp.  146-157;  and,  on  the  many  mathematical 
properties  of  the  series,  L.  E.  Dickson,  Theory  of  Numbers,  i,  pp.  393-411.  1919. 

Of  these  famous  and  fascinating  numbers  a  mathematical, friend  writes  to  me: 
"All  the  romance  of  continued  fractions,  linear  recurrence  relations,  surd  approxi- 
mations to  integers  and  the  rest,  lies  in  them,  and  they  are  a  source  of  endless 
curiosity.  How  interesting  it  is  to  see  them  striving  to  attain  the  unattainable, 
the  golden  ratio,  for  instance;   and  this  is  only  one  of  hundreds  of  such  relations." 

*  Proc.  R.S.E.  vn,  p.  391,  1872. 


xiv] 


OR  PHYLLOTAXIS 


925 


many  in  the  other.  Accordingly,  in  our  diagrammatic  construction, 
the  spirals  AO  and  aO  must,  and  always  can,  be  so  taken  that  m 
spirals  parallel  to  aO,  and  n  spirals  parallel  to  AO,  shall  separately 
include  all  the  leaves  upon  the  stem  or  cone. 

If  m  and  n  have  a  common  factor,  I,  it  can  easily  be  shewn  that 
the  arrangement  is  composite,  and  that  there  are  I  fundamental, 
or  genetic  spirals,  and  I  leaves  (including  A)  which  are  situated 


Fig.  451. 

exactly  on  the  line.  Aa.  That  is  to  say,  we  have  here  a  whorled 
arrangement,  which  we  have  agreed  to  leave  unconsidered  in  favour 
of  the  simpler  case.  We  restrict  ourselves,  accordingly,  to  the  cases 
where  there  is  but  one  genetic  spiral,  and  when  therefore  m  and  n 
are  prime  to  one  another. 

Our  fundamental,  or  genetic,  spiral,  as  we  have  seen,  is  that  which 
passes  from  A  (or  a)  to  the  leaf  which  is  situated  nearest  to  the 
base-line  Aa.  The  fundamental  spiral  will  thus  be  right-handed 
(A,  P,  etc.)  if  F,  which  is  nearer  to  A  than  to  a,  be  this  leaf — left- 
handed  if  it  be  jo.     That  is  to  say,  we  make  it  a  convention  that  we 


926  ON  LEAF-ARRANGEMENT  [ch. 

shall  always,  for  our  fundamental  spiral,  run  round  the  system,  from 
one  leaf  to  the  next,  hy  the  shortest  way. 

Now  it  is  obvious,  from  the  symmetry  of  the  figure  (as  further 
shewn  in  Fig.  451),-  that,  besides  the  spirals  running  along  AO  and 
aO,  we  have  a  series  running /rom  the  steps  on  aO  to  the  steps  on  ^0. 
In  other  words  we  can  find  a  leaf  (S)  upon  AO,  which,  like  the  leaf  0, 
is  reached  directly  by  a  spiral  series  from  A  and  from  a,  such  that 
aS  includes  n  steps,  and  AS  (being  part  of  the  old  spiral  fine  AO) 
now  includes  m  —  n  steps.  And,  since  m  and  n  are  prime  to  one 
another  (for  otherwise  the  system  would  have  been  a  composite  or 
whorled  one),  it  is  evident  that  we  can  continue  this  process  of 
convergence  until  we  come  down  to  a  1,  1  arrangement,  that  is  to 
say  to  a  leaf  which  is  reached  by  a  single  step,  in  opposite  directions 
from  A  and  from  a,,  which  leaf  is  therefore  the  first  leaf,  next  to  A, 
of  the  fundamental  or  generating  spiral. 

If  our  original  lines  along  AO  and  aO  contain,  for  instance,  13 
and  8  steps  respectively  (i.e.  m  =  13,  n=  8),  then  our  next  series, 
observable  in  the  same  cone,  will  be  8  and  (13  —  8)  or  5;  the  next 
5  and  (8  —  5)  or  3;  the  next  3,  2;  and  the  next  2,  1 ;  leading  to  the 
ultimate  condition  of  1,  1.  These  are  the  very  series  which  we  have 
found  to  be  common,  or  normal;  and  so  far  as  our  investigation 
has  yet  gone,  it  has  proved  to  us  that,  if  one  of  these  exists,  it  entails, 
ipso  facto,  the  presence  of  the  rest. 

In  following  down  our  series,  according  to  the  above  construction, 
we  have  seen  that  at  every  step  we  have  changed  direction,  the 
longer  and  the  shorter  sides  of  our  triangle  changing  places  every 
time.  Let  us  stop  for  a  moment,  when  we  come  to  the  1,  2  series, 
or  AT,  aT  of  Fig.  451.  It  is  obvious  that  there  is  nothing  to  prevent 
us  making  a  new  1,  3  series  if  we  please,  by  continuing  the  generating 
spiral  through  three  leaves,  and  connecting  the  leaf  so  reached 
directly  with  our  initial  one.  But  in  the  case  represented  in  Fig.  451, 
it  is  obvious  that  these  two  series  {A,  1,  2,  3,  etc.,  and  a,  3,  6,  etc.) 
will  be  running  in  the  same  direction;  i.e.  they  will  both  be  right- 
handed,  or  both  left-handed  spirals.  The  simple  meaning  of  this 
is  that  the  third  leaf  of  the  generating  spiral  was  distant  from  our 
initial  leaf  by  more  than  the  circumference  of  the  cyhndrical  stem; 
in  other  words,  that  there  were  more  than  two,  but  less  than  three 
leaves  in  a  single- turn  of  the  fundamental  spiral. 


XIV]  OR  PHYLLOTAXIS  927 

Less  tnan  two  there  can  obviously  never  be.  When  there  are 
exactly  two,  we  have  the  simplest  of  all  possible  arrangements, 
namely  that  in  which  the  leaves  are  placed  alternately  on  opposite 
sides  of  the  stem.  When  there  are  more  than  two,  but  less  than 
three,  we  have  the  elementary  condition  for  the  production  of  the 
series  which  we  have  been  considering,  namely  1,  2;  2,  3;  3,  5,  etc. 
To  put  the  latter  part  of  this  argument  in  more  precise  language, 
let  us  say  that:  If,  in  our  descending  series,  we  come  to  steps  1 
and  t,  where  t  is  determined  by  the  condition  that  1  and  t  +  1  would 
give  spirals  both  right-handed,  or  both  left-handed;  it  follows  that 
there  are  less  than  t  +  1  leaves  in  a  single  turn  of  the  fundamental 
spiral.  And,  determined  in  this  manner,  it  is  found  in  the  great 
majority  of  cases,  in  fir-cones,  and  a  host  of  other  examples  of 
phyllotaxis,  that  t=  2.  In  other  words,  in  the  great  majority  of 
cases,  we  have  what  corresponds  to  an  arrangement  next  in  order 
of  simphcity  to  the  simplest  case  of  all :  next,  that  is  to  say,  to  the 
arrangement  which  consists  of  opposite  and  alternate  leaves. 

"These  simple  considerations,"  as  Tait  says,  "explain  completely 
the  so-called  mysterious  appearance  of  terms  of  the  recurring  series 
1,  2,  3,  5,  8,  13,  etc.*  The  other  natural  series,  usually  but  mis- 
leadingly  represented  by  convergents  to  an  infinitely  extended 
continuous  fraction,  are  easily  explained,  as  above,  by  taking  ^  =  3, 
4,  5,  etc.,  etc."  Many  examples  of  these  latter  series  have  been 
recorded,  as  more  or  less  rare  abnormalities,  by  Dickson  f  and  other 
writers. 

We  have  now  learned,  among  other  elementary  facts,  that  wherever 
any  one  system  of  spiral  steps  is  present,  certain  others  invariably 
and  of  necessity  accompany  it,  and  are  definitely  related  to  it.  In 
any  diagram,  such  as  Fig.  451,  in  which  we  represent  our  leaf- 
arrangement  by  means  of  uniform  and  regularly  interspaced  dots, 
we  can  draw  one  series  of  spirals  after  another,  and  one  as  easily 

*  The  necessary  existence  of  these  recurring  spirals  is  also  proved,  in  a  somewhat 
different  way,  by  Leslie  Ellis,  On  the  theory  of  vegetable  spirals,  in  Mathematical 
and  other  Writings,  1863,  pp.  358-372.  Leslie  Ellis,  Whewell's  brother-in-law,  was 
a  man  of  great  originahty.  He  is  best  remembered,  perhaps,  for  his  views  on  the 
Theory  of  Probabilities  (cf.  J.  M.  Keynes,  Treatise  an  Probabilities,  1921,  p.  92), 
and  for  his  association  with  Stebbing  as  editor  of  Bacon. 

t  Proc.  R.S.E.  vn,  p.  397,  1872;   Trans.  E.S.E.  xxvi,  pp.  505-520,  1872. 


928  ON  LEAF-ARRANGEMENT  [ch. 

as  another.  In  a  fire-cone  one  particular  series,  or  rather  two 
conjugate  series,  are  always  conspicuous,  but  the  related  series  may 
be  sought  and  found  with  httle  difficulty.  The  spruce-fir  is  com- 
monly said  to  have  a  phyllotaxis  of  8/13;    but  we  may  count  still 


Fig.  452. 

steeper  and  nearly  vertical  rows  of  scales  to  the  number  of  13/21; 
and  if  we  take  pains  to  number  all  the  scales  consecutively,  we  may 
find  the  lower  series,  5/8,  3/5,  and  even  1/2,  with  ease  and  certainty. 
The  phenomenon  is  illustrated  by  Fig.  452,  a-d.  The  ground-plan 
of  all  these  diagrams  is  identically  the  same.     The  generating  spiral 


XIV]  PR  PHYLLOTAXIS  929 

in  each  case  represents  a  divergence  of  3/8,  or  135°  of  azimuth;  and 
the  points  succeed  one  another  at  the  same  successional  distances 
parallel  to  the  axis.  The  rectangular  outhnes,  which  correspond  to 
the  exposed  surface  of  the  leaves  or  cone-scales,  are  of  equal  area, 
and  of  equal  number.  Nevertheless  the  appearances  presented  by 
these  diagrams  are  very  different;  for  in  one  the  eye  catches  a  5/8 
arrangement,  in  another  a  3/5 ;  and  so  on,  down  to  an  arrangement 
of  1/1.  The  mathematical  side  of  this  very  curious  phenomenon 
I  have  not  attempted  to  investigate.  But  it  is  quite  obvious  that, 
in  a  system  within  which  various  spirals  are  imphcitly  contained, 
the  conspicuousness  of  one  set  or  another  does  not  depend  upon 
angular  divergence.  It  depends  on  the  relative  proportions  in  length 
and  breadth  of  the  leaves  themselves;  or,  more  strictly  speaking, 
on  the  ratio  of  the  diagonals  of  the  rhomboidal  figure  by  which 
each  leaf-area  is  circumscribed.  When,  as  in  the  fir-cone,  the  scales 
by  mutual  compression  conform  to  these  rhomboidal  outlines,  their 
inclined  edges  at  once  guide  the  eye  in  the  direction  of  some  one 
particular  spiral;  and  we  shall  not  fail  to  notice  that  in  such  cases 
the  usual  result  is  to  give  us  arrangements  corresponding  to  the 
middle  diagrams  in  Fig.  452,  which  are  the  configurations  in  which 
the  quadrilateral  outhnes  approach  most  nearly  to  a  rectangular 
form,  and  give  us  accordingly  the  least  possible  ratio  (under  the 
given  conditions)  of  sectional  boundary-wall  to  surface  area. 

The  manner  in  which  one  system  of  spirals  may  be  caused  to 
slide,  so  to  speak,  into  another,  has  been  ingeniously  demonstrated 
by  Schwendener  on  a  mechanical  model,  consisting  essentially  of 
a  framework  which  can  be  opened  or  closed  to  correspond  with  one 
another  of  the  above  series  of  diagrams*. 

The  same  curious  fact,  that  one  Fibonacci  series  leads  to,  or 
involves  the  rest,  is  further  shewn,  in  a  very  simple  way,  in  the 
following  diagrammatic  Table  (p.  930).  It  shews,  in  the  first 
instance,  the  numerical  order  of  the  scales  on  a  fir-cone,  in  so-called 
5/8  phyllotaxis;  that  is  to  say,  it  represents  the  cone  unwrapped, 
with  the  two  principal  spirals  lying  along  the  axes  of  a  rectangular 
system.  Starting  from  0,  the  abscissae  increase  by  5,  the  ordinates 
by  8;    or,  in  other  words,  any  given  number  m  =  bx  +  d>y\    it  is 

*  A  common  form  of  pail-shaped  waste-paper  basket,  with  wide  rhomboidal 
meshes  of  cane,  is  well-nigh  as  good  a  model  as  is  required. 


2 

3 

5 

8 

13 

21 

34 

55 

2 

-1 

1 

0 

1 

1 

2 

3 

-1 

1 

0 

1 

i 

2 

3 

5 

930  ON  LEAF-ARRANGEMENT  [ch. 

easy,  then,  to  number  the  entire  system.  The  generating  spiral, 
0,  1,  2,  3, ... ,  and  the  various  secondary  Fibonacci  spirals,  are  then 
easily  recognised. 

A  Fibonacci  series,  unwrapped  from  a  cone  or  cylinder.  m  =  6x-\-  Sy. 

1    6    11    16    21    26    31    36 

T    2    3'    8    13    18    23    28 

15    To     5"    0     5    10    15    20 

^r8l3'83    2    7    12 

■5T'2B2Tl6rT  6  T4 

The  place  of  the  first  scale  in  each  series  is  then  found  to  be  as 
follows : 

Series     1 

y~    2 

And  this  is  the  Fibonacci  series  over  again.  We  also  see  how  the 
several  spirals,  of  which  these  are  the  beginnings,  alternate  to  the 
right  and  left  of  an  asymptotic  line,  where  x/y  =  0-618. . . . 

The  Fibonacci  numbers,  so  conspicuous  in  the  fir-cone,  make  their 
appearance  also  in  the  flower.  The  commonest  of  floral  numbers 
are  3  and  5;  among  the  Composites  we  find  8  ray-florets  in  the 
single  dahlia,  13  in  the  ragwort,  21  in  the  ox-eye  daisy  or  the  mari- 
gold. In  the  last  two,  heads  with  34  ray-florets  are  apt  to  be 
produced  at  certain  times  or  in  certain  places*;  and  in  C.  segetum 
these  florets  are  said  to  vary  in  a  bimo(fe,l  curve  of  frequency,  with  a 
high  maximum  at  13  and  a  lower  at  21t.  The  simplest  explanation 
(though  perhaps  it  does  not  go  far)  is  to  suppose  that  a  ligulate 
floret  terminates,  or  tends  to  terminate,  each  of  the  principal  spiral 
series.  But  among  the  higher  numbers  these  numerical  relations 
are  only  approximate,  and  the  whole  matter  rests,  so  far,  on  some- 
what scanty  evidence. 

The  determination  of  the  precise  angle  of  divergence  of  two  con- 
secutive leaves  of  the  generating  spiral  does  not  enter  into  the  above 
general  investigation  (though  Tait  gives,  in  the  same  paper,  a  method 

*  Cf.  G.  Henslow,  On  the  origin  of  dimerous  and  trimerous  whorls  among  the 
flowers  of  Dicotyledons,  Trans.  Linn.  Soc.  (Bot.)  (2),  vii,  p.  161,  1908. 
f  Cf.  A.  Gravis,  Elements  de  Physiologic  vegetale,  1921,  p.  122. 


XIV]  OR  PHYLLOTAXIS  931 

by  which  it  may  be  easily  determined);  and  the  very  fact  that  it 
does  not  so  enter  shews  it  to  be  essentially  unimportant.  The 
determination  of  so-called  ''orthostichies,"  or  precisely  vertical  suc- 
cessions of  leaves,  is  also  unimportant.  We  have  no  means,  other 
than  observation,  of  determining  that  one  leaf  is  vertically  above 
another,  and  spiral  series  such  as  we  have  been  dealing  with  will 
appear,  whether  such  orthostichies  exist,  whether  they  be  near  or 
remote,  or  whether  the  angle  of  divergence  be  such  that  no  precise 
vertical  superposition  ever  occurs.  And  lastly,  the  fact  that  the 
successional  numbers,  expressed  as  fractions,  1/2,  2/3,  3/5,  represent 
a  convergent  series,  whose  final  term  is  equal  to  0-61803...,  the  sectio 
aurea  or  "golden  mean"  of  unity,  is  seen  to  be  a  mathematical 
coincidence,  devoid  of  biological  significance;  it  is  but  a  particular 
case  of  Lagrange's  theorem  that  the  roots  of  every  numerical  equation 
of  the  second  degree  can  be  expressed  by  a  periodic  continued  frac- 
tion. The  same  number  has  a  multitude  of  curious  arithmetical 
properties.  It  is  the  final  term  of  all  similar  series  to  that  with 
which  we  have  been  dealing,  such  for  instance  as  1/3,  3/4,  4/7,  etc., 
or  1/4,  4/5,  5/9,  etc.  It  is  a  number  beloved  of  the  circle-squarer, 
and  of  all  those  who  seek  to  find,  and  then  to  penetrate,  the  secrets 
of  the  Great  Pyramid.  It  is  deep-set  in  the  regular  pentagon  and 
dodecahedron,  the  triumphs  of  Pythagorean  or  Euclidean  geometry. 
It  enters  (as  the  chord  of  an  angle  of  36°)  into  the  thrice-isosceles 
triangle  of  which  we  have  spoken  on  p.  762 ;  it  is  a  number  which 
becomes  (by  the  addition  of  unity)  its  own  reciprocal — its  properties 
never  end.  To  Kepler  (as  Naber  tells  us)  it  was  a  symbol  of  Creation, 
or  Generation.  Its  recent  application  to  biology  and  art-criticism 
by  Sir  Theodore  Cook  and  others  is  not  new.  Naber' s  book,  already 
quoted,  is  full  of  it.     Zeising*,  in  1854,  found  in  it  the  key  to  all 

*  A.  Zeising,  Neue  Lehre  von  der  Proportion  des  menschlichen  Korpers  aus  einem 
bisher  unerkannt  gebliebenen  die  game  Natur  und  Kunst  durchdringenden  morpho- 
logischen  Grundgesetze  entwickelt,  Leipzig,  1854,  457  pp.;  ibid.  Deutsche  Viertel- 
jahrsschrift,  1868,  p.  261;  also,  posthumously,  Der  Goldene  Schnitt,  Leipzig,  1884, 
24  pp.  Cf.  S.  Gunther,  Adolph  Zeising  als  Mathematiker,  Ztschr.  f.  Math.  u. 
Physik.  {Hist.  Lit.  Abth.),  xxi,  pp.  157-165,  1876;  also  F.  X.  PfeiflFer,  Die  Propor- 
tionen  des  goldenen  Schnittes  an  den  Blattern  u.  Stengelen  der  Pflanzen,  Ztschr. 
f.  math.  u.  naturw.  Unterricht,  xv,  pp.  325-338,  1885.  For  other  references,  see 
R.  C.  Archibald,  op.  cit.  Among  modern  books  on  similar  lines,  the  following 
are  curious,  interesting  and  beautiful  (whether  we  agree  with  them  or  not):  Jay 
Hambidge,  Dynamic  Symmetry,  Yale,  1920;  C.  Arthur  Coan,  Nature's  Harmonic 
Unity,  New  York,  1912. 


932  ON  LEAF-ARRANGEMENT  [ch. 

morphology,  and  the  same  writer,  later  on,  declared  it  to  dominate 
both  architectm^e  and  music.  But  indeed,  to  use  Sir  Thomas 
Browne's  words  (though  it  was  of  another  number  that  he  spoke): 
"To  enlarge  this  contemplation  into  all  the  mysteries  and  secrets 
accommodable  unto  this  number,  were  inexcusable  Pythagorisme." 
That  this  number  has  any  serious  claim  at  all  to  enter  into  the 
biological  question  of  phyllotaxis  seems  to  depend  on  the  assertion, 
.first  made  by  Chauncey  Wright*,  that,  if  the  successive  leaves 
of  the  fundamental  spiral  be  placed  at  the  particular  azimuth  which 
divides  the  circle  in  this  "sectio  aurea,"  then  no  two  leaves  will 
ever  be  superposed f;  and  thus  we  are  said  to  have  "the  most 
thorough  and  rapid  distribution  of  the  leaves  round  the  stem,  each 
new  or  higher  leaf  falling  over  the  angular  space  between  the  two 
older  ones  which  are  nearest  in  direction,  so  as  to  divide  it  in  the 
same  ratio  (K),  in  which  the  first  two  or  any  two  successive  ones 
divide  the  circumference.  Now  5/8  and  all  successive  fractions 
differ  inappreciably  from  Z."  To  this  view  there  are  many  simple 
objections.  In  the  first  place,  even  5/8,  or  0-625,  is  but  a  moderately 
close  approximation  to  the  "golden  mean";  and  furthermore, 
the  arrangements  by  which  a  better  approximation  is  got,  such  as 
8/13,  13/21,  and  the  very  close  approximations  such  as  34/55,  55/89, 
89/144,  etc.,  are  comparatively  rare,  while  the  much  less  close 
approximations  of  3/5  or  2/3,  or  even  1/2,  are  extremely  common. 
Again,  the  general  type  of  argument  such  as  that  which  asserts 
that  the  plant  is  "aiming  at"  something  which  we  may  call  an 
"ideal  angle"  is  one  which  cannot  commend  itself  to  a  plain  student 
of  physical  science :  nor  is  the  hypothesis  rendered  more  acceptable 
when  Sir  T.  Cook  qualifies  it  by  telHng  us  that  "all  that  a  plant 
can  do  is  to  vary,  to  make  blind  shots  at  constructions,  or  to 
'  mutate '  as  it  is  now  termed :  and  the  most  suitable  of  these  con- 
structions will  in  the  long  run  be  isolated  by  the  action  of  Natural 
Selection."     Thirdly,  we  must  not  suppose  the  Fibonacci  numbers 

♦  On  the  uses  and  origin  of  the  arrangement  of  leaves  in  plants,  Mem.  Amer. 
Acadfix,  p.  380,  1871,  Cambridge,  Mass.  Cf.  J.  Wiesner,  Ueberdie  Beziehungen  der 
Stellungsverhdltnisse  der  Laubbldtter  zur  Beleuchtung,  Wien,  1902. 

t  This  is  what  Ruskin  spoke  of  as  "  the  vacant  space  " ;  Mod.  Painters,  v,  chap,  vi, 
p.  44,  1860.  Leonardo  had  in  like  manner  explained  the  leaf-arrangement  as 
serving  to  let  air  pass  between  the  leaves,  keep  one  from  overshadowing  another, 
and  let  rain-drops  fall  from  the  one  leaf  to  the  one  below. 


XIV]  OR  PHYLLOTAXIS  933 

to  liave  any  exclusive  relation  to  the  Golden  Mean;  for  arithmetic 
teaches  us  that,  beginning  with  any  two  numbers  whatsoever,  we  are 
led  by  successive  summations  toward  one  out.  of  innumerable  series 
of  numbers  whose  ratios  one  to  another  converge  to  the  Golden 
Mean*.  Fourthly,  the  supposed  isolation  of  the  leaves,  or  their  most 
complete  "distribution  to  the  action  of  the  surrounding  atmosphere" 
is  manifestly  very  little  affected  by  any  conditions  which  are  confined 
to  the  angle  of  azimuth.  For  if  it  be  (so  to  speak)  Nature's  object 
to  set  them  farther  apart  than  they  actually  are,  to  give  them  freer 
exposure  to  the  air  or  to  the  sunlight  than  they  actually  have,  then 
it  is  surely  manifest  that  the  simple  way  to  do  so  is  to  elongate 
the  axis,  and  to  set  the  leaves  farther  apart,  lengthways  on  the 
stem.  This  has  at  once  a  far  more  potent  effect  than  any  nice 
manipulation  of  the  "angle  of  divergence." 

Lastly,  and  this  seems  the  simplest,  the  most  cogent  and  most 
unanswerable  objection  of  them  all,  if  it  be  indeed  desirable  that 
no  leaf  should  be  superimposed  above  another,  the  one  condition 
necessary  is  that  the  common  angle  of  azimuth  should  not  be  a 
rational  multiple  of  a  right  angle — should  not  be   equivalent   to 

—  I  -  j .     One  irrational  angle  is  as  good  as  another :    there  is  no 

special  merit  in  any  one  of  them,  not  even  in  the  ratio  divina.  We 
come  then  without  more  ado  to  the  conclusion  that  while  the 
Fibonacci  series  stares  us  in  the  face  in  the  fir-cone,  it  does  so  for 
mathematical  reasons;  and  its  supposed  usefulness,  and  the  hypo- 
thesis of  its  introduction  into  plant-structure  through  natural 
selection,  are  matters  which  deserve  no  place  in  the  plain  study 
of  botanical  phenomena.  As  Sachs  shrewdly  recognised  years  ago, 
all  such  speculations  as  these  hark  back  to  a  school  of  mystical 
idealism. 

*  Thus,  instead  of  beginning  with  1,  1,  let  us  begin  1,  7.     The  summation-series 
is  then  1,  7,  8,  15,  23,  38,  61,  99,  160,  259,  ...,  etc.;    and  99/160=0-618...  and 
259/160=  1-619...;    and  so  on.     But  after  all,  the  old  Fibonacci  numbers  are  not 
far  away.     For  we  may  write  the  new  series  in  the  form : 
7  (0,  1,  1,  2,  3,  5,  8,  ...) 
+  1  (1,0,  1,  1,2,3,5,'...). 


CHAPTER  XV 

ON  THE  SHAPES  OF  EGGS,  AND  OF  CERTAIN  OTHER 
HOLLOW  STRUCTURES 

The  eggs  of  birds  and  all  other  hard-shelled  eggs,  such  as  those  of 
the  tortoise  and  the  crocodile,  are  simple  solids  of  revolution ;  but 
they  differ  greatly  in  form,  according  to  the  configuration  of  the 
plane  curve  by  the  revolution  of  which  the  egg  is,  in  a  mathematical 
sense,  generated.  Some  few  eggs,  such  as  those  of  the  owl,  the 
penguin,  or  the  tortoise,  are  spherical  or  very  nearly  so;  a  few  more, 
such  as  the  grebe's,  the  cormorant's  or  the  pelican's,  are  approxi- 
mately ellipsoidal,  with  symmetrical  or  nearly  symmetrical  ends, 
and  somewhat  similar  are  the  so-called  "cylindrical"  eggs  of  the 
megapodes  and  the  sand-grouse;  the  great  majority,  like  the  hen's 
egg,  are  "  ovoid,"  a  little  blunter  at  one  end  than  the  other ;  and  some, 
by  an  exaggeration  of  this  lack  of  antero-posterior  symmetry,  are 
blunt  at  one  end  but  characteristically  pointed  at  the  other,  as  is 
the  case  with  the  eggs  of  the  guillemot  and  puffin,  the  sandpiper, 
plover  and  curlew.  It  is  an  obvious  but  by  no  means  negligible  fact 
that  the  egg,  while  often  pointed,  is  never  flattened  or  discoidal; 
it  is  a  prolate,  but  never  an  oblate,  spheroid.  Its  oval  outhne  has 
one  maximal  and  two  minimal  radii  of  curvature,  one  minimum  being 
less  than  the  other.  The  evolute  to  a  curve  often  emphasises,  even 
exaggerates,  its  features ;  and  the  evolutes  to  a  series  of  eggs  (i.e. 
to  their  generating  curves)  are  more  conspicuously  different  than 
the  eggs  themselves  (Fig.  453)*. 

The  careful  study  and  collection  of  birds'  eggs  would  seem  to  have 
begun  with  the  Coujit  de  Marsiglil,  the  same  celebrated  naturalist 

*  Cf.  A.  Mallock,  On  the  shapes  of  birds'  eggs.  Nature,  cxvi,  p.  311,  1925.  The 
evolute  may  be  easily  if  somewhat  roughly  drawn  by  erecting  perpendiculars  on 
a  sufficient  nunrber  of  tangents  to  the  curve.  The  .evolute  then  appears  as  an 
envelope,  the  perpendiculars  all  being  tangents  to  it. 

t  De  avibus  circa  aquas  Danubii  vagantihus  et  de  ipsarum  nidis  (Vol.  v  of  the 
Danubius  Panonico-Mysicus),  Hagae  Com.  1726.  Count  Giuseppi  Ginanni,  or 
Zinanni,  came  soon  afterwards  with  his  book  Delle  uove  e  dei  nidi  degli  uccelli, 
Vcnezia,  1737. 


CH.  XV]  ON  THE  SHAPES  OF  EGGS  935 

who  first  studied  the  "flowers"  of  the  coral,  and  who  wrote  the 
Histoire  physique  de  la  mer;  and  the  specific  form  as  well  as  the 
colour  and  other  attributes  of  the  egg  have  been  again  and  again 
discussed,  and  not  least  by  the  many  dilettanti  naturalists  of  the 
•  eighteenth  century  who  soon  followed  in  Marsigli's  footsteps*. 

We  need  do  no  more  than  mention  Aristotle's  belief,  doubtless 
old  in  his  time,  that  the  more  pointed  egg  produces  the  male  chicken, 
and  the  blunter  egg  the  hen ;  though  this  theory  survived  into  modern 
times t  and  still  lingers  on  (cf.  p.  943).  Several  naturahsts,  such  as 
Giinther  (1772)  and  Biihle  (1818),  have  taken  the  trouble  to  disprove 
it  by  experiment.     A  more  modern  and  more  generally  accepted 


^ A 


.A- 


a  b  c  d  e 

Fig.  453.   Typical  forms  of  birds'  eggs:  from  A.  Mallock. 
The  figures  below  are  pinhole  photographs  of  the  eggs.  The  upper  figures  (drawn 
to  a  uniform  scale)  shew  the  generating  curves  and  their  evolutes. 

a   Green  plover.  c    Crow.  e   Kingfisher. 

b   Humming-bird.  d   Pheasant.  /  Owl. 

explanation  has  been  that  the  form  of  the  egg  is  in  direct  relation 
to  that  of  the  bird  which  has  to  be  hatched  within — a  view  that 
would  seem  to  have  been  first  set  forth  by  Naumann  and  Biihle, 
in  their  great  treatise  on  eggs  J,  and  adopted  by  Des  Murs§  and 
many  other  well-known  writers. 

In  a  treatise  by  de  Lafresnaye||,  an  elaborate  comparison  is  made 

*  But  Sir  Thomas  Browne  had  a  collection  of  eggs  at  Norwich  in  1671,  according 
to  Evelyn. 

f  Cf.  Lapierre,  in  Buffon's  Histoire  Naturelh,  ed.  Sonnini,  1800. 

%  Eier  der  Vogel  Deutschlands,  1818-28  (cit.  Des  Murs,  p.  36). 

§   Traite  dVologie,  1860. 

II  F.  de  Lafre.snaye,  Comparaison  des  ceufs  des  oiseaux  avec  leurs  squelettes, 
comme  seul  moyen  de  reconnaitre  la  cause  de  leurs  differentes  formes.  Rev.  Zool. 
1845,  pp.  180-187,  239-244. 


936  ON  THE  SHAPES  OF  EGGS  [ch. 

between  the  skeleton  and  the  egg  of  various  birds,  to  shew,  for 
instance,  how  those  birds  with  a  deep-keeled  sternum  laid  rounded 
eggs,  which  alone  could  accommodate  the  form  of  the  young. 
According  to  this  yiew,  that  "Nature  had  foreseen*"  the  form 
adapted  to  and  necessary  for  the  growing  embryo,  it  was  easy  to. 
correlate  the  owl  with  its  spherical  egg,  the  diver  with  its  elhptical 
one,  and  in  like  manner  the  round  egg  of  the  tortoise  and  the 
elongated  one  of  the  crocodile,  with  the  shape  of  the  creatures  which 
had  afterwards  to  be  hatched  therein.  A  few  writers,  such  as 
Thienemannf,  looked  at  the  same  facts  the  other  way,  and  asserted 
that  the  form  of  the  egg  was  determined  by  that  of  the  bird  by 
which  it  was  laid  and  in  whose  body  it  had  been  conformed. 

In  more  recent  times,  other  theories,  based  upon  the  principles 
of  Natural  Selection,  have  been  current  and  very  generally  accepted 
to  account  for  these  diversities  of  form.  The  pointed,  conical  egg 
of  the  guillemot  is  generally  supposed  to  be  an  adaptation,  advan- 
tageous to  the  species  in  the  circumstances  under  which  the  egg 
is  laid;  the  pointed  egg  is  less  apt  than  a  spherical  one  to  roll  off 
the  narrow  ledge  of  rock  on  which  this  bird  is  said  to  lay  its  solitary 
egg,  and  the  more  pointed  the  egg,  so  niuch  the  fitter  and  hkeher  is 
it  to  survive.  The  fact  that  the  plover  or  the  sandpiper,  breeding 
in  very  different  situations,  lay  eggs  that  are  also  conical,  ehcits 
another  explanation,  to  the  effect  that  here  the  conical  form  permits 
the  many  large  eggs  to  be  packed  closely  under  the  mother  bird  J. 
Whatever  truth  there  be  in  these  apparent  adaptations  to  existing 
circumstances,  it  is  oiily  by  a  very  hasty  logic  that  we  can  accept 
them  as  a  vera  causa,  or  adequate  explanation  of  the  facts ;  and  it  is 
obvious  that  in  the  bird's  egg  we  have  an  admirable  case  for  direct 
investigation  of  the  mechanical  or  physical  significance  of  its  form§. 

*  Cf.  Des  Murs,  p.  67:  "Elle  devait  encore  penser  au  moment  ou  ce  germe 
aurait  besoin  de  I'espace  necessaire  a  son  accroissement,  a  ce  moment  ou . .  .  il  devra 
remplir  exactement  I'intervalle  circonscrit  par  sa  fragile  prison,  etc." 

f  F.  A.  L.  Thienemann,  Syst.  Darstellung  der  Fortpflanzung  der  Vogel  Europas, 
Leipzig,  1825-38. 

X  Cf.  Newton's  Dictionary  of  Birds,  1893,  p.  191 ;  Szielasko,  Gestalt  der  Vogeleier, 
Journ.  f.  Ornith.  Lm,  pp.  273-297,  1905. 

§  Jacob  Steiner  suggested  a  Cartesian  oval,  r  +  mr'  —  c,  as  a  general  formula 
for  all  eggs  (cf.  Fechner,  Ber.  sacks.  Ges.  1849,  p.  57);  but  this  formula  (which 
fails  in  such  a  case  as  the  guillemot)  is  purely  empirical,  and  has  no  mechanical 
foundation. 


XV]  AND  OTHER  HOLLOW  STRUCTURES  937 

Of  all  the  many  naturalists  of  the  eighteenth  and  nineteenth 
centuries  who  wrote  on  the  subject  of  eggs,  only  two  (so  far  as 
I  am  aware)  ascribed  the  form  of  the  egg  to  direct  mechanical 
causes.  Giinther*,  in  1772,  declared  that  the  more  or  less  rounded 
or  pointed  form  of  the  egg  is  a  mechanical  consequence  of  the 
pressure  of  the  oviduct  at  a  time  when  the  shell  is  yet  unformed 
or  unsolidified ;  and  that  accordingly,  to  explain  the  round  egg  of 
the  owl  or  the  kingfisher,  we  have  only  to  admit  that  the  oviduct 
of  these  birds  is  somewhat  larger  than  that  of  most  others,  or 
less  subject  to  violent  contractions.  This  statement  contains,  in 
essence,  the  whole  story  of  the  mechanical  conformation  of  the  egg. 
A  hundred  and  twenty  years  after,  Dr  J.  Ryder  of  Philadelphia 
gave,  as  near  as  may  be,  the  same  explanation  f. 

Let  us  consider,  very  briefly,  the  conditions  to  which  the  egg  is 
subject  in  its  passage  down  the  oviduct. 

(1)  The  "egg,"  as  it  enters  the  oviduct,  consists  of  the  yolk  only, 
enclosed  in  its  vitelHne  membrane.  As  it  passes  down  the  first 
portion  of  the  oviduct  the  white  is  gradually  superadded,  and 
becomes  in  turn  surrounded  by  the  "shell-membrane."  About  this 
latter  the  shell  is  secreted,  rapidly  and  at  a  late  period:  the  egg 
having  meanwhile  passed  on  into  a  wider  portion  of  the  oviducal 
tube;  called  (by  loose  analogy,  as  Owen  says)  the  "uterus."  Here 
'the  egg  assumes  its  permanent  form,  here  it  ultimately  becomes 
rigid,  and  it  is  to  this  portion  of  the  oviduct  that  our  argument 
principally  refers. 

(2)  Both  the  yolk  and  the  entire  egg  tend  to  fill  completely  their 
respective  membranes,  and,  whether  this  be  due  to  growth  or 
imbibition  on  the  part  of  the  contents  or  to  contraction  on  the  part 
of  the  surrounding  membranes,  the  resulting  tendency  is  for  both 
yolk  and  egg  to  be,  in  the  first  instance,  spherical,  unless  or  until 
distorted  by  external  pressure. 

(3)  The  egg  is  subject  to  pressure  within  the  oviduct,  which  is 
an  elastic,  muscular  tube,  along  the  walls  of  which  pass  peristaltic 

*  F,  C.  Giinther,  Sammlung  von  Nestern  und  Eyern  verschiedener  Vogel,  Niirnb. 
1772.  Cf.  also  Raymond  Pearl,  Morphogenetic  activity  of  the  oviduct,  J.  Exp.  Zool. 
VI,  pp.  339-359,  1909.  - 

t  J.  Ryder,  The  mechanical  genesis  of  the  form  of  the  fowl's  egg,  Proc\  Amer. 
Philosoph.  Soc.  Philadelphia,  xxxi,  pp.  203-209,  1893;  cf.  A.  S.  Packard, 
Inheritance  of  acquired  characters,  'Proc.  Amer.  Acad.  1894,  p.  360. 


938  ON  THE  SHAPES  OF  EGGS  [ch. 

waves  of  contraction.  These  muscular  contractions  hiay  be  de- 
scribed as  the  contraction  of  successive  annuli  of  muscle,  giving 
annular  (or  radial)  pressure  to  successive  portions  of  the  egg ;  they 
drive  the  egg  forward  against  the  frictional  resistance  of  the  tube, 
while  tending  at  the  same  time  to  distort  its  form.  While  nothing 
is  known,  so  far  as  I  am  aware,  of  the  muscular  physiology  of  the 
oviduct,  it  is  well  known  in  the  case  of  the  intestine  that  the  presence 
of  an  obstruction  leads  to  the  development  of  violent  contractions 
in  its  rear,  which  waves  of  contraction  die  away,  and  are  scarcely 
if  at  all  propagated  in  advance  of  the  obstruction ;  indeed  in  normal 
intestinal  peristalsis  a  wave  of  relaxation  travels  close  ahead  of  the 
wave  of  constriction. 

(4)  The  egg  is,  to  all  intents  and  purposes,  a  solid  of  revolution ; 
in  other  words,  its  transverse  sections  are  all  but  perfect  circles, 
sd  nearly  perfect  that,  chucked  in  the  lathe,  an  egg  "runs  true." 
This  may  be  taken  to  shew  that  the  direct  pressure  of  the  oviduct, 
whether  elastic  or  muscular,*  is  large  compared  with  the  weight  of 
the  egg.  Even  in  ostrich  eggs,  where  if  anywhere  gravitational 
deformation  should  be  found,  the  greatest  and  least  equatorial 
diameters  do  not  differ  by  1  per  cent.,  and  sometimes  by  less  than 
one  part  in  a  thousand*. 

(5)  It  is  known  by  observation  that  a  hen's  egg  is  always  laid 
blunt  end  foremost  f. 

(6)  It  can  be  shewn,  at  least  as  a  very  common  rule,  that  those 
eggs  which  are  most  unsjnnmetrical,  or  most  tapered  off  posteriorly, 
are  also  eggs  of  a  large  size  relatively  to  the  parent  bird.  The 
guillemot  is  a  notable  case  in  point,  and  so  also  are  the  curlews, 
sandpipers,  phaleropes  and  terns.  We  may  accordingly  presume 
that  the  more  pointed  eggs  are  those  that  are  large  relatively  to 
the  tube  or  oviduct  through  which  they  have  to  pass,  or,  in  other 
words,  are  those  which  are  subject  to  the  greatest  pressure  while 

*  Cf.  Mailock,  op.  cit. 

t  This  was  known  to  Albertus  Magnus,  though  his  explanation  was  wrong, 
"Ova  autem  habentia  duos  colores  non  sunt  oranino  penitus  rotunda,  sed  ex  una 
parte  sunt  acuta  habentia  angulum  sphericum  acutum,  sicut  sunt  composita  ex 
duobus  seraispheris,  in  una  parte  extensis  ad  angulum  acutum  et  in  alia  parte 
sphericis  noo  extensis  in  loco  ubi  est  polus  ovi. . .  .  Et  in  exitu  ovi  acutus  angulus 
exit  uliimo,  eo  quod  ipse  porrectus  est  ad  interiora  matricis  versus  parietem  ubi 
ovum  cum  matrice  continuatur  in  sui  generatione"  {De  animalibas,  lib.  xvii, 
tract.  1,  c.  3). 


XV]  AND  OTHER  HOLLOW  STRUCTURES  939 

being  forced  along.  So  general  is  this  relation  that  we  may  go  still 
further,  and  presume  with  great  plausibihty  in  the  few  exceptional 
cases  (of  which  the  apteryx  is  the  most  conspicuous)  where  the  egg 
is  relatively  large  though  not  markedly  unsymmetrical,  that  in  these 
cases  the  oviduct  itself  is  in  all  probability  large  (as  Giinther  had 
suggested)  in  proportion  to  the  size  of  the  bird.  In  the  case  of  the 
common  fowl  we  can  trace  a  direct  relation  between  the  size  and 
shape  of  the  egg,  for  the  first  eggs  laid  by  a  young  pullet  are  usually 
smaller,  and  at  the  same  time  are  much  more  nearly  spherical  than  the 
later  ones ;  and,  moreover,  some  breeds  of  fowls  lay  proportionately 
smaller  eggs  than  others,  and  on  the  whole  the  former  eggs  tend  to 
be  rounder  than  the  latter*. 

We  may  now  proceed  to  enquire  more  particularly  how  the  form 
of  the  egg  is  controlled  by  the  pressures  to  which  it  is  subjected. 

The  egg,  just  prior  to  the  formation  of  the  shell,  is,  as  we  have 
seen,  a  fluid  body,  tending  to  a  spherical  shape  and  enclosed  within 
a  membrane. 

Our  problem,  then,  is:  Given  an  incompressible  fluid,  contained 
in  a  deformable  capsule,  which  is  either  (a)  entirely  inextensible,  or 
(6)  slightly  extensible,  and  which  is  placed  in  a  long  elastic  tube  the 
walls  of  which  are  radially  contractile,  to  determine  the  shape  under 
some  given  distribution  of  pressure.  We  may  assume,  at  least  to 
begin  with,  that  the  shell-membrane  is  homogeneous  and  isotropic — 
uniform  in  all  parts  and  in  all  directions. 

If  the  capsule  be  spherical,  inextensible,  and  completely  filled 
with  the  fluid,  absolutely  no  deformation  can  take  place.  The  few 
eggs  that  are  actually  or  approximately  spherical,  such  as  those  of 
the  tortoise  or  the  owl,  may  thus  be  alternatively  explained  as  cases 
where  little  or  no  deforming  pressure  has  been  applied  prior  to  the 
solidification  of  the  shell,  or  else  as  cases  where  the  capsule  was  so 

*  In  so  far  as  our  explanation  involves  a  shaping  or  moulding  of  the  egg  by 
the  uterus  or  oviduct  (an  agency  supplemented  by  the  proper  tensions  of  the 
egg),  it  is  curious  to  note  that  this  is  very  much  the  same  as  that  old  view  of 
Telesius  regarding  the  formation  of  the  embryo  {De  rerum  natura,  vi,  cc.  4  and  10), 
which  he  had  inherited  from  Galen,  and  of  which  Bacon  speaks  {Nov.  Org.  cap.  .50; 
cf.  Ellis's  note).  Bacon  expressly  remarks  that  "Telesius  should  have  been  able 
to  shew  the  like  formation  in  the  shells  of  eggs."  This  old  theory  of  embryonic 
modelling  survives  in  our  usage  of  the  term  "matrix"  for  a  "mould." 


940  ON  THE  SHAPES  OF  EGGS  [ch. 

little  capable  of  extension  and  so  completely  filled  as  to  preclude  the 
possibility  of  deformation. 

If  the  capsule  be  not  spherical,  but  be  inextensible,  then  only  such 
deformation  can  take  place  as  tends  to  make  the  shape  more  nearly 
spherical;  and  as  the  surface  area  is  thereby  decreased,  the  envelope 
must  either  shrink  or  pucker.  In  other  words,  an  incompressible 
fluid  contained  in  an  inextensible  envelope  cannot  be  deformed 
without  puckering  of  the  envelope. 

But  let  us  next  assume,  as  the  condition  by  which  this  result 
may  be  avoided,  that  the  envelope  is  to  some  extent  extensible  and 
that  deformation  is  so  far  permitted.  It  is  obvious  that,  on  the 
presumption  that  the  envelope  is  only  moderately  extensible,  the 
whole  structure  can  only  be  distorted  to  a  moderate  degree  away 
from  the  spherical  or  spheroidal  form. 

At  all  points  the  shape  is  determined  by  the  law  of  the  distribution 
of  radial  pressure  withih  the  given  region  of  the  tube,  surface  friction 
helping  to  maintain  the  egg  in  position.  If  the  egg  be  under 
pressure  from  the  oviduct,  but  without  any  marked  component 
either  in  a  forward  or  backward  direction,  the  egg  will  be  compressed 
in  the  middle,  and  will  tend  more  or  less  to  the  form  of  a  cylinder 
with  spherical  ends.  The  eggs  of  the  grebe,  cormorant,  or  crocodile 
may  be  supposed  to  receive  their  shape  in  such  circumstances. 

When  the  egg  is  subject  to  the  peristaltic  contraction  of  the 
oviduct  during  its  formation,  then  from  the  nature  and  direction  of 
motion  of  the  peristaltic  wave  the  pressure  will  be  greatest  some- 
where behind  the  middle  of  the  egg;  in  other  words,  the  tube  is 
converted  for  the  time  being  into  a  more  conical  form,  and  the 
simple  result  follows  that  the  anterior  end  of  the  -egg  becomes  the 
broader  and  the  posterior  end  the  narrower. 

The  peristalsis  of  the  oviduct  thus  plays  a  double  part,  in  pro- 
pelling the'  egg  down  the  oviduct  and  in  impressing  on  it  its  ovoid 
form;  but  the  whole  process  is  a  very  slow  one,  for  the  hen's  oviduct 
is  only  a  few  inches  long,  and  the  egg  is  some  ten  or  twelve  hours 
upon  its  way.  We  shall  consider  presently  certain  shells  which 
may  be  regarded  as  so  many  drops  or  vesicles  deformed  by  gravity; 
that  is  a  statical  problem.  Compared  with  it  the  problem  of  the 
egg  is  a  dynamical  one;  and  yet  it  becomes  a  quasi-statical  one, 
because  the  action  is  so  very  slow.     It  is  an  action  without  lag 


XV]  AND  OTHER  HOLLOW  STRUCTURES  941 

and  without  momentum;  and  the  question,  common  in  dynamical 
problems,  of  the  relation  between  the  period  of  the  appHcation  of 
the  force  and  the  free  period  of  response  or  adjustment  to  it  need 
not  concern  us  at  all. 

Again,  the  case  of  the  egg  is  somewhat  akin  to  a  hydrodynamical 
problem ;  for  as  it  lies  in  the  oviduct  we  may  look  on  it  as  a  stationary 
body  round  which  waves  are  flowing,  with  the  same  result  as  when 
a  body  moves  through  a  fluid  at  rest.  Thus  we  may  treat  it  as  a 
hydrodynamical  problem,  but  a  very  simple  one — simphfied  by  the 
absence  of  all  eddies  and  every  form  of  turbulence;  and  we  come 
to  look  on  the  egg  as  a  streamlined  structure,  though  its  streamlines 
are  of  a  very  simple  kind. 

The  mathematical  statement  of  the  case  begins  as  follows: 
In  our  egg,  consisting  of  an  extensible  membrane  filled  with  an 
incompressible  fluid  and  under  external  pressure,  the  equation  of 
the  envelope  is  ;p„  +  T  (1/r  +  \jr')  =  P,  where  p^  is  the  normal 
component  of  external  pressure  at  a  point  where  r  and  r'  are  the 
radii  of  curvature,  T  is  the  tension  of  the  envelope,  and  P  the 
internal  fluid  pressure.  This  is  simply  the  equation  of  an  elastic 
surface  where  T  represents  the  coefficient  of  elasticity;  in  other 
words,  a  flexible  elastic  shell  has  the  same  mathematical  properties 
as  our  fluid,  membrane-covered  egg.  And  this  is  the  identical 
equation  which  we  have  already  had  so  frequent  occasion  to  employ 
in  our  discussion  of  the  forms  of  cells;  save  only  that  in  these 
latter  we  had  chiefly  to  study  the  tension  T  (i.e.  the  surface-tension 
of  the  senii-fluid  cell)  and  had  little  or  nothing  to  do  with  the  factor 
of  external  pressure  (^„),  which  in  the  case  of  the  egg  becomes  of 
chief  importance. 

To  enquire  how  an  elastic  sphere  or  spheroid  will  be  deformed 
in  passing  down  a  peristaltic  tube  is  an  ill-defined  and  indeterminate 
problem ;  but  we  can  study  the  effect  produced  in  the  shape  of  any 
particular  egg,  and  so  far  infer  the  forces  which  have  been  in  action. 
We  need  only  study  a  single  meridian  of  the  egg,  inasmuch  as  we 
have  found  it  to  be  a  solid  of  revolution.  At  successive  points 
along  this  meridian,  let  us  determine  the  amount  of  curvature,  that 
is  to  say  the  principal  radii  of  curvature,  in  latitude  and  longitude, 
in  the  Gaussian  formula  P  =  pn  +  T  (1/r  4-  l/r') :  or,  as  we  may  write 
it  if  we  have  any  reason  to  doubt  the  uniformity  or  isotropy  of  the 


942  ON  THE  SHAPES  OF  EGGS  [ch. 

membrane,  Tjr  +  T'jr'.  The  sum  of  these  curvatures  varies  from  point 
to  point;  the  internal  or  hydrodynamical  pressure,  P,  is  constant;  and 
therefore  the  external  pressure,  jo„,  varies  from  point  to  point  with 
the  curvature,  and  is  a  direct  function  of  the  shape  of  the  egg. 

Some  few  eggs,  such  as  the  owl's  and  the  kingfisher's,  are  so  nearly 
spherical  that  we  are  apt  to  speak  of  them  as  spheres ;  but  they  are 
all  prolate  more  or  less,  and  no  egg  is  so  nearly  circular  in  meridional 
section  as  all  eggs  are  in  their  circles  of  latitude.  When  the  egg  is 
all  but  spherical  that  shape  may  be  due  (as  we  have  seen)  to  various 
causes :  to  a  relatively,  small  size  of  the  egg,  allowing  it  to  descend 
the  tube  under  a  minimum  of  peristaltic  pressure;  perhaps  to  an 
unusually  strong  shell-membrane,  resistant  of  deformation;  in 
general  terms,  to  a  possible  diminution  of  j?„ ,  or  a  possible  increase 
of  T.  But  all  eggs  have  approximately  spherical  ends,  and  the 
big  anterior  end  of  the  large  conical  eggs  of  plover  or  curlew  or 
guillemot  is  conspicuously  so.  Here  the  egg  projects  into  the  wide 
cavity  of  the  uncontracted  oviduct,  external  or  peristaltic  pressure 
does  not  exist,  the  shell-membrane  has  to  resist  internal  pressure 
without  further  external  support,  and  the  resultant  spherical  cur- 
vature is  an  indication  of  the  uniformity,  or  isotropy,  of  the  mem- 
brane. The  lesser  of  the  two  spherical  ends,  that  is  to  say  the 
posterior  end,  has  by  much  the  greater  curvature,  and  the  tension 
there  is  correspondingly  great.  It  would  seem  that  the  membrane 
ought  to  be  thicker  or  stronger  at  this  pointed  end  than  elsewhere, 
but  it  is  not  known  to  be  so.  In  any  case,  it  is  just  here,  in  this 
presumably  weakest  part,  that  we  are  most  apt  to  find  the  irregulari- 
ties and  deformities  of  misshapen  eggs. 

Within  the  egg  lies  the  yolk,  and  the  yolk  is  invariably  spherical 
or  very  nearly  so,  whatever  be  the  form  of  the  entire  egg.  The 
reason  is  simple,  and  lies  in  the  fact  that  the  fluid  yolk  is  itself 
enclosed  within  another  membrane,  between  which  and  the  shell- 
membrane  Hes  the  fluid  albumin,  which  transmits  a  uniform  hydro- 
static pressure  to  the  yolk*.  The  lack  of  friction  between  the  yolk- 
membrane  and  the  white  of  the  egg  is  indicated  by  the  well-known 
fact  that  the  "germinal  spot"  on  the  surface  of. the  yolk  is  always 

*  In  like  manner,  the  cell-nucleus  is  "usually  globular,  except  in  certain 
specialised  tissues,  or  when  it  degenerates"  (Darlington).  Whether  it  possesses 
a  membrane  is  matter  in  dispute,  but  it  at  all  events  possesses  a  surface,  with 
a  phase- difference  between  it  and  the  surrounding  cytoplasm.     Cf.  above,  p.  295. 


XV]  AND  OTHER  HOLLOW  STRUCTURES  943 

found  uppermost,  however  we  may  place  and  wherever  we  may  open 
the  egg ;  that  is  to  say,  the  yolk  easily  rotates  within  the  egg,  bringing 
its  lighter  pole  uppermost. 

In  its  passage  down  the  oviduct  the  egg  is  not  merely  thrust  but 
also  screwed  along;  and  its  spiral  course  leaves  traces  on  wellnigh 
all  its  structure  save  the  shell.  When  we  have  broken  the  shell  of 
a  hard-boiled  egg  the  shell-membrane  below  peels  off  in  spiral  strips, 
and  even  the  white  tends  to  flake  off  in  layers,  spirally.  In  the 
fresh  unboiled  egg  two  knotted  cords — the  treadles  or  chalazae — are 
connected  with  the  yolk,  and  lie  fore-and-aft  of  it,  loose  in  the 
albimien.  These  represent  the  free  ends  of  a  yolk-membrane,  which 
got  caught  in  the  constricted  oviduct  while  the  yolk  between  them 
was  being  screwed  along:  very  much  as  we  may  wrap  an  apple  in 
a  handkerchief,  hold  the  two  ends  fast,  and  twirl  the  apple  round. 

These,  then,  are  the  general  principles  involved  in,  and  illustrated 
by,  the  configuration  of  an  egg ;  and  they  take  us  as  far  as  we  can 
safely  go  without  actual  quantitative  determination,  in  each  par- 
ticular case,  of  the  forces  concerned*. 

In  certain  cases  among  the  invertebrates,  we  again  find  instances 
of  hard-shelled  eggs  which  have  obviously  been  moulded  by  the 
oviduct,  or  so-called  "ootype,"  in  which  they  have  lain:  and  not 
merely  in  such  a  way  as  to  shew  the  effects  of  peristaltic  pressure 
upon  a  uniform  elastic  envelope,  but  so  as  to  impress  upon  the 
egg  the  more  or  less  irregular  form  of  the  cavity  within  which  it 
had  been  for  a  time  contained  and  compressed. ,  After  this  fashion 
is  explained  the  curious  form  of  the  egg  in  Bilharzia  (Schistosoma) 
haematobium,  a  formidable  parasitic  worm  to  which  is  due  a  disease 
wide-spread  in  Africa  and  Arabia,  and  an  especial  scourge  of  the 
Mecca  pilgrims.  The  egg  in  this  worm  is  provided  at  one  end  with 
a  little  spine,  which  is  explained  as  having  been  moulded  within  a 
little  funnel-shaped  expansion  of  the  uterus,  just  where  it  communi- 
cates with  the  common  duct  leading  from  the  ovary  and  yolk-gland. 
Owing  to  some  anatomical  difference  in  the    uterus,    the   httle 

*  It  is  a  common  but  unfounded  belief  among  poultry-men  that  shape  and  size 
are  related  to  the  sex  of  the  egg,  the  longer  eggs  producing  mostly  male  chicks. 
That  there  is  no  such  correlation  between  sex  on  the  one  hand  and  weight,  length 
or  shape  on  the  other,  has  been  clearly  demonstrated.  Cf.  M.  A.  Jull  and  J.  P. 
Quinn,  Journ.  Agr.  Research,  xxix,  pp.  195-201,  1924. 


944  ON  THE  SHAPES  OF  EGGS  [ch. 

spine  may  be  at  the  end  or  towards  the  side  of  the  egg :  and  this 
-visible  difference  has  led  to  the  recognition  of  a  new  species,  S. 
mansoni*.  In  a  third  species,  S.  japonicum,  the  egg  is  described 
as  bulging  into  a  so-called  "calotte,"  or  bubble-like  convexity  at 
the  end  opposite  to  the  spine.  This,  I  think,  may,  with  very  httle 
doubt,  be  ascribed  to  hardening  of  the  egg-shell  having  taken  place 
just  at  the  time  when  partial  relief  from  pressure  was  being 
experienced  by  the  egg  in  the  neighbourhood  of  the  dilated  orifice 
of  the  oviduct. 

This  case  of  Bilharzia  is  not,  from  our  present  point  of  view,  a 
very  important  one,  but  nevertheless  it  is  interesting.  It  ascribes 
to  a  mechanical  cause  a  curious  pecuUarity  of  form ;  and  it  shews,  by 
reference  to  this  mechanical  principle,  how  two  simple  mechanical 
modifications  of  the  same  thing  may  not  only  seem  very  different 
to  the  systematic  naturaUst's  eye,  but  may  actually  lead  to  the 
recognition  of  a  new  species,  with  its  own  geographical  distribution, 
and  its  own  pathogenic  characteristics. 

On  the  form  of  sea-urchins 

As  a  corollary  to  the  problem  of  the  bird's  egg,  we  may  consider 
for  a  moment  the  forms  assumed  by  the  shells  of  the  sea-urchins. 
These  latter  are  commonly  divided  into  two  classes — the  Regujar 
and  the  Irregular  Echinids.  The  regular  sea-urchins,  save  in  sHglit 
details  which  do  not  affect  our  problem,  have  a  complete  axial 
symmetry.  The  axis  of  the  animal's  body  is  vertical,  with  mouth 
below  and  the  intestinal  outlet  above;  and  around  this  axis  the 
shell  is  built  as  a  symmetrical  system.  It  follows  that  in  horizontal 
section  the  shell  is  everywhere  circular,  and  we  need  only  consider 
its  form  as  seen  in  vertical  section  or  projection.  The  irregular 
urchins  (very  inaccurately  so-called)  have  the  anal  extremity  of 
the  body  removed  from  its  central,  dorsal  situation;  and  it  follows 
that  they  have  now  a  single  plane  of  symmetry,  about  which  the 
organism,  shell  and  all,  is  bilaterally  symmetrical.  We  need  not 
concern  ourselves  in  detail  with  the  shapes  of  their  shells,  which 
may  be  very  simply  interpreted,  by  the  help  of  radial  coordinates, 
as  deformations  of  the  circular  or  "tegular"  type. 

*  L.  W.  Sambon,  Proc.  Zool.  Soc.  1907  (i),  p.  283;  also  in  Journ.  Trop.  Med.  and 
Hygiene,  Sept.  15,  1926. 


XV]  AND  OF  SEA-URCHINS  945 

The  sea-urchin  shell  consists  of  a  membrane,  stiffened  into  rigidity 
by  calcareous  deposits,  which  constitute  a  beautiful  skeleton  of 
separate,  neatly  fitting  "ossicles."  The  rigidity  of  the  shell  is  more 
apparent  than  real,  for  the  entire  structure  is,  in  a  sluggish  way, 
plastic ;  inasmuch  as  each  Httle  ossicle  is  capable  of  growth,  and  the 
entire  shell  grows  by  increments  to  each  and  all  of  these  multi- 
tudinous elements,  whose  individual  growth  involves  a  certain 
amount  of  freedom  to  move  relatively  to  one  another ;  in  a  few  cases 
the  ossicles  are  so  little  developed  that  the  whole  shell  appears 
soft  and  flexible.  The  viscera  of  the  animal  occupy  but  a  small 
part  of  the  space  within  the  shell,  the  cavity  being  mainly  filled  by 
a  large  quantity  of  watery  fluid,  whose  density  must  be  very  near 
to  that  of  the  external  sea-water. 

Apart  from  the  fact  that  the  sea-urchin  continues  to  grow,  it 
is  plain  that  we  have  here  the  same  general  conditions  as  in  the 
egg-shell,  and  that  the  form  of  the  sea-urchin  is  subject  to  a  similar 
equihbrium  of  forces.  But  there  is  this  important  difference,  that 
an  external  muscular  pressure  (such  as  the  .oviduct  administers 
during  the  consolidation  of  the  egg-shell)  is  now  lacking.  In  its 
place  we  have  the  steady  continuous  influence  of  gravity,  and  there 
is  yet  another  force  which  in  all  probability  we  require  to  take  into 
consideration. 

While  the  sea-urchin  is  alive,  an  immense  number  of  delicate 
"tube-feet,"  with  suckers  at  their  tips,  pass  through  minute  pores 
in  the  shell,  and,  like  so  many  long  cables,  moor  the  animal  to  the 
ground.  They  constitute  a  symmetrical  system  of  forces,  with  one 
resultant  downwards,  in  the  direction  of  gravity,  and  another  out- 
wards in  a  radial  direction ;  and  if  we  look  upon  the  shell  as  originally 
spherical,  both  will  tend  to  depress  the  sphere  into  a  flattened  cake. 
We  need  not  consider  the  radial  component,  but  may  treat  the 
case  as  that  of  a  spherical  shell  symmetrically  depressed  under  the 
influence  of  gravity.  This  is  precisely  the  condition  which  we  have 
to  deal  with  in  a  drop  of  liquid  lying  on  a  plate ;  the  form  of  which 
is  determined  by  its  own  uniform  surface-tension,  plus  gravity, 
acting  against  the  internal  hydrostatic  pressure.  Simple  as  this 
system  is,  the  full  mathematical  investigation  of  the  form  of  a  drop 
is  not  easy,  and  we  can  scarcely  hope  that  the  systematic  study 
of  the  Echinodermata  will  ever  be  conducted  by  methods  based 


946  ON  THE  SHAPES  OF  EGGS  [ch. 

on  Laplace's  diiferential  equation*;  but  we  have  little  difficulty  in 
seeing  that  the  various  forms  represented  in  a  series  of  sea-urchin 
shells  are  no  other  than  those  which  we  may  easily  and  perfectly 
imitate  in  drops. 

In  the  case  of  the  drop  of  water  (or  of  any  other  particular  liquid) 
the  specific  surface-tension  is  always  constant,  and  the  pressure 
varies  inversely  as  the  radius  of  curvature;  therefore  the  smaller 
the  drop  the  more  nearly  is  it  able  to  conserve  the  spherical  form, 
and  the  larger  the  drop  the  more  does  it  become  flattened  under 
gravityf.  We  can  imitate  this  phenomenon  by  using  india-rubber 
balls  filled  with  water,  of  different  sizes ;  the  httle  ones  will  remain 
very  nearly  spherical,  but  the  larger  will  fall  down  "of  their  own 
weight,"  into  the  form  of  more  and  more  flattened  cakes;  and  we 
see  the  same  thing  when  we  let  drops  of  heavy  oil  (such  as  the 
orthotoluidene  spoken  of  on  p.  370)  fall  through  a  tall  column  of 
water,  the  httle  ones  remaining  round,  and  the  big  ones  getting 
more  and  more  flattened  as  they  sink.  In  the  case  of  the  sea-urchin, 
the  same  series  of  forms  may  be  assumed  to  occur,  irrespective  of 
size,  through  variations  in  T,  the  specific  tension,  or  "strength" 
of  the  enveloping  shell.  Accordingly  we  may  study,  entirely  from 
this  point  of  view,  such  a  series  as  the  following  (Fig.  454).  In 
a  very  few  cases,  such  as  the  fossil  Paheechinus,  we  have  an 
approximately  spherical  shell,  that  is  to  say  a  shell  so  strong  that 
the  influence  of  gravity  becomes  negligible  as  a  cause  of  deformation, 
just  as  (to  compare  small  things  with  great)  the  surface  tension  of 
mercury  is  so  high  that  small  drops  of  it  seem  perfectly  spherical  {. 
The  ordinary  species  of  Echinus  begin  to  display  a  pronounced 
depression,  and  this  reaches  its  maximum  in  such  soft-shelled  flexible 
forms  as  Phormosoma.     On  the  general  question  I  took  the  oppor- 

♦  Cf.  Bashforth  and  Adams,  Theoretical  Forms  of  Drops,  etc.,  Cambridge,  1883. 

t  The  drops  must  be  spherical,  or  very  nearly  so,  to  produce  a  rainbow.  But  the 
bow  is  said  to  be  always  better  defined  near  the  top  than  down  below;  which  seems 
to  shew  that  the  lower  and  larger  raindrops  are  the  less  perfect  spheres.  (Cf. 
T.  W.  Backhouse,  Symons's  M.  Met.  Mag.  1879,  p.  25.)  For  the  small  round 
droplets  in  the  cloud  tend  to  cannon  off  one  another,  and  remain  small  and  spherical. 
But  when  there  comes  a  diflference  of  potential  between  cloud  and  cloud,  or  be- 
tween earth  and  sky,  then  the  spherules  become  distorted,  one  droplet  coalesces 
with  another,  and  the  big  drops  begin  to  fall. 

X  Cf.  A.  Ferguson,  On  the  theoretical  shape  of  large  bubbles  and  drops,  Phil.  Mag. 
(6),  XXV,  pp.  507-520,  1913. 


XV] 


AND  OF  SEA-UKCHINS 


947 


tunity  of  consulting  Mr  C.  R.  Darling,  who  is  an  acknowledged 
expert  in  drops,  and  he  at  once  agreed"  with  me  that  such  forms  as  are 
represented  in  Fig.  454  are  no  other  than  diagrammatic  illustrations 
of  various  kinds  of  drops,  "most  of  which  can  easily  be  reproduced 
in  outline  by  the  aid  of  liquids  of  approximately  equal  density  to 
water,  although  some  of  them  are  fugitive."  He  found  a  difficulty 
in  the  case  of  the  outline  which  represents  Asthenosoma,  but  the 
reason  for  the  anomaly  is  obvious;   the  flexible  shell  has  flattened 


Fig.  454.  Diagrammatic  vertical  outlines  of  various  sea-urchins:  A,  Palaeechinus; 
B,  Echinus  acutits;  C,  Cidaris;  D,  D',  Coelopleurus;  E,  E',  Genicopatagus ; 
F,  Phormosoma  liiculenter ;   G,  P.  tenuis;   H,  Asthenosoma;   I,  Urechinus. 

down  until  it  has  come  in  contact  with  the  hard  skeleton  of  the  jaws, 
or  "  Aristotlci's  lantern,"  within,  and  the  curvature  of  the  outline 
is  accordingly  disturbed.  The  elevated,  conical  shells  such  as  those 
of  Urechinus  and  Coelopleurus  evidently  call  for  some  further  ex- 
planation; for  there  is  here  some  cause  at  work  to  elevate,  rather 
than  to  depress  the  shell.  Mr  Darling  tells  me  that  these  forms 
"are  nearly  identical  in  shape  with  globules  I  have  frequently 
obtained,  in  which,  on  standing,  bubbles  of  gas  rose  to  the  summit 
and  pressed  the  skin  upwards,  without  being  able  to  escape."  The 
same  condition  may  be  at  work  in  the  sea-urchin;    but  a  similar 


948      ON  THE  FORM  AND  BRANCHING      [ch. 

tendency  would  also  be  manifested  by  the  presence  in  the  upper 
part  of  the  shell  of  any  accumulation  of  substance  Hghter  than  water, 
such  as  is  actually  present  in  the  masses  of  fatty,  oily  eggs. 

On  the  form  and  branching  of  blood-vessels 

Passing  to  what  may  seem  a  very  different  subject,  we  may 
investigate  a  number  of  interesting  points  in  connection  with  the 
form  and  structure  of  the  blood-vessels,  and  we  shall  find  ourselves 
helped,  at  least  in  the  outset,  by  the  same  equations  as  those  we 
have  used  in  studying  the  egg-shell. 

We  know  that  the  fluid  pressure  (P)  within  the  vessel  is  balanced 
by  (1)  the  tension  (T)  of  the  wall,  divided  by  the  radius  of  curvature, 
and  (2)  the  external  pressure  (pn),  normal  to  the  wall:  according  to 
our  formula 

P  =  y„+r(l/r+l//). 

If.  we  neglect  the  external  pressure,  that  is  to  say  any  support 
which  may  be  given  to  the  vessel  by  the  surrounding  tissues,  and 
if  we  deal  only  with  a  cylindrical  vein  or  artery,  this  formula 
becomes  simpUfied  to  the  form  P  =  T/R.  That  is  to  say,  under 
constant  pressure,  the  tension  varies  as  the  radius.  But  the  tension, 
per  unit  area  of  the  vessel,  depends  upon  the  thickness  of  the  wall, 
that  is  to  say  on  the  amount  of  membranous  and  especially  of 
muscular  tissue  of  which  it  is  Qomposed.  Therefore,  so  long  as  the 
pressure  is  constant,  the  thickness  of  the  wall  should  vary  as  the 
radius,  or  as  the  diameter,  of  the  blood-vessel. 

But  it  is  not  the  case  that  the  pressure  is  constant,  for  it 
gradually  falls  off,  by  loss  through  friction,  as  we  pass  from  the 
large  arteries  to  the  small;  and  accordingly  we  find  that  while,  for 
a  time,  the  cross-sections  of  the  larger  and  smaller  vessels  are 
symmetrical  figures,  with  the  wall-thickness  proportional  to  the  size 
of  the  tube,  this  proportion  is  gradually  lost,  and  the  walls  of  the 
small  arteries,  and  still  more  of  the  capillaries,  become  exceedingly 
thin,  and  more  so  than  in  strict  proportion  to  the  narrowing  of  the 
tube. 

In  the  case  of  the  heart  we  have,  within  each  of  its  cavities,  a 
pressure  which,  at  any  given  moment,  is  constant  over  the  whole  wall- 


I 


XV]  OF  BLOOD-VESSELS  949 

area,  but  the  thickness  of  the  wall  varies  very  considerably.  For 
instance,  in  the  left  ventricle  the  apex  is  by  much  the  thinnest  por- 
tion, as  it  is  also  that  with  the  greatest  curvature.  We  may  assume, 
therefore  (or  at  least  suspect),  that  the  formula,  ^  (1/r  +  1/r')  =  C, 
holds  good ;  that  is  to  say,  that  the  thickness  {tj  of  the  wall  varies 
inversely  as  the  mean  curvature.  This  may  be  tested  experimentally, 
by  dilating  a  heart  with  alcohol  under  a  known  pressure,  and  then 
measuring  the  thickness  of  the  walls  in  various  parts  after  the  whole 
organ  has  become  hardened.  By  this  means  it  is  found  that,  for 
each  of  the  cavities,  the  law  holds  good  with  great  accuracy*. 
Moreover,  if  we  begin  by  dilating  the  right  ventricle  and  then  dilate 
the  left  in  like  manner,  until  the  whole  heart  is  equally  and  sym- 
metrically dilated,  we  find  (1)  that  we  have  had  to  use  a  pressure 
in  the  left  ventricle  from  six  to  seven  times  as  great  as  in  the  right 
ventricle,  and  (2)  that  the  thickness  of  the  walls  i»  just  in  the  same 
proportion  f. 

Many  problems  of  a  hydrodynamical  kind  arise  in  connection 
with  the  flow  of  blood  through  the  blood-vessels;  and  while  these 
are  of  primary  importance  to  the  physiologist  they  interest  the 
morphologist  in  so  far  as  they  bear  on  questions  of  structure  and  form. 
As  an  example  of  such  mechanical  problems  we  may  take  the  con- 
ditions which  go  to  determine  the  manner  of  branching  of  an  artery, 
or  the  angle  at  which  its  branches  are  given  off;  for,  as  John  Hunter 
said  J,  "To  keep  up  a  circulation  sufficient  for  the  part,  and  no  more, 
Nature  has  varied  the  angle  of  the  origin  of  the  arteries  accordingly." 
This  is  a  vastly  important  theme,  and  leads  us  a  deal  farther  than 
does  the  problem,  petty  in  comparison,  of  the  shape  of  an  egg.  For 
the  theorem  which  John  Hunter  has  set  forth  in  these  simple  words 
is  no  other  than  thaf  " principle  of  minimal  work"  which  is  funda- 
mental in  physiology,  and  which  some  have  deemed  the  very  criterion 

*  R.  H.  Woods,  On  a  physical  theorem  applied  to  tense  membranes,  Journ. 
of  Anat.  and  Phys.  xxvi,  pp.  362-371,  1892.  A  similar  investigation  of  the 
tensions  in  the  uterine  wall,  and  of  the  varying  thickness  of  its  muscles,  was 
attempted  by  Haughton  in  his  Animal  Mechanics,  1873,  pp.  151-158. 

t  This  corresponds  with  a  determination  of  the  normal  pressures  (in  systole) 
by  Knohl,  as  being  in  the  ratio  of  1  :  6-8. 

%  Essays,  edited  by  Owen,  i,  p.  134,  1861.  The  subject  greatly  interested  Keats. 
See  his  Notebook,  edited  by  M.  B.  Forman,  1932,  p,  7;  and  cf.  Keats  as  a  Medical 
Student,  by  Sir  Wm  Hale- White,  in  Guy's  Hospital  Reports,  lxxiii,  pp.  249-262, 
1925. 


950      ON  THE  FORM  AND  BRANCHING     [ch. 

of  "organisation*."  For  the  principle  of  Lagrange,  the  "principle 
of  virtual  work,"  is  the  key  to  physiological  equilibrium,  and 
physiology  itself  has  been  called  a  problem  in  maxima  and  minima  f. 

This  principle,  overflowing  into  morphology,  helps  to  bring  the 
morphological  and  the  physiological  concepts  together.  We  have 
dealt  with  problems  of  maxima  and  minima  in  many  simple  con- 
figurations, where  form  alone  seemed  to  be  in  question;  and  we 
meet  with  the  same  principle  again  wherever  work  has  to  be  done 
and  mechanism  is  at  hand  to  do  it.  That  this  mechanism  is  the 
best  possible  under  all  the  circumstaaces  of  the  case,  that  its  work 
is  done  with  a  maximum  of  efiiciency  and  at  a  minimum  of  cost, 
may  not  always  lie  within  oui;  range  of  quantitative  demonstration, 
but  to  believe  it  to  be  so  is  part  of  our  common  faith  in  the  perfection 
of  Nature's  harndiwork.  All  the  experience  and  the  very  instinct  of 
the  physiologist  tells  him  it  is  true ;  he  comes  to  use  it  as  a  postulate, 
or  meihodus  inveniendi,  and  it  does  not  lead  him  astray.  The  dis- 
covery of  the  circulation  of  the  blood  was  implicit  in,  or  followed 
quickly  after,  the  recognition  of  the  fact  that  the  valves  of  heart 
and  veins  are  adapted  to  a  one-way  circulation ;  anA  we  may  begin 
likewise  by  assuming  a  perfect  fitness  or  adaptation  in  all  the  minor 
details  of  the  circulation. 

As  part  of  our  concept  of  organisation  we  assume  that  the  cost 
of  operating  a  physiological  system  is  a  minimum,  what  we  mean 
by  cost  being  measurable  in  calories  and  ergs,  units  whose  dimensions 
are  equivalent  to  those  of  work.  The  circulation  teems  with  illustra- 
tions of  thi^  great  and  cardinal  principle.  "  To  keep  up  a  circulation 
sufiicient  for  the  part  and  no  more"  Nature  has  not  only  varied  the 
angle  of  branching  of  the  blood-vessels  to  suit  her  purpose,  she  has 
regulated  the  dimensions  of  every  branch  and  stem  and  twig  and 
capillary;  the  normal  operation  of  the  heart  is  perfection  itself, 
even  the  amount  of  oxygen  which  enters  and  leaves  the  capillaries 
is  such  that  the  work  involved  in  its  exchange  and  transport  is 
a  minimum.  In  short,  oxygen  transport  is  the  main  object  of  the 
circulation,  and  it  seems  that  through  all  the  trials  and  errors  of 

*  Cf.  Cecil  D.  Murray,  The  physiological  principle  of  minimal  work,  in  the  vascular 
system,  and  the  cost  of  blood-volume,  Proc.  Acad.  Nat.  Sci.  xii,  pp.  207-214,  1926; 
The  angle  of  branching  of  the  arteJries,  Journ.  Gen.  Physiol,  ix,  pp.  835-841,  1926; 
On  the  branching-angles  of  trees,  ibid,  x,  p.  72;'),  1927. 

t  By  Dr  F.  H.  Pike,  quoted  by  C.  D.  Murray. 


XV]  OF  BLOOD-VESSELS  951 

growth  and  evolution  an  efficient  mode  of  transport  has  been  attained. 
To  prove  that  it  is  the  very  best  of  all  possible  modes  of  transport 
may  be  beyond  our  powers  and  beyond  our  needs;  but  to  assume 
that  it  is  perfectly  economical  is  a  sound  working  hypothesis*.  And 
by  this  working  hypothesis  we  seek  to  understand  the  form  and 
dimensions  of  this  structure  or  that,  in  terms  of  the  work  which  it 
has  to  do. 

The  general  principle,  then,  is  that  the  form  and  arrangement 
of  the  blood-vessels  is  such  that  the  circulation  proceeds  with  a 
minimum  of  effort,  and  with  a  minimum  of  wall-surface,  the  latter 
condition  leading  to  a  minimum  of  friction  and  being  therefore 
included  in  the  first.  What,  then,  should  be  the  angle  of  branching, 
such  that  there  shall  be  the  least  possible  loss  of  energy  in  the  course 
of  the  circulation?  In  order  to  solve  this  problem  in  any  particular 
case  we  should  obviously  require  to  know  (1)  how  the  loss  of  energy 
depends  upon  the  distance  travelled,  and  (2)  how  the  loss  of  energy 
varies  with  the  diameter  of  the  vessel.  The 
loss  of  energy  is  evidently  greater  in  a 
narrow  tube  than  in  a  wide  one,  and  greater, 
obviously,  in  a  long  journey  than  a  short. 
If  the  large  artery,  AB,  gives  off  a  com- 
paratively narrow  branch  leading  to  P 
(such  as  CP,  or  DP),  the  route  ACP  is 
evidently  shorter  than  ADP,  but  on  the 
other  hand,  by  the  latter  path,  the  blood  has 
tarried  longer  in  the  wide  vessel  ^B,  and  has    '^  ^.    ^g 

had  a  shorter  course  in  the  narrow  branch. 

The  relative  advantage  of  the  two  paths  will  depend  on  the  loss 
of  energy  in  the  portion  CD,  as  compared  with  that  in  the  alternative 
portion  CD',  the  one  being  short  and  narrow,  the  other  long  and 
wide.     If  we  ask,  then,  which  factor  is  the  more  important,  length 

*  Cf.  A.  W.  Volkmann,  Die  Haemodynamik  nach  Verauchen,  Leipzig,  1850 
(a  work  of  great  originality);  G.  Schwalbe,  Ueber.  .  .die  Gestaltung  des  Arterien- 
systems,  Jen.  Zeitschr.  xii,  p.  267,  1878;  W.  Hess,  Eine  mechanischbedingte  Gesetz- 
massigkeit  im  Bau  des  Blutgefasssystems,  A.f.  Entw.  Me'ch.  xvi,  p.  632, 1903 ;  Ueber 
die  peripherische  Reguiierung  der  Blutzirkulation,  Pfluger's  Archiv,  CLXvm,  pp. 
439-490,  1917;  R.  Thoma,  Die  mittlere  Durchflussmengen  der  Arteriert  des 
Menschen  ais  Funktion  des  Gefassradius,  ibid,  clxxxix,  pp.  282-310,  cxcra, 
pp.  385-406,  1921-22;  E.  Blum,  Querschnittsbeziehungen  zwischen  Stamm  u.  Asten 
im  Arteriensystem,  ibid,  clxxv,  pp.  1-19,  1919. 


952 


ON  THE  FORM  AND  BRANCHING 


[CH. 


or  width,  we  may  safely  take  it  that  the  question  is  one  of  degree ; 
and  that  the  factor  of  width  will  become  the  more  important  of 
the  two  wherever  artery  and  branch  are  markedly  unequal  in  size. 
In  other  words,  it  would  seem  that  for  small  branches  a  large  angle 
of  bifurcation,  and  for  large  branches  a  small  one,  is  always  the 
better.  Roux  has  laid  down  certain  rules  in  regard  to  the  branching 
of  arteries,  which  cojrespond  with  the  general  conclusions  which  we 
have  just  arrived  at.  The  most  important  of  these  are  as  follows: 
(1)  If  an  artery  bifurcates  into  two  equal  branches,  these  branches 
come  off  at  equal  angles  to  the  main  stem.  (2)  If  one  of  the  two 
branches  be  smaller  than  the  other,  then  the  main  branch,  or 
continuation  of  the  original  artery,  makes  with  the  latter  a  smaller 
angle  than  does  the  smaller  or  "lateral"  branch.  And  (3)  all 
branches  which  are  so  small  that  they  scarcely  seem  to  weaken  or 
diminish  the  main  stem  come  off  from  it  at  a  large  angle,  from 
about  70°  to  90°. 

We  may  follow  Hess  in  a  further  investigation  of  the  phenomenon. 
Let  AB  be  an  artery,  from  which  a  branch  has  to  be  given  off  so 

as  to  reach  P,  and  let  ACP,  ADP,  etc., 
be  alternative  courses  which  the  branch 
may  follow:  CD,  DE,  etc.,  in  the 
diagram,  being  equal  distances  (=  I) 
along  AB.  Let  us  call  the  angles  PCD, 
PDE,  Xi,  X2,  etc.:  and  the  distances 
CD',  DE',  by  which  each  branch  exceeds 
the  next  in  length,  we  shall  call  li,l2,  etc. 
Now  it  is  evident  that,  of  the  courses 
shewn,  ACP  is  the  shortest  which  the 
blood  can  take,  but  it  is  also  that  by 
which  its  transit  through  the  narrow 
branch  is  the  longest.  We  may  reduce 
its  transit  through  the  narrow  branch 
more  and  more,  till  we  come  to  CGP,  or 
rather  to  a  point  where  the  branch  comes 
off  at  right  angles  to  the  main  stem ;  but 
in  so  doing  we  very  considerably  increase  the  whole  distance 
travelled.  We  may  take  it  that  there  will  be  some  intermediate  point 
which  will  strike  the  balance  of  advantage. 


Fig.  456. 


XV]  OF  BLOOD-VESSELS  953 

Now  it  is  easy  to  shew  that  if,  in  Fig.  456,  the  route  ADP  and 
AEP  (two  contiguous  routes)  be  equally  favourable,  then  any  other 
route  on  either  side  of  these,  such  as  ACP  or  AFP,  must  be  less 
favourable  than  either.  Let  ADP  and  AEP,  then,  be  equally 
favourable;  that  is  to  say,  let  the  loss  of  energy  which  the  blood 
suffers  in  its  passage  along  these  two  routes  be  equal.  Then,  if 
we  make  the  distance  DE  very  small,  the  angles  Xg  and  x^  are  nearly 
equal,  and  may  be  so  treated.  And  again,  if  DE  be  very  small, 
then  DE'E  becomes  a  right  angle,  and  I2  (or  DE')  =  I  cos  Xg.  But 
if  L  be  the  loss  of  energy  per  unit  distance  in  the  wide  tube  AB, 
and  U  be  the  corresponding  loss  of  energy  in  the  narrow  tube  DP, 
etc.,  then  IL  =  I2L' ,  because,  as  we  have  assumed,  the  loss  of  energy 
on  the  route  DP  is  equal  to  that  on  the  whole  route  DEP.  Therefore 
IL  =  IL  cos  Xg,  and  cos  x^  =  LjU .  That  is  to  say,  the  most  favour- 
able angle  of  branching  will  be  such  that  the  cosine  of  the  angle  is 
equal  to  the  ratio  of  the  loss  of  energy  which  the  blood  undergoes, 
per  unit  of  length,  in  the  main  vessel,  as  compared  with  that  which 
it  undergoes  in  the  branch.  The  path  of  a  ray  of  light  from  one 
refractive  medium  to  another  is  an  analogous  but  much  more  famous 
problem;  and  the  analogy  becomes  a  close  one  when  we  look  upon 
the  branching  artery  as  the  special  case  of  "grazing  incidence." 

After  thus  dealing  with  the  most  suitable  angle  of  branching, 
we  have  still  to  consider  the  appropriate  cross-section  of  the  branches 
compared  with  the  main  trunk,  for  instance  in  the  special  case  where 
a  main  artery  bifurcates  into  two.  That  the  sectional  area  of  the  two 
branches  may  together  equal  the  area  of  the  parent  trunk,  it  is  (of 
course)  only  necessary  that  the  diameters  of  trunk  and  branch  should 
be  as\/2  :  1,  or  (say)  as  14  :  10,  or  (still  more  roughly)  as  10  :  7;  and 
in  the  great  vessels,  this  simple  ratio  comes  very  nearly  true.  We 
have,  for  instance,  the  following  measurements  of  the  common  ihac 
arteries,  into  which  the  abdominal  aorta  subdivides: 

Internal  diameter  of  abdominal  arteries* 


Aorta  abdom.  (mm.) 

15-2             120 

141 

13-9 

Iliaca  comm.  d. 
Iliaca  comm.  s. 

10-8              8-8 
10-7              8-6 

10-4 
9-5 

8-6 
10-0 

Mean  of  do. 

10-8              8-7 

100 

9-3 

Ratio 

71               72 

69 
Av.  70 

p.c 

67  p.c. 

* 

From  R.  Thoma,  op.  cit. 

P- 

388. 

954      ON  THE  FORM  AND  BRANCHING      [ch. 

But  the  increasing  surface  of  the  branches  soon  means  increased 
friction,  and  a  slower  pace  of  the  blood  travelling  through;  and 
therefore  the  branches  must  be  more  capacious  than  at  first  appears. 
It  becomes  a  question  not  of  capacity  but  of  resistance;  and  in 
general  terms  the  answer  is  that  the  ratio  of  resistance  to  cross- 
section  shall  be  equal  in  every  part  of  the  system,  before  and  after 
bifurcation,  as  a  condition  of  least  possible  resistance  in  the  whole 
system;  the  total  cross-section  of  the  branches,  therefore,  must  be 
greater  than  that  of  the  trunk  in  proportion  to  the  increased 
resistance. 

An  approximate  result,  familiar  to  students  of  hydrodynamics,  is 
that  the  resistance  is  a  minimum,  and  the  condition  an  optimum,  when 
the  cross-section  of  the  main  stem  is  to  the  sum  of  the  cross-sections 
of  the  branches  as  1  :  ^2,  or  1  :  1-26.  Accordingly,  in  the  case  of 
a  blood-vessel  bifurcating  into  two  equal  branches,  the  diameter 
of  each  should  be  to  that  of  the  main  stem  (approximately)  as 


J 


1-26 

-^-il,     or  (say)  8:  10. 


While  these  statements  are  so  far  true,  and  while  they  undoubtedly 
cover  a  great  number  of  observed  facts,  yet  it  is  plain  that,  as  in 
all  such  cases,  we  must  regard  them  not  as  a  complete  explanation, 
but  as  factors  in  a  complicated  phenomenon :  not  forgetting  that 
(as  one  of  the  most  learned  of  all  students  of  the  heart  and  arteries, 
Dr  Thomas  Young,  said  in  his  Croonian  lecture*)  all  such  questions 
as  these,  and  all  matters  connected  with  the  muscular  and  elastic 
powers  of  the  blood-vessels,  "belong  to  the  most  refined  departments 
of  hydrauhcs";    and  Euler  himself  had  commented  on  the  "in- 

*  On  the  functions  of  the  heart  and  arteries,  Phil.  Trans.  1809,  pp.  1-31, 
of.  1808,  pp.  164-186;  Collected  Works,  i,  pp.  511-534,  1855.  The  same  lesson  is 
conveyed  by  all  such  work  as  that  of  Volkmann,  E.  H.  Weber  and  Poiseuille. 
Cf.  Stephen  Hales's  Statical  Essays,  ii,  Introduction:  "Especially  considering 
that  they  [i.e.  animal  Bodies]  are  in  a  manner  framed  of  one  continued  Maze  of 
innumerable  Canals,  in  which  Fluids  are  incessantly  circulating,  some  with  great 
Force  and  Rapidity,  others  with  very  different  Degrees  of  rebated  Velocity: 
Hence,  etc."  Even  Leonardo  had  brought  his  knowledge  of  hydrodynamics  to 
bear  on  the  valves  of  the  heart  and  the  vortex-like  eddies  of  the  blood.  Cf. 
J.  Playfair  McMurrich,  L.  da  Vinci,  the  Anatomist,  1930,  p.  165;  etc.  How  com- 
plicated the  physiological  aspect  of  the  case  becomes  may  be  judged  by  Thoma's 
papers  quoted  above. 


XV]  OF  BLOOD-VESSELS  955 

superable  difficulties"  of  this  sort  of  problem*.  Some  other 
explanation  must  be  sought  in  order  to  account  for  a  phenomenon 
which  particularly  impressed  John  Hunter's  mind,  namely  the 
gradually  altering  angle  at  which  the  successive  intercostal  arteries 
are  given  off  from  the  thoracic  aorta:  the  special  interest  of  this 
case  arising  from  the  regularity  and  symmetry  of  the  series,  for 
"there  is  not  another  set  of  arteries  in  the  body  whose  origins  are 
so  much  the  same,  whose  offices  are  so  much  the  same,  whose  dis- 
tances from  their  origin.to  the  place  of  use,  and  whose  uses  [?  sizes]  f 
are  so  much  the  same." 

The  mechanical  and  hydrodynamical  aspect  of  the  circulation 
was  as  plain  to  John  Hunter's  mind  as  it  had  been  to  William 
Harvey  or  to  Stephen  Hales,  or  as  it  was"  afterwards  to  Thomas 
Young;  but  it  was  not  always  plain  to  other  men.  When  a  turtle's 
heart  has  been  removed  from  its  body,  the  blood  may  still  be  seen 
moving  in  the  capillaries  for  some  short  while  thereafter;  and 
Haller,  seeing  this,  "attributed  it  to  some  unknown  power  which 
he  conceived  to  be  exerted  by  the  solid  tissues  on  the  blood  and  also 
by  the  globules  of  the  blood  on  each  other;  to  which  power,  until 
further  investigation  should  elucidate  its  nature,  he  gave  the  name 
of  attraction''  So  said  WilHam  Sharpey,  the  father  of  modern 
Enghsh  physiology;  and  Sharpey  went  on  to  say  that  "many 
physiologists  accordingly  maintain  the  existence  of  a  peculiar  pro- 
pulsive power  in  the  coats  of  the  capillary  vessels  different  from 
contractility,  or  that  the  globules  of  the  blood  are  possessed  of  the 
power  of  spontaneous  motion."  Alison,  great  physician  and  famous 
vitalist,  "extended  this  view,  in  so  far  as  he  regards  the  motion 
of  the  blood  in  the  capillaries  as  one  of  the  effects  produced  by  what 
he  calls  vital  attraction  and  repulsion,  powers  which  he  conceives 
to  be  general  attributes  of  hving  matter."  But  Sharpey 's  own 
clear  insight  so  far  overcame  his  faith  in  Alison  that  he  found  it 
"not  impossible  that  a  certain  degree  of  agitation  might  be  occa- 
sioned in  the  blood  by  the  elastic  resilience  of  the  vessels  reacting 
on  it,  after  the  distending  force  of  the  heart  has  been  withdrawn"; 
and,  in  short,  that  the  evidence  in  the  case  did  not  "warrant  the 

*  In  a  tract  entitled  Principia  pro  motu  sanguinis  per  arterias  determinando 
Op.  posth.  XI,  pp.  814-823,  1862. 

f  "Sizes"  is  Owen's  editorial  emendation,  which  seems  amply  justified. 


956      ON  THE  FORM  AND  BRANCHING      [ch. 

assumption  of  a  peculiar  power  acting  on  the  blood,  of  whose 
existence  in  the  animal  economy  we  have  as  yet  no  other  evidence  *," 

Sir  Charles  Bell,  whose  anatomical  skill  was  great  but  his  mathe- 
matical insight  small,  drew  the  conclusion,  of  no  small  historic 
interest,  that  "the  laws  of  hydraulics,  though  illustrative,  are  not 
strictly  applicable  to  the  explanation  of  the  circulation  of  the  blood, 
nor  to  the  actions  of  the  living  frame."  He  goes  on  to  say: 
*' Although  \^e  perceive  admirable  mechanism  in  the  heart,  and  in 
the  adjustment  of  the  tubes  on  hydraulic  principles:  and  although 
the  arteries  and  veins  have  form,  calibre  and  curves  suited^  to  the 
conveyance  of  fluid,  according  to  our  knowledge  of  hydraulic  engines: 
yet  the  laws  of  life,  or  of  physiology,  are. essential  to  the  explanation 
of  the  circulation  of  the  blood.  And  this  conclusion  we  draw,  not 
only  from  the  extent  and  minuteness  of  the  vessels,  but  also  from 
the  peculiar  nature  of  the  blood  itself.  Life  is  in  both,  and  a  mutual 
influence  prevails  f."  This  peculiar  form  of  vitalism  savours  more 
of  Bichat  and  the  French  school  than  of  the  teaching  of  John 
Hunter  or  Thomas  Young.  It  is  precisely  that  idea  of  "organic 
control"  or  "organic  coordination,"  which  the  physiologists  are 
always  reluctant  to  accept,  always  unwilling  to  abandon:  which 
is  said  to  be  inherent  in  every  process  or  operation  of  the  body,  and 
to  differentiate  biology  from  all  the  physical  sciences :  and  of  which 
in  our  own  day  Haldane  has  been  the  chief  and  great  protagonist. 
But  it  is  a  subject  with  which  this  book  is  not  concerned.    , 

To  conclude,  we  may  now  approach  the  question  of  economical 
size  of  the  blood-vessels  in  a  broader  way.  They  must  not  be  too 
small,  or  the  work  of  driving  blood  through  them  will  be  too  great; 

*  See  Sharpey's  article  on  Cilia,  in  Todd's  Cyclopaedia,  i,  p.  637,  1836;  also 
Allen  Thomson's  admirable  article  on  the  Circulation,  ibid.  p.  672.  Alison's  views 
were  based  not  only  on  Haller,  but  largely  on  Dr  James  Black's  Essay  on  the 
Capillary  Circulation,  London,  1825. 

t  Practical  Essays,  1842,  p.  88.  When  Sir  Charles  Bell  declared  that  hydraulic 
principles  were  not  enough,  but  that  "the  laws  of  life"  were  needed  to  explain  the 
circulation  of  the  blood,  he  was  right  from  his  point  of  view.  He  was  slow  to  see, 
and  unwilling  to  admit,  that  hydrodynamical  principles  suffice  to  explain  a  large, 
essential  part  of  the  problem ;  but  as  a  physiologist  he  had  every  reason  to  know 
that  that  part  was  not  the  whole.  He  may  have  had  many  things  in  mind:  the 
arrest  of  the  circulation  in  inflammation,  as  we  see  it  in  a  frog's  web ;  that  a  cut 
artery  bleeds  to  death  while  a  torn  one  does  not  bleed  at  all;  that  blood  does  not 
coagulate  when  stagnant  within  its  own  vessels — a  fact  which,  as  John  Hunter  said, 
"has  ever  appeared  to  me  the  most  interesting  fact  in  physiology." 


XV]  OF  BLOOD-VESSELS  957 

they  must  not  be  too  large,  or  they  will  hold  more  blood  than  is 
needed — and  blood  is  a  costly  thing.  We  rely  once  more  on 
Poiseuille's  Law*,  which  tells  us  the  amount  of  work  done  in  causing 
so  much  fluid  to  flow  through  a  tube  against  resistance,  the  said 
resistance  being  measured  by  the  viscosity  of  the  fluid,  the  coefficient 
of  friction  and  the  dimensions  of  the  tube;  but  we  have  also  to 
account  for  the  blood  itself,  whose  maintenance  requires  a  share 
of  the  bodily  fuel,  and  whose  cost  per  c.c.  may  (in  theory  at  least) 
be  expressed  in  calories,  or  in  ergs  per  day.  The  total  cost,  then, 
of  operating  a  given  section  of  artery  will  be  measured  by  (1)  the 
work  done  in  overcoming  its  resistance,  and  (2)  the  work,  done 
in  providing  blood  to  fill  it;  we  have  come  again  to  a  differential 
equation,  leading  to  an  equation  of  maximal -efficiency.  The  general 
result!  is  as  follows:  it  can  be  made  a  quantitative  one  by  intro- 
ducing known  experimental  values.  Were  blood  a  cheaper  thing 
than  it  is  we  might  expect  all  arteries  to  be  uniformly  larger  than 
they  are,, for  thereby  the  burden  on  the  heart  (the  flow  remaining 
equal)  would  be  greatly  reduced — thus  if  the  blood-vessels  were 
doubled  in  diameter,  and  their  volume  thereby  quadrupled,  the 
work  of  the  heart  would  be  reduced  to  one-sixteenth.  On  the  other 
hand,  were  blood  a  scarcer  and  still  costlier  fluid,  narrower  blood- 
vessels would  hold  the  available  supply;  but  a  larger  and  stronger 
heart  would  be  needed  to  overcome  the  increased  resistance. 

*  Owing  to  faulty  determination  of  the  fall  of  pressure  in  the  capillaries, 
Poiseuille's  equation  used  to  be  deemed  inapplicable  to  them ;  but  Krogh's  recent 
work  removes,  or  tends  to  remove,  the  inconsistency  {Anatomy  and  Physiology  of 
the  Capillaries,  Yale  University  Press,  1922). 

t  Cf.  C.  D.  Murray,  op.  cit.  p.  211. 


CHAPTER  XVI 

ON  FORM  AND  MECHANICAL  EFFICIENCY 

There  is  a  certain  large  class  of  morphological  problems  of  which 
we  have  not  yet  spoken,  and  of  which  we  shall  be  able  to  say  but 
Uttle.  Nevertheless  they  are  so  important,  so  full  of  deep  theoretical 
significance,  and  so  bound  up  with  the  general  question  of  form 
and  its  determination  as  a  result  of  growth,  that  an  essay  on 
growth  and  form  is  bound  to  take  account  of  them,  however  im- 
perfectly and  briefly.  The  phenomena  which  I  have  in  mind  are 
just  those  many  cases  where  adaptation  in  the  strictest  sense  is 
obviously  present,  in  the  clearly  demonstrable  form  of  mechanical 
fitness  for  the  exercise  of  some  particular  function  or  action  which 
has  become  inseparable  from  the  life  and  well-being  of  the  organism. 

When  we  discuss  certain  so-called  "adaptations"  to  outward 
circumstance,  in  the  way  of  form,  colour  and  so  forth,  we  are  often 
apt  to  use  illustrations  convincing  enough  to  certain  minds  but 
unsatisfying  to  others — in  other  words,  incapable  of  demonstration. 
With  regard  to  coloration,  for  instance,  it  is  by  colours  "cryptic," 
"\Yarning,"  "signalling,"  "mimetic,"  and  so  on*,  that  we  prosaically 
expound,  and  slavishly  profess  to  justify,  the  vast  AristoteUan 
synthesis  that  Nature  makes  all  things  with  a  purpose  and  "does 
nothing  in  vain."  Only  for  a  moment  let  us  glance  at  some  few 
instances  by  which  the  modern  teleologist  accounts  for  this  or  that 
manifestation  of  colour,  and  is  led  on  and  on  to  beUefs  and  doctrines 
to  which  it  becomes  more  and  more  difficult  to  subscribe. 

Some  dangerous  and  mahgnant  animals  are  said  (in  sober  earnest) 
to  wear  a  perpetual  war-paint,  in  order  to  "remind  their  enemies 
that  they  had  better  leave  them  alonef."     The  wasp  and  the  hornet, 

*  For  a  more  elaborate  classification,  into  colours  cryptic,  procryptic,  anti- 
cryptic,  apatetic,  epigamic,  sematic,  episematic,  aposematic,  etc.,  see  Poulton's 
Colours  of  Animals  (Int.  Scientific  Series,  Lxvin,  1890;  cf.  also  R.  Meldola, 
Variable  protective  colouring  in  insects,  P.Z.S.  1873,  pp.  153-162;  etc.  The  subject 
is  well  and  fully  set  forth  by  H.  B.  Cott,  Adaption  coloration  in  Animals,  1940. 

t  Dendy,  Evolutionary  Biology,  1912,  p.  336. 


CH.  XVI]    THE  INTERPRETATION  OF  COLOUR         959 

in  gallant  black  and  gold,  are  terrible  as  an  army  with  banners; 
and  the  Gila  Monster  (the  poison-lizard  of  the  Arizona  desert)  is 
splashed  with  scarlet — its  dread  and  black  complexion  stained  with 
heraldry  more  dismal.  But  the  wasp-Hke  livery  of  the  noisy,  idle 
hover-flies  and  drone-flies  is  but  stage  armour,  and  in  their  tinsel 
suits  the  little  counterfeit  cowardly  knaves  mimic  the  fighting  crew. 

The  jewelled  splendour  of  the  peacock  and  the  humming-bird, 
and  the  less  effulgent  glory  of  the  lyre-bird  and  the  Argus  pheasant, 
are  ascribed  to  the  unquestioned  prevalence  of  vanity  in  the  one  sex 
and  wantonness  in  the  other*. 

The  zebra  is  striped  that  it  may  graze  unnoticed  on  the  plain,  the 
tiger  that  it  may  lurk  undiscovered  in  the  jungle;  the  banded 
Chaetodont  and  Pomacentrid  fishes  are  further  bedizened  to  the 
hues  of  the  coral-reefs  in  which  they  dwell  f-  The  tawny  lion  is 
yellow  as  the  desert  sand;  but  the  leopard  wears  its  dappled  hide 
to  blend,  as  it  crouches  on  the  branch,  with  the  sun-flecks  peeping 
through  the  leaves. 

The  ptarmigan  and  the  snowy  owl,  the  arctic  fox  and  the  polar 
bear,  are  white  among  the  snows;  but  go  he  north  or  go  he  south, 
the  raven  (Uke  the  jackdaw)  is  boldly  and  impudently  black. 

The  rabbit  has  his  white  scut,  and  sundry  antelopes  their  piebald 
flanks,  that  one  timorous  fugitive  may  hie  after  another,  spying  the 
warning  signal.  The  primeval  terrier  or  colHe-dog  had  brown  spots 
over  his  eyes  that  he  might  seem  awake  when  he  was  sleeping  J : 
so  that  an  enemy  might  let  the  sleeping  dog  lie,  for  the  singular 
reason  that  he  imagined  him  to  be  awake.     And  a  flock  of  flamingos, 

*  Delight  in  beauty  is  one  of  the  pleasures  of  the  imagination;  there  is  no 
limit  to  its  indulgence,  and  no  end  to  the  results  which  we  may  ascribe  to  its 
exercise.  But  as  for  the  particular  "standard  of  beauty"  which  the  oird  (for 
instance)  admires  and  selects  (as  Darwin  says  in  the  Origin,  p.  70,  edit.  1884), 
we  are  very  much  in  the  dark,  and  we  run  the  risk  of  arguing  in  a  circle ;  for  wellnigh 
all  we  can  safely  say  is  what  Addison  says  (in  the  412th  Spectator) — that  each  different 
species  "is  most  aflFected  with  the  beauties  of  its  own  kind, . .  ,Hinc  merula  in  nigro 
86  oblectat  nigra  marito; .  .  .hinc  noctua  tetram  Canitiem  alarum  et  glaucos  miratur 
ocellos." 

t  Cf.  T.  W.  Bridge,  Cambridge  Natural  History  (Fishes),  vn,  p.  173,  1904;  also 
K.  V.  Frisch,  Ueber  farbige  Anpassung  bei  Fische,  Zool.  Jahrh.  {Abt.  Allg.  ZooL), 
xxxn,  pp.  171-230,  1914.  But  Reighard,  in  what  Raymond  Pearl  calls  "one  of 
the  most  beautiful  experimental  studies  of  natural  selection  which  has  ever  been 
made,"  found  no  relation  between  the  colours  of  coral-reef  fishes  and  their  elimination 
by  natural  enemies  {Carnegie  Inst.  Publication  103,  pp.  257-325,  1908). 

}  Nature,  L,  p.  672;  u,  pp.  33,  57,  533,  1894-95. 


960    ON  FORM  AND  MECHANICAL  EFFICIENCY   [ch. 

wearing  on  rosy  breast  and  crimson  wings  a  garment  of  in- 
visibility, fades  away  into  the  sky  at  dawn  or  sunset  like  a  cloud 
incarnadine*. 

To  buttress  the  theory  of  natural  selection  the  sa'me  instances 
of  "adaptation"  (and  many  more)  are  used,  as  in  an  earher  but  not 
distant  age  testified  to  the  wisdom  of  the  Creator  and  revealed  to 
simple  piety  the  immediate  finger  of  God.  In  the  words  of  a  certain 
learned  theologian f,  ''The  free  use  of  final  causes  to  explain  what 
seems  obscure  was  temptingly  easy-. . . .  Hence  the  finalist  was  often 
the  man  who  made  a  liberal  use  of  the  ignava  ratio,  or  lazy  argument: 
when  you  failed  to  explain  a  thing  by  the  ordinary  process  of 
causahty,  you  could  'explain'  it  by  reference  to  some  purpose  of 
nature  or  of  its  Creator.  This  method  lent  itself  with  dangerous 
facility  to  the  well-meant  endeavours  of  the  older  theologians  to 
expound  and  emphasise  the  beneficence  of  the  divine  purpose." 
Mutatis  mutandis,  the  passage  carries  its  plain  message  to  the 
naturahst. 

The  fate  of  such  arguments  or  illustrations  is  always  the  same. 
They  attract  and  captivate  for  awhile;  they  go  to  the  building  of 
a  creed,  which  contemporary  orthodoxy  %  defends  under  its  severest 
penalties:  but  the  time  comes  when  they  lose  their  fascination, 
they  somehow  cease  to  satisfy  and  to  convince,  their  foundations 

*  They  are  "wonderfully  fitted  for  'vanishment'  against  the  flushed,  rich- 
coloured  skies  of  early  morning  and  evening. .  .their  chief  feeding  -  times " ;  and 
"look  like  a  real  sunset  or  dawn,  repeated  on  the  opposite  side  of  the  heavens — 
either  east  or  west  as  the  case  may  be"  (Thayer,  Concealing -coloration  in  the  Animal 
Kingdom,  New  York,  1909,  pp.  154-155).  This  hypothesis,  like  the  rest,  is  not 
free  from  difficulty.  Twilight  is  apt  to  be  short  in  the  homes  of  the  flamingo; 
moreover,  Mr  Abel  Chapman  watched  them  on  the  Guadalquivir  feeding  by  day,  as 
I  also  have  seen  them  at  Walfisch  Bay. 

f  Principal  Galloway,  Philosophy  of  Religion,  1914,  p.  344. 

J  Professor  D.  M.  S.  Watson,  addressing  the  British  Association  in  1929  on 
Adaptation,  parted  company  with  what  I  had  called  contemporary  orthodoxy  in 
1917.  Speaking  of  such  morphological  differences  as  "have  commonly  been 
assumed  to  be  of  an  adaptive  nature,"  he  said:  "That  these  structural  diflferences 
are  adaptive  is  for  the  most  part  pure  assumption. . .  .There  is  no  branch  of  zoology 
in  which  assumption  has  played  a  greater  part,  or  evidence  a  less  part,  than  in 
the  study  of  such  presumed  adaptations. ' '  Hume,  in  his  Dialogue  concerning  Natural 
Religion,  shewed  similar  caut-ion:  "Steps  of  a  stair  are  plainly  constructed  that  human 
legs  may  use  them  in  mounting;  and  this  inference  is  certain  and  infallible.  Human 
legs  are  also  contrived  for  walking  and  mounting ;  and  this  inference,  I  allow,  is  not 
altogether  so  certain." 


XVI]  THE  PROBLEM  OF  ADAPTATION  %1 

are  discovered  to  be  insecure,  and  in  the  end  no  man  troubles  to 
controvert  them*. 

But  of  a  very  different  order  from  all  such  "adaptations"  as 
these  are  those  very  perfect  adaptations  of  form  which,  for  instance, 
fit  a  fish  for  swimming  or  a  bird  for  flight.  Here  we  are  far  above 
the  region  of  mere  hypothesis,  for  we  have  to  deal  with  questions 
of  mechanical  efficiency  where  statical  and  dynamical  considerations 
can  be  appHed  and  established  in  detail.  The  naval  architect  learns 
a  great  part  of  his  lesson  from  the  stream-lining  of  a  fish|;  the 
yachtsman  learns  that  his  sails  are  nothing  more  than  a  great  bird's 
wing,  causing  the  slender  hull  to  fly  along  % ;  and  the  mathematical 
study  of  the  stream-lines  of  a  bird,  and  of  the  principles  underlying 
the  areas  and  curvatures  of  its  wings  and  tail,  has  helped  to  lay  the 
very  foundations  of 'the  modern  science  of  aeronautics. 

We  know,  for  example,  how  in  strict  accord  with  theorv  i\t  was 
George  Cayley  who  explained  it  first)  the  wing,  whether  of  bird  or 
insect,  stands  stiff  along  its  "leading  edge,"  like  the  mast  before  the 
sail;  and  how,  conversely,  it  thins  out  exquisitely  fine  along  its  rear 
or  "trailing  edge,"  where  sharp  discontinuity  favours  the  formation 
of  uplifting  eddies.  And  pQ  see  how,  alike  in  the  flying  wing,  in 
the  penguin's  swimming  wing  and  in  the  whale's  flipper,  the  same 
design  of  stiff  fore-edge  and  thin  fine  trailing  edge,  both  curving  away 
evenly  to  meet  at  the  tip,  is  continually  exemphfied. 

We  learn  how  lifting  power  not  only  depends  on  area  but  has 
a  linear  factor  besides,  such  that  a  long  narrow  wing  is  more  stable 
and  effective  both  for  speedy  and  for  soaring  flight  than  a  short 
and  broad  one  of  equal  area ;  and  how  in  this  respect  the  hawkmoth 
differs  from  the  butterfly,  the  swallow  from  the  thrush.  We  are 
taught  how  every  wing,  and  every  kite  or  sail,  must  have  a  certain 

*  The  influence  of  environment  on  coloration  is  one  thing,  and  the  hypothesis 
of  protective  colouring  is  quite  another.  That  arctic  animals  are  often  white, 
and  desert  animals  sandy-hued  or  isabelline,  are  simple  and  undisputed  facts; 
but  such  field-naturalists  as  Theodore  Roosevelt,  Selous  (in  his  African  Nature 
Notes),  Buxton  {Animal  Life  in  Deserts),  and  Abel  Chapman  {Savage  Sudan,  and 
Retrospect)  reject  with  one  accord  the  theory  of  colour-protection. 

t  No  creature  shews  more  perfect  stream-lining  than  a  fur-seal  swimming. 
Every  curve  is  a  continuous  curve,  the  very  ears  and  eye-slits  and  whiskers  falling 
into  the  scheme,  and  the  flippers  folding  close  against  the  body. 

J  Cf.  Manfred  Curry,  Yacht  Racing,  and  the  Aerodynamics  of  Sails,  London, 
1928. 


962    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

amount  of  arch*  or  "belly,"  slight  in  the  rapid  fliers,  deeper  in  the 
slow,  flattened  in  the  strong  wind,  bulging  in  the  gentler  breeze; 
and  how  advantageous  is  all  possible  stiffening  of  sail  or  wing,  and 
why  accordingly  the  yachtsman  inserts  "battens"  and  the  Chinaman 
bamboos  in  his  sail.  We  are  shewn  by  Lilienthal  himself  how  a 
powerful  eddy,  the  so-called  "ram,"  forms  under  the  fore-edge,  and 
is  sometimes  caught  in  a  pocket  of  the  bird's  under  wing-coverts 
and  made  use  of  as  a  forward  drive. 

We  have  lately  learned  how  the  gaps  or  slots  between  the  primary 
wing-feathers  of  a  crow,  and  a  slight  power  of  the  wing-feathers 
to  twist,  like  the  slats  of  a  Venetian  blind,  play  their  necessary 


1 

Fig.  457.  Ligaments  in  a  swan's  wing.  1,  2,  3,  remiges;  A,  B,  longitudinal 
ligaments,  with  their  oblique  branches;  C,  small  subcutaneous  ligaments. 
From  Marey,  after  Pettigrew. 

part  under  certain  conditions  in  the  perfect  working  of  the  machine. 
Nothing  can  be  simpler  than  the  mechanism  by  whi(!h  all  this -is 
done.  Delicate  Ugaments  run  along  the  base  of  the  wing  from 
feather  to  feather,  and  send  a  branch  to  every  quill  (Fig.  457);  by 
these,  as  the  wing  extends,  the  quills  are  raised  into  their  places, 
and  kept  at  their  due  and  even  distances  apart.  Not  only  that,  but 
every  separate  ligamentous  strand  curls  a  httle  way  round  its  feather 
where  it  is  inserted  into  it;  and  thereby  the  feather  is  not  only 
elevated  into  its  place,  but  is  given  the  little  twist  which  brings  it 
to  its  proper  and  precise  obliquity. 

*  On  the  curvatura  veli,  cf.  J.  Bernoulli,  Acta  Ertidit.  Lips.  1692,  p.  202.     Studied 
also  by  Eiffel,  Resistance  de  Vair  et  V aviation,  Paris,  1910. 


XVI]  THE  MECHANISM  OF  FLIGHT  963 

All  this  is  part  of  the  automatic  mechanism  of  the  wing,  than  which 
there  is  nothing  prettier  in  all  anatomy. '  The  triceps  muscle,  massed 
on  the  shoulder,  extends  the  elbow- j  oint  in  the  usual  way.  But  another 
muscle  (a  long  flexor  of  the  wrist)  has  its  origin  above  the  elbow  and 
its  insertion  below  the  wrist.  It  passes  over  two  j  oints ;  it  is  what 
German  anatomists  call  a  zweigelenkiger  Muskel.  It  transmits  to  the 
one  joint  the  movements  of  the  other;  and  in  birds  of  powerful  flight 
it  becomes  less  and  less  muscular,  more  and  more  tendinous,  and  so 
more  and  more  completely  automatic.  The  wing  itself  is  kept  hght; 
its  chief  muscle  is  far  back  on  the  shoulder ;  a  contraction  of  that  re- 
mote muscle  throws  the  whole  wing  into  gear.  The  little  ligaments  we 
have  been  speaking  of  are  so  linked  up  with  the  rest  of  the  mechanism 
that  we  have  only  to  hold  a  bird's  wing  by  the  arm-boue  and  extend 
its  elbow- joint,  to  see  the  whole  wing  spring  into  action,  with  every 
joint  extended,  and  every  feather  tense  and  in  its  place*. 

Again  on  the  fore-edge  of  the  wing  there  lies  a  tiny  mobile  "  thumb," 
whose  little  tuft  of  stiff,  strong  feathers  forms  the  so-called  "bastard 
wing."  We  used  to  look  on  it  as  a  "vestigial  organ,"  a  functionless 
rudiment,  a  something  which  from  ancient  times  had  "lagged 
superfluous  on  the  stage";  until  a  man  of  genius  saw  that  it  was 
just  the  very  thing  required  to  break  the  leading. vortices,  keep  the 
flow  stream-lined  at  a  larger  angle  of  incidence  than  before,  and 
thereby  help  the  plane  to  land.  So  he  invented,  to  his  great  profit 
and  advantage,  the  "slotted  wing." 

We  learn  many  and  many  another  interesting  thing.  How  a 
stiff  "comb"  along  the  leading  edge,  a  broad  soft  fringe  along 
the  trailing  edge  (the  fringe  acting  as  a  damper  and  preventing 
"fluttering"),  and  a  soft,  downy  upper  surface  of  the  wing,  all  help 
as  silencers,  and  give  the  owl  her  noiseless  flight.  How  the  wing- 
loading  of  the  owls  is  lower  than  in  any  other  birds,  lower  even 
than  in  the  eagles;  and  how  owl  and  eagle  have  power  to  spare  to 
carry  easily  their  prey  of  mouse  or  mountain-hare.  How  the  deep 
terminal  wing-slots  aid  the  heavy  rook  or  heron  in  their  slow 


*  The  mole's  forelimb  has  a  somewhat  similar  action,  by  which,  as  the  arm  and 
hand  are  pulled  violently  backward,  the  claws  are  powerfully  and  automatically 
flexed  for  digging.  The  "suspensory  ligament "  of  the  horse,  which  is,  or  was,  a  short 
flexor  of  the  digit,  is  an  analogous  mechanism.  See  a  paper  of  mine  On  the  nature  and 
action  of  certain  ligaments,  Journ.  of  Anat.  and  Physiol,  xviii,  pp.  406-410,  1884. 


964    ON  FOKM  AND  MECHANICAL  EFFICIENCY  [ch. 

flapping  flight ;  how  the  sea-gull  does  not  need  them,  for  his  load  is 
lighter  and  his  wings  move  slowly.  There  is  never  a  discovery 
made  in  the  theory  of  aerodynamics  but  we  find  it  adopted  already 
by  Nature,  and  exemplified  in  the  construction  of  the  wing*. 

We  may  illustrate  some  few  of  the  principles  ijivolved  in  the  con- 
struction of  the  bird's  wing  with  a  half-sheet  of  paper,  whose  laws, 
as  it  planes  or  glides  downwards,  Clerk  Maxwell  explained  many 
years  ago|. 

To  improve  this  first  and  roughest  of  models,  we  see  that  its 
leading  edge  had  better  be  as  long  as  possible,  and  that  sharp 


1 

/ 

/ 

1 
\ 

/ 

\ 

/ 

\ 

L 

i 

Fig.  458.     A  diagrammatic  bird. 

corners  are  bound  to  cause  disturbance ;  let  us  get  rid  of  the  corners 
and  turn  the  leading  edge  into  a  continuous  curve  (Fig.  458).  The 
leading  edge  is  now  doing  most  of  the  ^^ork,  and  the  area  within 
and  behind  is  doing  little  good.  Vorticoid  air-currents  are  beating 
down  on  either  side  on  this  inner  area;  moreover,  air  is  ''sliding 
out"  below,  and  tending  to  curl  round  the  tip  and  edges  of  the 

*  Note  that  the  aeroplane  copies  the  beetle  rather  than  the  bird,  as  Lilienthal 
himself  points  out,  in  Vom  Gleitflug  zu  Segelflug,  Berlin,  1923. 

f  On  a  particular  case  of  the  descent  of  a  heavy  body  in  a  resisting  medium, 
Camb.  and  Dublin  Math.  Jonrn.  ix,  pp.  145-148,  1854;  Sci.  Papers,  i,  pp.  115-118. 
This  elegant  and  celebrated  little  paper  was  written  by  Clerk  Maxwell  while  an 
undergraduate  at  Trinity  College,  Cambridge. 


XVI]  OF  STREAM-LINES  965 

wing,  all  with  so  much  waste  of  energy*.  In  short  the  broad  wing 
is  less  efficient  than  the  narrow;  and  on  either  side  of  our  sheet 
of  paper  we  may  cut  out  a  portion  whiph  is  useless  and  in  the  way : 
for  the  same  reasons  we  may. cut  out  the  middle  part  of  the  tail, 
which  also  is  doing  more  harm  than  goodf. 

Hard  as  the  problem  is,  and  harder  as  it  becomes,  we  may  venture 
on.  In  aeronautics,  as  in  hydrodynamics,  we  try  to  determine  the 
resistances  encountered  by  bodies  of  various  shapes,  moving  through 
various  fluids  at  various  speeds;  and  in  so  doing  we  learn  the 
enormous,  the  paramount  importance  of  "stream-lining."  There 
would  be  no  need  for  stream-lining  in  a  "perfect  fluid,"  but  in  air  or 
water  it  makes  all  the  difference  in  the  world.  Stream-lining  impHes 
a  shape  round  which  the  medium  streams  so  smoothly  that  resistance 
is  at  last  practically  nil;  there  only  remains  the  slight  "skin- 
friction,"  which  can  be  reduced  or  minimised  in  various  ways.  But 
the  least  imperfection  of  the  stream-lining  leads  to  whirls  and 
"pockets"  of  dead  water  or  dead  air,  which  mean  large  resistance 
and  waste  of  energy.  The  converse  and  more  general  problem  soon 
emerges,  of  how  in  natural  objects  stream-lining  comes  to  be;  and 
whether  or  no  the  more  or  l^ss  stream-lined  shape  tends  to  be 
impressed  on  a  deformable  or  plastic  body  by  its  own  steady  motion 
through  a  fluid  J.  The  principle  of  least  action,  the  "loi  de  repos," 
is  enough  to  suggest  that  the  stream  will  tend  to  impress  its  stream- 
lines on  the  plastic  body,  causing  it  to  yield  or  "give,"  until  it  ends 
by  ofl'ering  a  minimum  of  resistance;  and  experiment  goes  some 
way  to  support  the  hypothesis.  A  bubble  of  mercury,  poised  in 
a  tube  through  which  air  is  blown,  assumes  a  stream-lined  shape, 
in  so  far  as  the  forces  due  to  the  moving  current  avail  against  the 

*  This  is  why  "slotting"  so  improves  the  broad  wing  of  the  crow.  See  on  this 
and  other  matters,  R.  R.  Graham's  papers  on  Safety  devices  in  the  wings  of 
birds,  in  British  Birds,  xxiv,  1930. 

t  Pettigrew  shewed  long  ago  {Tr.  R.S.E.  xxvi,  p.  361,  1872)  that  the  wing-area 
(in  insects)  "is  usually  greatly  in  excess  of  what  is  absolutely  required  for  flight," 
and  that  the  posterior  or  traihng  edge  could  be  largely  trimmed  away  without  the 
power  of  flight  being  at  all  diminished.  We  see  how  in  the  swallow  this  trailing  edge 
is  "trimmed  away"  till  a  bare  minimum  is  left,  and  how  (at  least  for  a  certain  kind 
of  flight)  the  wing  is  thereby  greatly  improved. 

X  Cf.  Enoch  Farrer,  The  shape  assumed  by  a  deformable  body  immersed  in 
a  moving  fluid,  Journ.  Franklin  Inst.  1921,  pp.  737-756;  also  Vaughan  Cornish, 
on  Waves  of  Sand  and  Snow. 


966    ON  FORM 'AND  MECHANICAL  EFFICIENCY  [cii. 

other  forces  of  restraint.  We  have  seen  how  the  egg  is  automatically 
stream-lined,  after  a  simple  fashion,  by  the  muscular  pressure  which 
drives  it  on  its  way.  The  contours  of  a  snowdrift,  of  a  wind- 
swept sand-dune,  even  of  the  flame  of  a  lamp,  shew  endless 
illustrations  of  stream-lines  or  eddy-curves  which  the  stream  itself 
imposes,  and  which  are  oftentimes  of  great  elegance  and  complexity. 
Always  the  stream  tends  to  mould  the  bodies  it  streams  over, 
facilitating  its  own  flow;  and  the  same  principle  must  somehow 
come  into  play,  at  least  as  a  contributory  factor,  in  the  making  of 
a  fish  or  of  a  bird.  But  it  is  obvious  in  both  of  these  that  even 
though  the  stream-lining  be  perfected  in  the  individual  it  is  also 
an  inheritance  of  the  race ;  and  the  twofold  problem  of  accumulated 
inheritance,  and  of  perfect  structural  adaptation,  confronts  us  once 
again  and  passes  all  our  understanding*. 

When,  after  attempting  to  comprehend  the  exquisite  adaptation 
of  the  swallow  or  the  albatross  to  the  navigation  of  the  air,  we  try 
to  pass  beyond  the  empirical  study  and  contemplation  of  such  per- 
fection of  mechanical  fitness,  and  to  ask  how  such  fitness  came  to 
be,  then  indeed  we  may  be  excused  if  we  stand  wrapt  in  wonderment, 
and  if  our  minds  be  occupied  and  even  satisfied  with  the  conception 
of  a  final  cause.  And  yet  all  the  while,  with  no  loss  of  wonderment 
nor  lack  of  reverence,  do  we  find  ourselves  constrained  to  believe 
that  somehow  or  other,  in  dynamical  principles  and  natural  law,' 
there  lie  hidden  the  steps  and  stages  of  physical  causation  by  which 
the  material  structure  was  so  shapen  to  its  endsf. 

The  problems  associated  with  these  phenomena  are  difficult  at 
every  stage,  even  long  before  we  approach  to  the  unsolved  secrets 
of  causation ;  and  for  my  part  I  confess  I  lack  the  requisite  know- 
ledge for  even  an  elementary  discussion  of  the  form  of  a  fish,  or  of 
an  insect,  or  of  a  bird.     But  in  the  form  of  a  bone  we  have  a  problem 

*  Mechanical  perfection  has  often  little  to  do  with  immunity  from  accident  or 
with  capacity  to  survive.  Legs  and  wings  of  locust  or  mayfly  are  indescribably 
perfect  for  their  brief  spell  of  life  and  narrow  sphere  of  toil;  but  they  may  be  torn 
asunder  in  a  moment,  and  whole  populations  perish  in  an  hour.  Careful  of  the 
type,  but  careless  of  the  single  life,  Nature  seems  ruthless  and  indiscriminate 
in  the  sacrifice  of  these  little  lives. 

t  Cf.  Professor  Flint,  in  his  Preface  to  Affleck's  translation  of  Janet's  Catises 
finales:  "We  are,  no  doubt,  still  a  long  way  from  a  mechanical  theory  of  organic 
growth,  but  it  may  be  said  to  be  the  quaesitum  of  modern  science,  and  no  one 
can  say  that  it  is  a  chimaera." 


XVI]  THE  STRENGTH  OF  MATERIALS  967 

of  the  same  kind  and  order,  so  far  simplified  and  particularised  that 
we  may  to  some  extent  deal  with  it,  and  may  possibly  even  find, 
in  our  partial  comprehension  of  it,  a  partial  clue  to  the  principles 
of  causation  underlying  this  whole  class  of  phenomena. 

Before  we  speak  of  the  form  of  a  bone,  let  us  say  a  word  about 
the  mechanical  properties  of  the  material  of  which  it  is  built*,  in 
relation  to  the  strength  it  has  to  manifest  or  the  forces  it  has  to 
resist:  understanding  always  that  we  mean  thereby  the  properties 
of  fresh  or  living  bone,  with  all  its  organic  as  well  as  inorganic 
constituents,  for  dead,  dry  bone  is  a  very  different  thing.  In  all 
the  structures  raised  by  the  engineer,  in  beams,  pillars  and  girders 
of  every  kind,  provision  has  to  be 
made,  somehow  or  other,  for  strength 
of  two  kinds,  strength  to  resist  com- 
pression or  crushing,  and  strength  to 
resist  tension  or  pulling  asunder.  The 
evenly  loaded  column  is  designed  with 
a  view  to  supporting  a  downward 
pressure^  the  wire-rope,  like  the  tendon  ^^' 

of  a  muscle,  is  adapted  only  to  resist  a  tensile  stress ;  but  in  many  or 
most  cases  the  two  functions  are  very  closely  inter-related  and  com- 
bined. The  case  of  a  loaded  beam  is  a  familiar  one;  though,  by  the 
way,  we  are  now  told  that  it  is  by  no  means  so  simple  as  it  looks,  and 
indeed  that  "the  stresses  and  strains  in  this. log  of  timber  are  so 
complex  that  the  problem  has  not  yet  been  solved  in  a  manner  that 
.reasonably  accords  with  the  known  strength  of  the  beam  as  found 
by  actual  experiment!."  However,  be  that  as  it  may.  we  know, 
roughly,  that  when  the  beam  is  loaded  in  the  middle  and  supported 
at  both  ends,  it  tends  to  be  bent  into  an  arc.  in  which  condition 
its  lower  fibres  are  being  stretched,  or  are  undergoing  a  tensile 

*  Cf.  Sir  Donald  MacAlister,  How  a  bone  is  built,  Engl.  III.  Mag.  1884, 
t  Professor  Claxton  Fidler,  On  Bridge  Construction,  p.  22  (4th  ed.),  1909;  cf. 
{int.  al.)  Love's  Elasticity,  p.  20  {Historical  Introduction),  2nd  ed.,  1906,  where  the 
bending  of  the  beam,  and  the  distortion  or  warping  of  its  cross-section,  are  studied 
after  the  manner  of  St  Venant,  in  his  Memoir  on  Torsion  (1855).  How  complex  the 
question  has  become  may  be  judged  from  such  papers  as  Price,  On  the  structure 
of  wood  in  relation  to  .its  elastic  properties,  Phil.  Trans.  (A),  ccvni,  1928;  or 
D.  B.  Smith  and  R.  V.  Southwell,  On  the  stresses  induced  by  flexure  in  a  deep 
rectangular  beam,  Proc.  U.S.  (A),  cXLiii,  pp.  271-285,  1934. 


968    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

stress,  while  its  upper  fibres  are  undergoing  compression.  It  follows 
that  in  some  intermediate  layer  there  is  a  "neutral  zone,"  where 
the  fibres  of  the  wood  are  subject  to  no  stress  of  either  kind. 

The  phenomenon  of  a  compression-member  side  by  side  with  a  tension- 
member  may  be  illustrated  in  many  simple  ways.  Ruskin  (in  Deucalion) 
describes  it  in  a  glacier.  He  then  bids  us  warm  a  stick  of  sealing-wax  and 
bend  it  in  a  horseshoe:  "you  will  then  see,  through  a  lens  of  moderate  power, 
the  most  exquisite  facsimile  of  glacier  fissures  produced  by  extension  on  its 
convex  surface,  and  as  faithful  an  image  of  glacier  surge  produced  by  com- 
pression on  its  concave  side."  A  still  more  beautiful  way  of  exhibiting  the 
distribution  of  strain  is  to  use  gelatin,  into  which  bubbles  of  gas  have  been 
.introduced  with  the  help  of  sodium  bicarbonate.  A  bar  of  such  gelatin,  when 
bent  into  a  hoop,  shews  on  the  one  side  the  bubbles  elongated  by  tension  and 
on  the  other  those  shortened  by  compression*. 

In  like  manner  a  vertical  pillar,  if  unevenly  loaded  (as  for  in- 
stance the  shaft  of  our  thigh-bone  normally  is),  will  tend  to  bend, 
and  so  to  endure  compression  on  its  concave,  and  tensile  stress 
upon  its  convex  side.  In  many  cases  it  is  the  business  of  the 
engineer  to  separate  out,  as  far  as  possible,  the  pressure-lines  from 
the  tension-lines,  in  order  to  use  separate  modes  of  construction, 
or  even  different  materials  for  each.  In  a  suspension-bridge,  for 
instance,  a  great  part  of  the  fabric  is  subject  to  tensile  strain  only, 
and  is  built  throughout  of  ropes  or  wires;  but  the  massive  piers 
at  either  end  of  the  bridge  carry  the  weight  of  the  whole  structure 
and  of  its  load,  and  endure  all  the  "compression-strains"  which 
are  inherent  in  the  system.  Very  much  the  same  is  the  case  in 
that  wonderful  arrangement  of  struts  and  ties  which  constitute,  or 
complete,  the  skeleton  of  an  animal.  The  "skeleton,"  as  we  see  it 
in  a  Museum,  is  a  poor  and  even,  a  misleading  picture  of  mechanical 
efficiency f.  From  the  engineer's  point  of  view,  it  is  a  diagram 
shewing  all  the  compression-lines,  but  by  no  means  all  the 
tension-lines  of  the  construction;  it  shews  all  the  struts,  but  few 
of  the  ties,  and  perhaps  we  might  even  say  none  of  the  principal 

*  Cf.  Emil  Hatschek,  Gestait  und  Orientirung  von  Gasblasen  in  Gelen,  KoUoid- 
Ztschr.  XV,  pp.  226-234,  1914. 

t  In  preparing  or  "macerating"  a  skeleton,  the  naturalist  nowadays  carries 
on  the  process  till  nothing  is  left  but  the  whitened  bones.  But  the  old  anatomists, 
whose  object  was  not  the  study  of  "comparative  morphology"  but  the  wider 
theme  of  comparative  physiology,  were  wont  to  macerate  by  easy  stages;  and  in 
many  of  their  most  instructive  preparations  the  ligaments  were  intentionally  left 
in  connection  with  the  bones,  and  as  part  of  the  "skeleton." 


XVI]  THE  STRENGTH  OF  M]A  T|E  R I A  L  S  969 

ones;  it  falls  all  to  pieces  unless  we  clamp  it  together,  as  ])est  we 
can,  in  a  more  or  less  clumsy  and  immobilised  way.  But  in  life, 
that  fabric  of  struts  is  surrounded  and  interwoven  with  a  compli- 
cated system  of  ties — "its  living  mantles  jointed  strong,  With 
ghstering  band  and  silvery  thong*":  ligament  and  membrane, 
muscle  and  tendon,  run  between  bone  and  bone;  and  the  beauty 
and  strength  of  the  mechanical  construction  he  not  in  one  part  or 
in  another,  but  in  the  harmonious  concatenation  which  all  the  parts, 
soft  and  hard,  rigid  and  flexible,  tension-bearing  and  pressure-bearing, 
make  up  together  j*. 

However  much  we  may  find  a  tendency,  whether  in  Nature  or 
art,  to  separate  these  two  constituent  factors  of  tension  and  com- 
pression, we  cannot  do  so  completely ;  and  accordingly  the  engineer 
seeks  for  a  material  which  shall,  as  nearly  as  possible,  offer  equal 
resistance  to  both  kinds  of  strain  J. 

From  the  engineer's  point  of  view,  bone  may  seem  weak  indeed ; 
but  it  has  the  great  advantage  that  it  is  very  nearly  as  good  for  a  tie 
as  for  a  strut,  nearly  as  strong  to  withstand  rupture,  or  tearing  asunder, 
as  to  resist  crushing.  The  strength  of  timber  varies  with  the  kind, 
but  it  always  stands  up  better  to  tension  than  to  compression,  and 
wrought  iron,  with  its  greater  strength,  does  much  the  same;  but 
in  cast-iron  there  is  a  still  greater  discrepancy  the  other  way,  for  it 
makes  a  good  strut  but  a  very  bad  tie  indeed.  Mild  steel,  which  has 
displaced  the  old-fashioned  wrought  iron  in  all  engineering  construc- 
tions, is  not  only  a  much  stronger  material,  but  it  also  possesses, 
like- bone,  the  two  kinds  of  strength  in  no  very  great  relative  dis- 
proportion §. 

*  See  Oliver  Wendell  Holmes'  Anatomist's  Hymn. 

t  In  a  few  anatomical  diagrams,  for  instance  in  some  of  the  drawings  in 
Schmaltz's  Atlas  der  Anatomie  des  Pferdes,  we  may  see  the  system  of  "ties" 
diagrammatically  inserted  in  the  figure  of  the  skeleton.  Cf.  W.  K.  Gregory,  On  the 
principles  of  quadrupedal  locomotion,  Ann.  N.  Y.  Acad,  of  Sciences,  xxii,  p.  289, 
1912. 

X  The  strength  of  materials  is  not  easy  to  discuss,  and  is  still  harder  to  tabulate. 
The  wide  range  of  qualities  in  each  material,  in  timber  the  wide  differences  according 
to  the  direction  in  which  the  block  is  cut,  and  in  all  cases  the  wide  difference  between 
yield-point  and  fracture-point,  are  some  of  the  difficulties  in  the  way  of  a  succinct 
statement. 

§  In  the  modern  device  of  "reinforced  concrete,"  blocks  of  cement  and  rods 
of  steel  are  so  combined  together  as  to  resist  both  compression  and  tension  in 
due  or  equal  measure. 


970    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

When  the  engineer  constructs  an  iron  or  steel  girder,  to  take 
the  place  of  the  primitive  wooden  beam,  we  know  that  he  takes 
advantage  of  the  elementary  principle  we  have  spoken  of,  and  saves 
weight  and  economises  material  by  leaving  out  as  far  as  possible 
all  the  middle  portion,  all  the  parts  in  the  neighbourhood  of  the 
"neutral  zone";  and  in  so  doing  he  reduces  his  girder  to  an  upper 
and  lower  "flange,"  connected  together  by  a  "web,"  the  whole 
resembling,  in  cross-section,  an  I  or  an  I  . 

But  it  is  obvious  that,  if  the  strains  in  the  two  flanges  are  to 
be  equal  as  well  as  opposite,  and  if  the  material  be  such  as  cast-iron 
or  wrought-iron,  one  or  other  flange  must  be  made  much  thicker 
than  the  other  in  order  that  they  may  be  equally  strong* ;  and  if  at 
times  the  two  flanges  have,  as  it  were,  to  change  places,  or  play 
each  other's  parts,  then  there  must  be  introduced  a  margin  of 
safety  by  making  both  flanges  thick  enough  to  meet  that  kind  of 
stress  in  regard  to  which  the  material  happens  to  be  weakest. 
There  is  great  economy,  then,  in  any  material  which  is,  as  nearly 
as  possible,  equally  strong  in  both  ways;  and  so  we  see  that,  from 
the  engineer's  or  contractor's  point  of  view,  bone  is  a  good  and 
suitable  material  for  purposes  of  construction. 

The  I  or  the  H-girder  or  rail  is  designed  to  resist  bending  in  one 
particular  direction,  but  if,  as  in  a  tall  pillar,  it  be  necessary  to 
resist  bending  in  all  directions  alike,  it  is  obvious  that  the  tubular 
or  cylindrical  construction  best  meets  the  case;  for  it  is  plain  that 
this  hollow  tubular  pillar  is  but  the  I-girder  turned  round  every 
way,  in  a  "solid  of  revolution,"  so  that  on  any  two  opposite  sides 
compression  and  tension  are  equally  met  and  resisted,  and  there  is 
now  no  need  for  any  substance  at  all  in  the  way  of  web  or  "filHng" 
within  the  hollow  core  of  the  tube.  And  it  is  not  only  in  the 
supporting  pillar  that  such  a  construction  is  useful ;  it  is  appropriate 
in  every  case  where  stiffness  is  required,  where  bending  has  to  be 
resisted.  A  sheet  of  paper  becomes  a  stiff  rod  when  you  roll  it  up, 
and  hollow  tubes  of  thin  bent  wood  withstand  powerful  thrusts  in 
aeroplane  construction.  The  long  bone  of  a  bird's  wing  has  Httle 
or  no  weight  to  carry,  but  it  has  to  withstand  powerful  bending 

*  This  principle  was  recognized  as  soon  as  iron  came  into  common  use  as  a 
structural  material.  The  great  suspension  bridges  only  became  possible,  in  Telford's 
hands,  when  wrought  iron  became  available. 


XVI]  OF  TUBULAR  STRUCTURES  971 

moments;  and  in  the  arm-bone  of  a  long-winged  bird,  such  as 
an  albatross,  we  see  the  tubular  construction  manifested  in  its 
perfection,  the  bony  substance  being  reduced  to  a  thin,  perfectly 
cylindrical,  and  almost  empty  shell*.  The  quill  of  the  bird's 
feather,  the  hollow  shaft  of  a  reed,  the  thin  tube  of  the  wheat- 
straw  bearing  its  heavy  burden  in  the  ear,  are  all  illustrations  which 
Gahleo  used  in  his  account  of  this  mechanical  principle  f;  and  the 
working  of  his  practical  mind  is  exemplified  by  this  catalogue  of 
varied  instances  w^hich  one  demonstration  suffices  to  explain. 

The  same  principle  is  beautifully  shewn  in  the  hollow  body  and 
tubular  limbs  of  an  insect  or  a  crustacean ;  and  these  complicated 
and  elaborately  jointed  structures  have  doubtless  many  constructional 
lessons  to  teach  us.  We  know,  for  instance,  that  a  thin  cylindrical 
tube,  under  bending  stress,  tends  to  flatten  before  it  buckles,  and 
also  to  become  "lobed"  on  the  compression  side  of  the  bend;  and 
we  often  recognise  both  of  these  phenomena  in  the  joints  of  a 
crab's  legj. 

Two  points,  both  of  considerable  importance,  present  themselves 
here,  and  w^e  may  deal  with  them  before  we  go  further  on.  In  the 
first  place,  it  is  not  difficult  to  see  that  in  our  bending  beam  the 
stress  is  greatest  at  its  middle ;  if  we  press  our  walking-stick  hard 
against  the  ground,  it  will  tend  to  snap  midway.  Hence,  if  our 
cylindrical  column  be  exposed  to  strong  bending  stresses,  it  will 
be  prudent  and  economical  to  make  its  walls  thickest  in  the  middle 
and  thinning  off  gradually  towards  the  ends;  and  if  we  look  at 
a  longitudinal  section  of  a  thigh-bone,  we  shall  see  that  this  is  just 
what  Nature  has  done.  The  presence  of  a  "danger-point"  has  been 
avoided,  and  the  thickness  of  the  walls  becomes  nothing  less  than 

*  Marsigli  (op.  cit.)  was  acquainted  with  the  hollow  wfng-bones  of  the  pelican; 
and  Bulfon  deals  with  the  whole  subject  in  his  Discours  sur  la  nature  des  oiseaux. 

f  Galileo,  Dialogues  concerning  Two  New  Sciences  (1638).  Crew  and  Salvio's 
translation.  New  York,  1914,  p.  150;  Opere,  ed.  Favaro,  viii,  p.  186.  (According 
to  R,  A.  Millikan,  "we  owe  crur  present  day  civilisation  to  Galileo.")  Cf.  Borelli, 
De  Motu  Animaliiim,  i,  prop,  clxxx,  168.5.  Cf.  also  P.  Camper.  La  structure  des 
OS  dans  les  oiseaux,  0pp.  in,  p.  459,  ed.  1803;  A.  Rauber,  Gahleo  iiber  Knochen- 
formen,  Morphol.  Jahrb.  vii,  pp.  327,  328,  1881;  Paolo  Enriques,  Delia  economia 
di  sostanza  nelle  osse  cave,  Arch.  f.  Entw.  Mech.  xx,  pp.  427-465,  1906.  Galileo's 
views  on  the  mechanism  of  the  human  body  are  also  discussed  by  0.  Fischer,  in  his 
article  on  Physiologische  Mechanik,  in  the  Encycl.  d.  mathem.  Wissenschaften,  1904. 

X  Cf.  L:  G.  Brazier,  On  the  flexure  of  thin  cylindrical  shells,  etc.,  Proc.  R.S.  (A), 
cx^^,  p.  104,  1927. 


972    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

a  diagram,  or  "graph,"  of  the  bending-moments  from  one  point  to 
another  along  the  length  of  the  bone. 

The  second  point  requires  a  Uttle  more  explanation.  If  we 
imagine  our  loaded  beam  to  be  supported  at  one  end  only  (for 
instance,  by  being  built  into  a  wall),  so  as  to  form  what  is  called 
H  i  j      a  "bracket"  or  "cantilever,"  then  we  can 

see,  without  much  difficulty,  that  the  lines  of 
stress  in  the  beam  run  somewhat  as  in  the 
accompanying  diagram.  Immediately  under 
the  load,  the  "compression-lines "  tend  to  run 
■  ^    vertically  downward,  but  where  the  bracket  is 

^^'      '  fastened  to  the  wall  there  is  pressure  directed 

horizontally  against  the  wall  in  the  lower  part  of  the  surface  of 
attachment;  and  the  vertical  beginning  and  the  horizontal  end  of 
these  pressure-lines  must  be  continued  into  one  another  in  the  form 
of  some  even  mathematical  curve — which,  as  it  happens,  is  part 
of  a  parabola.  The  tension-lines  are  identical  in  form  with  the 
compression-lines,  of  which  they  constitute  the  "mirror-image"; 
and  where  the  two  systems ,  intercross  they  do  so  at  right  angles, 
or  "orthogonally"  to  one  another.  Such  systems  of  stress-lines  as 
these  we  shall  deal  with  again;  but  let  us  take  note  here  of  the 
important  though  well-nigh  obvious  fact,  that  while  in  the  beam 
they  both  unite  to  carry  the  load,  yet  it  is  often  possible  to  weaken 
one  set  of  lines  at  the  expense  of  the  other,  and  in  some  cases  to  do 
altogether  away  with  one  set  or  the  other.  For  example,  when  we 
replace  our  end-supported  beam  by  a  curved  bracket,  bent  upwards 
or  downwards  as  the  case  may  be,  we  have  evidently  cut  away 
in  the  one  case  the  greater  part  of  the  tension-hnes,  and  in  the 
other  the  greater  part  of  the  compression-hnes.  And  if  instead  of 
bridging  a  stream  with  our  beam  of  wood  we  bridge  it  with  a  rope, 
it  is  evident  that  this  new  construction  contains  all  the  tension-hnes, 
but  none  of  the  compression-hnes  of  the  old.  The  biological  interest 
connected  with  this  principle  Hes  chiefly  in  the  mechanical  construc- 
tion of  the  rush  or  the  straw,  or  any  other  typically  cylindrical 
stem.  The  material  of  which  the  stalk  is  constructed  is  very  weak 
,to  withstand  compression,  but  parts  of  it  have  a  very  great  tensile 
strength.  Schwendener,  who  was  both  botanist  and  engineer,  has 
elaborately  investigated  the  factor  of  strength  in  the  cylindrical 


XVI]  OF  STRESS  AND  STRAIN  973 

stem,  which  Gahleo  was  the  first  to  call  attention  to.  Schwendener* 
shewed  that  its  strength  was  concentrated  in  the  little  bundles  of 
"bast-tissue,"  but  that  these  bast-fibres  had  a  tensile  strength  per 
square  mm.  of  section  not  less,  up  to  the  limit  of  elasticity,  than 
that  of  steel-wire  of  such  quality  as  was  in  use  in  his  day. 

For  instance,  we  see  in  the  following  table  the  load  which  various 
fibres,  and  various  wires,  were  found  capable  of.  sustaining,  not 
up  to  the  breaking-point  but  up  to  the  "elastic  Hmit,"  or  point 
beyond  which  complete  recovery  to  the  original  length  took  place 
no  longer  after  release  of  the  load. 


Stress,  or  load  in 

Strain, 

gms.  per  sq.  nun., 
at  Limit  of 
Elasticity 

Do.,  in  tons 
per  sq.  inch 

or  amount 

of  stretching, 

per  mille 

Secale  cereale 

15-20 

9-4-12-5 

4-4 

Lilium  auratum 

19 

11-8 

7-6 

Phormium  tenax 

20 

12-5 

130 

Papyrus  antiquorum 
Molinia  coerulea 

20 
22 

12-5 
13-8 

15-2 
110 

Pincenectia  recurvata 

25 

15-6 

14-5 

Copper  wire 
Brass       „ 

121 
13-3 

7-6 

8-5 

10 
1-35 

Iron         „ 

21-9 

13-7 

10 

Steel 

24-&* 

15-4 

1-2 

*  This  hgure  should  be  considerably  higher  for  the  best  modern  steel. 

In  other  respects,  it  is  true,  the  plant-fibres  were  inferior  to  the 
wires;  for  the  former  broke  asunder  very  soon  after  the  limit  of 
elasticity  was  passed,  while  the  iron- wire  could  stand,  before  snapping, 
about  twice  the  load  which  was  measured  by  its  limit  of  elasticity : 
in  the  language  of  a  modern  engineer,  the  bast-fibres  had  a  low 
"yield-point,"  little  above  the  elastic  limit.  Nature  seems  content, 
as  Schwendener  puts  it,  if  the  strength  of  the  fibre  be  ensured  up 
to  the  elastic  limit;  for  the  equihbrium  of  the  structure  is  lost  as 
soon  as  that  limit  is  passed,  and  it  then  matters  httle  how  far  off 
the  actual  breaking-point  may  bef.  But  nevertheless,  within  cer- 
tain limits,  plant-fibre  and  wire  were  just  as  good  and  strong  one 

*  S.  Schwendener,  Das  mechanische  Princip  im  anatomischen  Bau  der  Monocotyleen, 
Leipzig,  1874;  Zur  Lehre  von  der  Festigkeit  der  Gewachse,  Sb.  Berlin.  Akad.  1884, 
pp.  1045-1070. 

t  The  great  extensibility  of  the  plant-fibre  is  due  to  the  spiral  arrangement  of 
the  ultramicroscopic  micellae  of  which  the  bast-fibre  is  built  up :  the  spiral  untwisting 
as  the  fibre  stretches,  in  a  right  or  left-hand  spiral  according  to  the  species.  Cf. 
C.  Stein bruck,  Die  Micellartheorie  auf  botanischem  Gebiete,  Biol.  Centralbl.  1925, 
p.  1. 


974    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

as  the  other.  And  then  Schwendener  proceeds  to  shew,  in  many 
beautiful  diagrams,  the  various  ways  in  which  these  strands  of  strong 
tensile  tissue  are  arranged  in  various  stems :  sometimes,  in  the  simpler 
cases,  forming  numerous  small  bundles  arranged  in  a  peripheral 
ring,  not  quite  at  the  periphery,  for  a  certain  amount  of  space  has 
to  be  left  for  living  and  active  tissue;  sometimes  in  a  sparser  ring 
of  larger  and  stronger  bundles;  sometimes  with  these  bundles 
further  strengthened  by  radial  balks  or  ridges;  sometimes  with  all 
the  fibres  set  close  together  in  a  continuous  hollow  cylinder.     In 

the  case  figured  in  Fig.  461,  Schwendener 
calculated  that  the  resistance  to  bending 
was  at  least  twenty-five  times  as  great  as 
it  would  have  been  had  the  six  main 
bundles  been  brought  close  together  in  a 
solid  core.  In  many  cases  the  centre  of 
the  stem  is  altogether  empty ;  in  all  other 
cases  it  is  filled  with  soft  tissue,  suitable 
for  various  functions,  but  never  such  as 
to  confer  mechanical  rigidity.  In  a  tall 
Fig.  461.  conical  stem,  such  as  that  of  a  palm-tree, 

we  can  see  not  only  these  principles  in  the  construction  of  the 
cylindrical  trunk,  but  we  can  observe,  towards  the  apex,  the  bundles 
of  fibre  curving  over  and  intercrossing  orthogonally  with  one  another, 
exactly  after  the  fashion  of  our  stress-lines  in  Fig.  460;  but  of 
course,  in  this  case,  we  are  still  dealing  with  tensile  members,  the 
opposite  bundles  taking  on  in  turn,  as  the  tree  sways,  the  alternate 
function  of  resisting  tensile  strain*. 

*  For  further  botanical  illustrations,  see  {int.  al.)  R.  Hegler,  Einfiuss  der  Zug- 
kraften  auf  die  Festigkeit  und  die  Ausbildung  meehanischer  Gewebe  in  Pflanzen, 
SB.  sacks.  Ges.  d.  Wiss.  1891,  p.  638;  Einfluss  des  mechanischen  Zuges  auf  das 
Wachstum  der  Pflanze,  Cohn's  Beitrdge,  vi,  pp.  383-432,  1893;  0.  M.  Ball,  Einfluss 
von  Zug  auf  die  Ausbildung  der  Festigkeitsgewebe,  Jakrb.  d.  wiss.  Bot.  xxxix, 
pp.  305-341,  1903;  L.  Kny,  Einfluss  von  Zug  und  Druck  auf  die  Richtung  der 
Scheidewande  in  sich  teilenden  Pflanzenzellen,  Ber.  d.  hot.  Gesellsch.  xiv,  pp.  378- 
391,  1896;  Sachs,  Mechanomorphose  und  Phylogenie,  Flora,  Lxxvin,  1894;  of.  also 
Pfliiger,  Einwirkung  der  Schwerkraft,  etc.,  uber  die  Richtung  der  ZeUtheilung, 
Archiv,  xxxiv,  1884;  G.  Haberlandt's  Physiological  Plant  Anatomy,  tr.  by  Montagu 
Drummond,  1914,  pp.  1.50-213.  On  the  engineering  side  of  the  case,  see  Angus  R. 
Fulton,  Experiments  to  show  how  failure  under  stress  occurs  in  timber,  etc.,  Trans. 
R.S.E.  XLViii,  pp.  417-440,  1912;  Fulton  shews  {int.  a/.)  that  "the  initial  cause  of 
fracture  in  timbers  lies  in  the  medullary  rays." 


XVI]  THE  STRUCTURE  OF  BONE  975 

The  Forth  Bridge,  from  which  the  anatomist  may  learn  many 
a  lesson,  is  built  of  tubes,  which  correspond  even  in  detail  to  the 
structure  of  a  cylindrical  branch  or  stem.  The  main  diagonal  struts 
are  tubes  twelve  feet  in  diameter,  and  withm  the  wall  of  each  of 
these  lie  six  T-shaped  "stiffeners,"  corresponding  precisely  to  the 
fibro- vascular  bundles  of  Fig.  461 ;  in  the  same  great  tubular  struts 
the  tendency  to  "buckle"  is  resisted,  just  as  m  the  jointed  stem  of 
a  bamboo,  by  "stiffening  rings,"  or  perforated  diaphragms  set 
twenty  feet  apart  within  the  tube.  We  may  draw  one  more  curious, 
albeit  parenthetic,  comparison.  An  engineering  construction,  no 
less  than  the  skeleton  of  plant  or  animal,  has  to  grow;  but  the 
living  thing  is  in  a  sense  complete  during  every  phase  of  its  existence 
while  the  engineer  is  often  hard  put  to  it  to  ensure  sufficient  strength 
in  his  unfinished  and  imperfect  structure.  The  young  twig  stands 
more  upright  than  the  old,  and  between  winter  and  summer  the 
weight  of  leafage  affects  all  the  curving  outlines  of  the  tree.  A 
slight  upward  curvature,  a  matter  of  a  few  inches,  was  deliberately 
given  to  the  great  diagonal  tubes  of  the  bridge  during  their  piecemeal 
construction;  and  it  was  a  triumph  of  engineering  foresight  to  see 
how,  like  the  twig,  as  length  and  weight  increased,  they  at  last  came 
straight  and  true. 

Let  us  now  come,  at  last,  to  the  mechanical  structure  of  bone, 
of  which  we  find  a  well-known  and  classical  illustration  in  the 
various  bones  of  the  human  leg.  In  the  case  of  the  tibia,  the  bone 
is  somewhat  widened  out  aboVe,  and  its  hollow  shaft  is  capped  by 
an  almost  flattened  roof,  on  which  the  weight  of  the  body  directly 
rests.  It  is  obvious  that,  under  these  circumstances,  the  engineer 
would  find  it  necessary  to  devise  means  for  supporting  this  flat  roof, 
and  for  distributing  the  vertical  pressures  which  impinge  upon  it 
to  the  cylindrical  walls  of  the  shaft. 

In  the  long  wing-bones  of  a  bird  the  hollow  of  the  bone  is  empty, 
save  for  a  thin  layer  of  living  tissue  lining  the  cylinder  of  bone; 
but  in  our  own  bones,  and  all  weight-carrying  bones  in  general,  the 
hollow  space  is  filled  with  marrow,  blood-vessels  and  other  tissues; 
and  amidst  these  living  tissues  lies  a  fine  lattice-work  of  little 
interlaced  "trabeculae"  of  bone,  forming  the  so-called  "cancellous 
tissue."     The  older  anatomists  were  content  to  describe  this  can- 


976    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

cellous  tissue  as  a  sort  of  spongy  network  or  irregular  honeycomb  * ; 
but  at  length  its  orderly  construction  began  to  be  perceived,  and 
attempts  were  made  to  find  a  meaning  or  "purpose"  in  the  arrange- 
ment. Sir  Charles  Bell  had  a  glimpse  of  the  truth  when  he  asserted  f 
that  "this  minute  lattice- work,  or  the  cancelli  which  constitute 
the  interior  structure  of  bone,  have  still  reference  to  the  forces 
acting  on  the  bone";  but  he  did  not  succeed  in  shewing  what  these 
forces  are,  nor  how  the  arrangement  of  the  cancelli  is  related  to  them. 
Jeffries  Wyman,  of  Boston,  came  much  nearer  to  the  truth  in 
a  paper  long  neglected  and  forgotten  {.  He  gives  the  gist  of  the 
whole  matter  in  two  short  paragraphs:  "1.  The  cancelli  of  such 
bones  as  assist  in  supporting  the  weight  of  the  body  are  arranged 
either  in  the  direction  of  that  weight,  or  in  such  a  manner  as  to 
support  and  brace  those  cancelli  which  are  in  that  direction.  In 
a  mechanical  point  of  view  they  may  be  regarded  in  nearly  all  these 
bones  as  a  series  of  'studs'  and  'braces.'  2.  The  direction  of  these 
fibres  in  some  of  the  bones  of  the  human  skeleton  is  characteristic 
and,  it  is  believed,  has  a  definite  relation  to  the  erect  position  which 
is  naturally  assumed  by  man  alone."  A  few  years  afterwards  the 
story  was  told  again,  and  this  time  with  convincing  accuracy.  It 
was  shewn  by  Hermann  Meyer  (and  afterwards  in  greater  detail  by 
Julius  Wolff  and  others)  that  the  trabeculae,  as  seen  in  a  longitudinal 
section  of  the  femur,  spread  in  beautiful  curving  lines  from  the  head 
to  the  hollow  shaft  of  the  bone ;  and  that  these  linear  bundles  are 
crossed  by  others,  with  so  nice  a  regularity  of  arrangement  that 
each  intercrossing  is  as  nearly  as  possible  an  orthogonal  one:  that 
is  to  say,  the  one  set  of  fibres  or  cancelli  cross  the  other  everywhere 
at  right  angles.  A  great  engineer.  Professor  Culmann  of  Zurich, 
to  whom  by  the  way  we  owe  the  whole  modern  method  of  "graphic 
statics,"  happened  (in  the  year  1866)  to  come  into  his  colleague 
Meyer's  dissecting-room,  where  the  anatomist  was  contemplating 

*  Sir  John  Herschel  described  a  bone  as  a  "framework  of  the  most  curious 
carpentry:  in  which  occurs  not  a  single  straight  line  nor  any  known  geometrical 
curve,  yet  all  evidently  systematic,  and  constructed  by  rules  which  defy  our 
research"  (On  the  Study  of  Natural  Philosophy,  1830,  p.  203). 

f  In  Animal  Mechanics,  or  Proofs  of  Design  in  the  Animal  Frame,  1827. 

X  Animal  mechanics :  on  the  cancellated  structure  of  spme  of  the  bones  of  the 
human  body,  Boston  Soc.  of  Nat.  Hist.  1849.  Reprinted,  together  with  Sir  C.  Bell's 
work,  by^  Morrill  Wyman,  Cambridge,  Mass.,  1902. 


i 


XVI]  THE  STRUCTURE  OF  BONE  977 

the  section  of  a  bone*.  The  engineer,  who  had  been  busy  designing 
a  new  and  powerful  crane,  saw  in  a  moment  that  the  arrangement 
of  the  bony  trabeculae  was  nothing  more  nor  less  than  a  diagram 
of  the  lines  of  stress,  or  directions  of  tension  and  compression,  in 
the  loaded  structure:  in  short,  that  Nature  was  strengthening  the 
bone  in  precisely  the  manner  and  directioi^  in  which  strength  was 
required;    and  he  is  said  to  have  cried  out,  "That's  my  crane!" 


Fig.  462.     Head  of  the  human  femur  in  section.     After  Schafer,  from 
a  photo  by  Professor  A.  Robinson. 

In  the  accompanying  diagram  of  Culmann's  crane-head,  we  recognise 
a  simple  modification,  due  entirely  to  the  curved  shape  of  the 
structure,  of  the  still  simpler  Hues  of  tension  and  compression  which 
we  have  already  seen  in  our  end-supported  beam,  as  represented 
in  Fig.  460.     In  the  shaft  of  the  crane  the  concave  or  inner  side, 

*  The  first  metatarsal,  rather  than  the  femur,  is  said  to  have  been  the  bone 
which  Meyer  was  demonstrating  when  Culmann  first  recognised  the  orthogonal 
intercrossing  of  the  cancelli  in  tension  and  compression;  cf.  A.  Kirchner,  Architektur 
der  Metatarsalien  des  Menschen,  Arch.  f.  Entw.  Mech.  xxiv,  pp.  539-^16,  1907. 


978    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

overhung  by  the  loaded  head,  is  the  "compression-member";  the 
outer  side  is  the  "tension-member";  the  pressure-Unes,  starting 
from  the  loaded  surface,  gather  themselves  together,  always  in  the 
direction  of  the  resultant  pressure,  till  they  form  a  close  bundle 
running  down  the  compressed  side  of  the  shaft :  while  the  tension- 
lines,  running  upwards  along  the  opposite  side  of  the  shaft,  spread 
out  through  the  head,  orthogonally  to,  and  linking  together,  the 
system  of  compression-lines.     The  head  of  the  femur  (Fig.  462)  is 


Fig.  463.     Crane-head  and  femur.     After  Culmann  and  J.  Wolff. 

a  little  more  complicated  in  form  and  a  little  less  symmetrical  than 
Culmann's  diagrammatic  crane,  from  which  it  chiefly  differs  in  the 
fact  that  its  load  is  divided  into  two  parts,  that  namely  which  is 
borne  by  the  head  of  the  bone,  and  that  smaller  portion  which 
rests  upon  the  great  trochanter ;  but  this  merely  amounts  to  saying 
that  a  notch  has  been  cut  out  of  the  curved  upper  surface  of  the 
structure,  and  we  have  no  difficulty  in  seeing  that  the  anatomical 
arrangement  of  the  trabeculae  follows  precisely  the  mechanical 
distribution  of  compressive  and  tensile  stress  or,  in  other  words, 
accords  perfectly  with  the  theoretical  stress-diagram  of  the  crane. 


XVI]  THE  STRUCTURE  OF  BONE  979 

The  lines  of  stress  are  bundled  close  together  along  the  sides  of 
the  shaft,  and  lost  or  concealed  there-  in  the  substance  of  the  solid 
wall  of  bone;  but  in  and  near  the  head  of  the  bone,  a  peripheral 
shell  of  bone  does  not  suffice  to  contain  them,  and  they  spread  out 
through  the  central  mass  in  the  actual  concrete  form  of  bony 
trabeculae*. 

Mutatis  mutandis,  the  same  phenomenon  may  be  traced  in  any 
other  bone  which  carries  weight  and  is  liable  to  flexure;  and  in 
the  OS  calcis  and  the  tibia,  and  more  or  less  in  all  the  bones  of  the 
lower  limb,  the  arrangement  is  found  to  be  very  simple  and  clear. 

Thus,  in  the  os  calcis,  the  weight  resting  on  the  head  of  the  bone 
has  to  be  transmitted  partly  through  the  backward-projecting  heel 
to  the  ground,  and  partly  forwards  through  its  articulation  with 

*  Among  other  works  on  the  mechanical  construction  of  bone  see:  Bourgery, 
Traite  de  Vanatomie  {I.  Osteologie),  1832  (with  admirable  illustrations  of  trabecular 
structure);  L.  Fick,  Die  Ursachen  der  Knochenformen,  Gottingen,  1857;  H.  Meyer, 
Die  Architektur  der  Spongiosa,  Arch.  f.  Anat.  und  Physiol,  xlvii,  pp.  615-628, 
1867;  Statik  u.  Mechanik  des  menschlichen  Knochengeriistes,  Leipzig,  1873;  H.  Wolfer- 
mann,  Beitrag  zur  K.  der  Architektur  der  Knochen,  Arch.  f.  Anat.  und  Physiol. 
1872,  p.  312;  J.  Wolff,  Die  innere  Architektur  der  Knochen,  Arch.  f.  Anat.  und 
Phya.  L,  1870;  Das  Gesetz  der  Transformation  bei  Knochen,  1892;  Y.  Dwight, 
The  significance  of  bone-architecture,  Mem.  Boston  Soc.  N.H.  iv,  p.  1,  1886; 
V.  von  Ebner,  Der  feinere  Bau  der  Knochensubstan^,  Wiener  Bericht,  Lxxn,  1875; 
Anton  Rauber,  Elastizitdt  und  Festigkeit  der  Knochen,  Leipzig,  1876;  0.  Meserer, 
EUlsI.  u.  Festigk.  d.  menschlichen  Knochen,  Stuttgart,  1880;  Sir  Donald  MacAlister, 
How  a  bone  is  buil^,  English  Illustr.  Mag.  1884,  pp.  640-649;  Rasumowsky, 
Architektonik  des  Fussskelets,  Int.  Monatsschr.  f.  ^na<.  .1889,  p.  197;  Zschokke, 
Weitere  Unters.  uher  das  Verhdltnis  der  Knochenhildung  zur  Statik  und  Mechanik 
des  Vertehratenskelets,  Ziirich,  1892;  W.  Roux,  Ges.  Ahhandlungen  uher  Entuncklungs- 
mechanik  der  Organismen,  Bd.  I,  Funktionelle  Anpassung,  Leipzig,  1895;  J.  Wolff, 
Die  Lehre  von  der  funktionellen  Knochengestalt,  Virchow's  Archiv,  clv,  1899; 
R.  Schmidt,  Vergl.  anat.  Studien  liber  den  mechanischen  Bau  der  Knochen  und 
seine  Vererbung,  Z.  f.  w.  Z.  lxv,  p.  65,  1899;  B.  Solger,  Der  gegenwartige  Stand 
der  Lehre  von  der  Knochenarchitektur,  in  Moleschott's  Unters.  z.  Naturlehre  des 
Menschen.  xvi,  p.  187,  1899;  H.  Triepel,  Die  Stossfestigkeit  der  Knochen,  Arch, 
f.  Anat.  und  Phys.  1900;  Gebhardt,  Funktionellwichtige  Anordnungsweisen  der 
feineren  und"  groberen  Bauelemente  des  Wirbelthierknochens,  etc.  Arch.  f.  ErUw. 
Mech.  1900-10;  Revenstorf,  Ueber  die  Transformation  der  Calcaneus -architektur. 
Arch.  f.  Entw.  Mech.  xxiii,  p.  379,  1907;  H.  Bernhardt,  Vererbung  der  inneren 
Knochenarchitektur  beim  Menschen,  und  die  Teleologie  bei  J.  Wolff,  Inaug.  Diss., 
Miinchen,  1907;  Herm.  Triepel,  Die  trajectoriellen  Structuren  (in  Einf.  in  die 
Physikalische  Anatomie,  1908);  A.  F.  Dixon,  Architecture  of  the  cancellous  tissue 
forming  the  upper  end  of  the  femur,  Journ.  Anai.  and  Phys.  (3),  xliv,  pp.  223-230, 
1910;  A.  Benninghoff,  Ueber  Leitsystem  der  Knochencompacta ;  Studien  zur 
Architektur  der  Knochen,  Beitr.  z.  Anat.  funktioneller  Systeme,  i,  1930. 


980    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

the  cuboid  bone,  to  the  arch  of  the  foot.  We  thus  have,  very  much 
as  in  a  triangular  roof- tree,,  two  compression-members  sloping  apart 
from  one  another;  and  these  have  to  be  bound  together  by  a  "tie" 
or  tension-member,  corresponding  to  the  third,  horizontal  member 
of  the  truss. 

It  is  a  simple  corollary,  confirmed  by  observation,  that  the 
trabeculae  have  a  very  different  distribution  in  animals  whose 
actions  and  attitudes  are  materially  different,  as  in  the  aquatic 
mammals,  such  as  the  beaver  and  the  seal*.  And  in  much  less 
extreme  cases  there  are  lessons  to  be  learned  from  a  study  of  the 


Fig,  464.     Diagram  of  stress-lines  in  the  human  foot.     From 
Sir  D.  MacAlister,  after  H.  Meyer. 

same  bone  in  different  animals,  as  the  loads  alter  in  direction  and 
magnitude.  The  gorilla's  heelbone  resembles  man's,  but  the  load 
on  the  heel  is  much  less,  for  the  erect  posture  is  imperfectly  achieved : 
in  a  common  monkey  the  heel  is  carried  high,  and  consequently 
the  direction  of  the  trabeculae  is  still  more  changed.  The  bear 
walks  on  the  sole  of  his  foot,  though  less  perfectly  than  does  man, 
and  the  lie  of  the  trabeculae  is  plainly  analogous  in  the  two;  but 
in  the  bear  more  powerful  strands  than  in  the  os  calcis  of  man 
transmit  the  load  forward  to  the  toes,  and  less  of  it  through  the 
heel  to  the  ground.  In  the  leopard  we  see  the  full  effect  of  tip-toe, 
or  digitigrade,  progression.     The  long  hind  part  (or  tuberosity)  of 

*  Cf.  G.  de  M.  Rudolf,  Habit  and  the  architecture  of  the  mammalian  femur, 
Jovrn.  Anatomy,  lvi,  pp.  139-146,  1922. 


xvi] 


OF  WARREN'S  TRUSS 


981 


the  heel  is  now  more  a  mere  lever  than  a  pillar  of  support;  it  is 
little  more  than  a  stiffened  rod,  with  compression-members  and 
tension-members  in  opposite  bundles,  inosculating  orthogonally  at 
the  two  ends*. 

In  the  bird  the  small  bones  of  the  hand,  dwarfed  as  they  are  in 
size,  have  still  a  deal  to  do  in  carrying  the  long  primary  flight- 
feathers,  and  in  forming  a  rigid  axis  for  the  terminal  part  of  the 
wing.  The  simple  tubular  construction,  which  answers  well  for 
the  long,  slender  arm-bones,  does  not  suffice  where  a  still  more 
efficient  stiffening  is  required.     In  all  the  mechanical ^ide  of  anatomy 


Fig.  465.     Metacarpal  bone  from  a  vulture's  wing;  stiffened  after  the  manner 
of  a  Warren's  truss.     From  0.  Prochnow,  Formenkunst  der  Natur, 

nothing  can  be  more  beautiful  than  the  construction  of  a  vulture's 
metacarpal  bone,  as  figured  here  (Fig.  465).  The  engineer  sees  in 
it  a  perfect  Warren's  truss,  just  such  a  one  as  is  often  used  for  a  main 
rib  in  an  aeroplane.  Not  only  so,  but  the  bone  is  better  than  the 
truss;  for  the  engineer  has  to  be  contefit  to  set  his  V-shaped  struts 
all  in  one  plane,  while  in  the  bone  they  are  put,  with  obvious  but 
inimitable  advantage,  in  a  three-dimensional  configuration. 


So  far,  dealing  wholly  with  the  stresses  and  strains  due  to  tension 
and  compression,  we  have  omitted  to  speak  of  a  third  very  im- 
portant factor  in  the  engineer's  calculations,  namely  what  is  known 
as   "shearing   stress."     A  shearing  force  is   one   which   produces 

*  Cf.  Fr.  Weidenreich,  Ueber  formbestimmende  Ursachen  am  Skelett,  und  die 
Erblichkeit  der  Knochenform,  Arch.  f.  Entw.  Mech.  li,  pp.  438-481,  1922. 


982    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

"angular  distortion"  in  a  figure,  or  (what  comes  to  the  same  thing) 
which  tends  to  cause  its  particles  to  slide  over  one  another,  A 
shearing  stress  is  a  somewhat  complicated  thing,  and  we  must  try- 
to  illustrate  it  (however  imperfectly)  in  the  simplest  possible  way. 
If  we  build  up  a  pillar,  for  instance,  of  flat  horizontal  slates,  or  of 
a  pack  of  cards,  a  vertical  load  placed  upon  it  will  produce  com- 
pression, but  will  have  no  tendency  to  cause  one  card  to  slide,  or 
shear,  upon  another;  and  in  like  manner,  if  we  make  up  a  cable 
of  parallel  wires  and,  letting  it  hang  vertically,  load  it  evenly  with 


Fig.  466.     Trabecular  structure  of  the  os  calcis.     From  MacAIister. 

a  weight,  again  the  tensile  stress  produced  has  no  tendency  to  cause 
one  wire  to  slip  or  shear  upon  another.  But  the  case  would  haye 
been  very  different  if  we  had  built  up  our  pillar  of  cards  or  slates 
lying  obliquely  to  the  lines  of  pressure,  for  then  at  once  there  would 
have  been  a  tendency  for  the  elements  of  the  pile  to  slip  and  shde 
asunder,  and  to  produce  what  the  geologists  call  "a  fault"  in  the 
structure. 

Somewhat  more  generally,  if  AB  be  a  bar,  or  pillar,  of  cross-section  a 
under  a  direct  load  P,  giving  a  direct  and  uniformly  distributed  stress  per  unit 
area  =p,  then  the  whole  pressure  P=pa.  Let  CD  be  an  oblique  section, 
inclined  at  an  angle  6  to  the  cross-section ;  the  pressure  on  CD  will  evidently 
be  =pa  cos  6.  But  at  any  point  0  in  CD,  the  pressure  P  may  be  resolved  into 
the  shearing  force  Q  acting  along  CD,  and  the  direct  force  N  perpendicular  to 
i  t :    where  N  =  P  cos  6  =  pa  cos  6,  and  ^  =  P  sin  ^ = ^a  sin  6.    The  shearing  force 


XVI 


OF  SHEARING  STRESS 


Q  upon  CD  =  q .  area  of  CD,  which  is  =  g .  a/cos  6.  Therefore  qa/ cos  6=  pa  sin  By 
therefore  q=p  sin  6  cos  6  =  ^p  sin  2^.  Therefore  when  sin  20=  1,  that  is,  when 
6  =  45°,  g  is  a  maximum,  and  =p/2;  and  when  sin 20  =  0,  that  is  when  0  =  0°  or 
90°,  then  q  vanishes  altogether. 

This  is  as  much  as  to  say,  that  under  this  form  of  loading  there  is 
no  shearing  stress  along  or  perpendicular  to  the  lines  of  principal 
stress,  or  along  the  Unes  of  maximum  compression  or  tension;  but 
shear  has  a  definite  value  on  all  other  planes,  and  a  maximum  value 
when  it  is  inclined  at  45°  to  the  cross-section.  This  may  be  further 
illustrated  in  various  simple  ways.  When 
we  submit  a  cubical  block  of  iron  to 
compression  in  the  testing  machine,  it 
does  not  tend  to  give  way  by  crumbling 
all  to  pieces,  but  always  disrupts  by 
shearing,  and  along  some  plane  approxi- 
mately at  45°  to  the  axis  of  compression ; 
this  is  known  as  Coulomb's 'Theory  of 
Fracture,  and,  while  subject  to  many 
qualifications,  it  is  still  an  important 
first  approximation  to  the  truth.  Again, 
in  the  beam  which  we  have  already  con- 
sidered under  a  bending  moment,  we 
know  that  if  we  substitute  for  it  a  pack  of  cards,  they  will  be  strongly 
sheared  on  one  another ;  and  the  shearing  stress  is  greatest  in  the 
"neutral  zone,"  where  neither  tension  nor  compression  is  manifested: 
that  is  to  say  in  the  Hne  which  cuts  at  equal  angles  of  45°  the 
orthogonally  intersecting  lines  of  pressure  and  tension. 

In  short  we  see  that,  while  shearing  stresses  can  by  no  means 
be  got  rid  of,  the  danger  of  rupture  or  breaking-down  under  shearing 
stress  is  lessened  the  more  we  arrange  the  materials  of  our  con- 
struction along  the  pressure-lines  and  tension-lines  of  the  system; 
for  along  these  lines  there  is  no  shear*. 

To  apply  these  principles  to  the  growth  and  development  of 
our  bone,  we  have  only  to  imagine  a  little  trabecula  (or  group  of 
trabeculae)  being  secreted  and  laid  down  fortuitously  in  any  direction 
within  the  substance  of  the  bone.  If  it  lie  in  the  direction  of  one  of 
the  pressure-lines,  for  instance,  it  will  be  in  a  position  of  comparative 


Fig.  467. 


It  is  also  obvious  that  a  free  surface  is  always  a  region  of  zero-shear. 


984    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

equilibrimn,  or  minimal  disturbance ;  but  if  it  be  inclined  obliquely 
to  the  pressure-lines,  the  shearing  force  will  at  once  tend  to  act 
upon  it  and  move  it  away.  This  is  neither  more  nor  less  than  what 
happens  when  we  comb  our  hair,  or  card  a  lock  of  wool :  filaments 
lying  in  the  direction  of  the  comb's  path  remain  where  they  were ; 
but  the  others,  under  the  influence  of  an  obUque  component  of 
pressure,  are  sheared  out  of  their  places  till  they  too  come  into 
coincidence  with  the  Unes  of  force.  So  straws  show  how  the  wind 
blows — or  rather  how  it  has  been  blowing.  For  every  straw  that 
lies  askew  to  the  wind's  path  tends  to  be  sheared  into  it;  but  as 
soon  as  it  has  come  to  lie  the  way  of  the  wind  it  tends  to  be 
disturbed  no  more,  save  (of  course)  by  a  violence  such  as  to  hurl 
it  bodily  away. 

In  the  biological  aspect  of  the  case,  we  must  always  remember 
that  our  bone  is  not  only  a  living,  but  a  highly  plastic  structure; 
the  httle  trabeculae  are  constantly  being  formed  and  deformed, 
demohshed  and  formed  anew.  Here,  for  once,  it  is  safe  to  say  that 
"heredity"  need  not  and  cannot  be  invoked  to  account  for  the 
configuration  and  arrangement  of  the  trabeculae:  for  we  can  see 
them  at  any  time  of  hfe  in  the  making,  under  the  direct  action 
and  control  of  the  forces  to  which  the  system  is  exposed.  If  a  bone 
be  broken  and  so  repaired  that  its  parts  lie  somewhat  out  of  their 
former  place,  so  that  the  pressure-  and  tension-lines  have  now  a  new 
distribution,  before  many  weeks  are  over  the  trabecular  system  will 
be  found  to  have  been  entirely  remodelled,  so  as  to  fall  into  line 
with  the  new  system  of  forces.  And  as  Wolff  pointed  out,  this 
process  of  reconstruction  extends  a  long  way  off  from  the  seat  of 
injury,  and,  so  cannot  be  looked  upon  as  a  mere  accident  of  the 
physiological  process  of  healing  and  repair;  for  instance,  it  may 
happen  that,  after  a  fracture  of  the  shaft  of  a  long  bone,  the 
trabecular  meshwork  is  wholly  altered  and  reconstructed  within  the 
distant  extremities  of  the  bone.  Moreover,  in  cases  of  transplantation 
of  bone,  for  example  when  a  diseased  metacarpal  is  repaired  by 
means  of  a  portion  taken  from  the  lower  end  of  the  ulna,  with 
astonishing  quickness  the  plastic  capabilities  of  the  bony  tissue  are 
so  manifested  that  neither  in  outward  form  nor  inward  structure 
can  the  old  portion  be  distinguished  from  the  new. 

Herein  then  lies,  so  far  as  we  can  discern  it,  a  great  part  at  least 


XVI]  ON  STRESS  AND  STRAIN  985 

of  the  physical  causation  of  what  at  first  sight  strikes  us  as  a  purely 
functional  adaptation:  as  a  phenomenon,  in  other  words,  whose 
physical  cause  is  as  obscure  as  its  final  cause  or  end  is  apparently 
manifest. 

Partly  associated  with  the  same  phenomenon,  and  partly  to  be 
looked  upon  (meanwhile  at  least)  as  a  fact  apart,  is  the  very  im- 
portant physiological  truth  that  a  condition  of  strain,  the  result 
of  a  stress,  is  a  direct  stimulus  to  growth  itself.  This  indeed  is  no 
less  than  one  of  the  cardinal  facts  of  theoretical  biology.  The  soles 
of  our  boots  wear  thin,  but  the  soles  of  our  feet  grow  thick,  the 
more  we  walk  upon  them:  for  it  would  seem  that  the  living  cells 
are  "stimulated"  by  pressure,  or  by  what  we  call  "exercise," -to 
increase  and  multiply.  The  surgeon  knows,  when  he  bandages  a 
broken  limb,  that  his  bandage  is  doing  something  more  ^an  merely 
keeping  the  parts  together:  and  that  the  even,  constant •  pressure 
which  he  skilfully  applies  is  a  direct  encouragement  of  growth  and 
an  active  agent  in  the  process  of  repair.  In  the  classical  experiments 
of  Sedillot*,  the  greater  part  ef  the  shaft  of  the  tibia  was  excised 
in  some  young  puppies,  leaving  the  whole  weight  of  the  body  to 
rest  upon  the  fibula.  The  latter  bone  is  normally  about  one-fifth 
or  sixth  of  the  diameter  of  the  tibia ;  but  under  the  new  conditions, 
and  under  the  "stimulus"  of  the  increased  load,  it  grew  till  it  was 
as  thick  or  even  thicker  than  the  normal  bulk  of  the  larger  bone. 
Among  plant  tissues  this  phenomenon  is  very  apparent,  and  in 
a  somewhat  remarkable  way;  for  a  strain  caused  by  a  constant  or 
increasing  weight  (such  as  that  in  the  stalk  of  a  pear  while  the  pear 
is  growing  and  ripening)  produces  a  very  marked  increase  of  strength 
without  any  necessary  increase  of  bulk,  but  rather  by  some  histo- 
logical, or  molecular,  alteration  of  the  tissues.  Hegler,  Pfeifer,  and 
others  have  investigated  this  subject,  by  loading  the  young  shoot 
of  a  plant  nearly  to  its  breaking  point,  and  then  redetermining  the 
breaking-strength  after  a  few  days.  Some  young  shoots  of  the 
sunflower  were  found  to  break  with  a  strain  of  160  gm.;  but  when 
loaded  with  150  gm.,  and  retested  after  two  days,  they  were  able 
to  support  250  gm. ;   and  being  again  loaded  with  something  short 

*  Sedillot,  De  I'influence  des  fonctions  sur  la  structure  et  la  forme  des  organes, 
C.R.  Lix,  p.  539,  1864;   ef.  lx,  p.  97,  1865;  Lxvm,  p.  1444,  1869. 


986    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

of  this,  by  next  day  they  sustained  300  gm.,  and  a  few  days  later 
even  400  gm.* 

The  kneading  of  dough  is  an  analogous  phenomenon.  The 
viscosity  and  perhaps  other  properties  of  the  stuff  are  affected  by 
the  strains  to  which  we  have  submitted  it,  and  may  thus  be  said 
to  depend  not  only  on  the  nature  of  the  substance  but  on  its 
history  f .  It  is  a  long  way  from  this  simple  instance,  but  we  stretch 
across  it  easily  in  imagination,  to  the  experimental  growth  of  a 
nerve-fibre  within  a  mass  of  clotted  lymph :  where,  when  we  draw 
out  the  clot  in  one  direction  or  another  we  lay  down  traction-lines, 
or  tension-lines,  and  make  of  them  a  path  for  growth  to  follow  J. 

Such  experiments  have  been  amply  confirmed,  but  so  far  as  I  am 
aware  we  do  not  know  much  more  about  the  matter:  we  do  not 
know,  for  instance,  how  far  the  change  is  accompanied  by  increase 
in  number  of  the  bast-fibres,  through  transformation  of  other  tissues ; 
or  how  far  it  is  due  to  increase  in  size  of  these  fibres;  or  whether 
it  be  not  simply  due  to  strengthening  of  the  original  fibres  by  some 
molecular  change.  But  I  should  be  much  inclined  to  suspect  that 
this  last  had  a  good  deal  to  do  with  the  phenomenon.  We  know 
nowadays  that  a  railway  axle,  or  any  other  piece  of  steel,  is  weakened 
by  a  constant  succession  of  frequently  interrupted  strains;  it  is 
said  to  be  "fatigued,"  and  its  strength  is  restored  by  a  period 
of  rest.  The  converse  effect  of  continued  strain  in  a  uniform  direc- 
tion may  be  illustrated  by  a  homely  example.  The  confectioner 
takes  a  mass  of  boiled  sugar  or  treacle  (in  a  particular  molecular 
condition  determined  by  the  temperature  to  which  it  has  been 
raised),  and  draws  the  soft  sticky  mass  out  into  a  rope;  and  then, 
folding  it  up  lengthways,  he  repeats  the  process  again  and  again. 
At  first  the  rope  is  pulled  out  of  the  ductile  mass  without  difficulty ; 
but  as  the  work  goes  on  it  gets  harder  to  do,  until  all  the  man's 
force  is  used  to  stretch  the  rope.     Here  we  have  the  phenomenon 

*  Op.  (it.  Hegler's  results  are  criticised  by  0.  M.  Ball,  Einfluss  von  Zug  aiif  die 
Ausbildung  der  Festigungsgewebe,  Jb.  d.  wiss.  Botanik,  xxxix,  pp.  305-341,  1903, 
and  by  H.  Keller,  Einfluss  von  Belastung  und  Lage  auf-die  Ausbildung  des  Gewebes 
in  Fruchtstielen,  Inaug.  Diss.  Kiel,  1904. 

t  Cf.  R.  K.  Schofield  and  G.  W.  S.  Blair,  On  dough,  Proc.  R.S.  (A),  cxxxviii, 
p.  707;  cxxxix,  p.  557,  1932-33;  also  Nadai  and  Wahl's  Plasticity,  1931.  For 
analogous  properties  of  hairs  and  fibres,  see  Shorter,  Journ.  Textile  Inst,  xv, 
1824;  etc. 

X  Cf.  Ross  Harrison's  Croonian  Lecture,  1933. 


XVI]  ON  STRESS  AND  STRAIN  987 

of  increasing  strength,  following  mechanically  on  a  rearrangement 
of  molecules,  as  the  original  isotropic  condition  is  transmuted  more 
and  more  into  molecular  asymmetry  or  anisotropy;  and  the  rope 
apparently  "adapts  itself"  to  the  increased  strain  which  it  is  called 
on  to  bear,  all  after  a  fashion  which  at  least  suggests  a  parallel  to 
the  increasing  strength  of  the  stretched  and  weighted  fibre  in  the 
plant.  For  increase  of  strength  by  rearrangement  of  the  particles 
we  have  already  a  rough  illustration  in  our  lock  of  wool  or  hank 
of  tow.  The  tow  will  carry  but  little  weight  while  its  fibres  are 
tangled  and  awry:  but  as  soon  as  we  have  carded  or  "hatchelled" 
it  out,  and  brought  all  its  long  fibres  parallel  and  side  by  side,  we 
make  of  it  a  strong  and  useful  cord*. 

But  the  lessons  which  we  learn  from  dough  and  treacle  are 
nowadays  plain  enough  in  steel  and  iron,  and  become  immensely 
more  important  in  these.  For  here  again  plasticity  is  associated 
with  a  certain  capacity  for  structural  rearrangement,  and  increased 
strength  again  results  therefrom.  Elaborate  processess  of  rolling, 
drawing,  bending,  hammering,  and  so  on,  are  regularly  employed  to 
toughen  and  strengthen  the  material.  The  "mechanical  structure" 
of  solids  has  become  an  important  subject.  And  when  the  engineer 
talks  of  repeated  loading,  of  elastic  fatigue,  of  hysteresis,  and  other 
phenomena  associated  with  plasticity  and  strain,  the  physiological 
analogues  of  these  physical  phenomena  are  perhaps  not  far  away. 

In  some  such  ways  as  these,  then,  it  would  seem  that  we  may 
coordinate,  or  hope  to  coordinate,  the  phenomenon  of  growth  with 
certain  of  the  beautiful  structural  phenomena  which  present  them- 
selves to  our  eyes  as  "provisions,"  or  mechanical  adaptations f,  for 
the  display  of  strength  where  strength  is  most  required.  That  is 
to  say  the  origin,  or  causation,  of  the  phenomenon  would  seem  to 
lie  partly  in  the  tendency  of  growth  to  be  accelerated  under  strain : 
and  partly  in  the  automatic  effect  of  shearing  strain,  by  which  it 
tends  to  displace  parts  which  grow  obliquely  to  the  direct  lines  of 
tension  and  of  pressure,  while  leaving  those  in  place  which  happen 
to  he  parallel  or  perpendicular  to  those  lines :    an  automatic  effect 

*  Cf.  Sir  Charles  Bell's  Animal  Mechanics,  chap,  v,  "Of  the  tendons  compared 
with  cordage." 

f  So  P.  Enriques  (op.  cit.  supra,  p.  5),  writing  on  the  economy  of  material  in  the 
construction  of  a  bone,  admits  that  "una  certa  impronta  di  teleologismo  qua  e  la 
e  rimasta,  mio  malgrado,  in  questo  scritto." 


988    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

which  we  can  probably  trace  as  working  on  all  scales  of  magnitude, 
and  as  accounting  therefore  for  the  rearrangement  of  minute  particles 
in  the  metal  or  the  fibre,  as  well  as  for  the  bringing  into  Hne  of 
the  fibres  within  the  plant,  or  of  the  trabeculae  within  the  bone. 

But  we  may  now  attempt  to  pass  from  the  study  of  the  individual 
bone  to  the  much  wider  and  not  less  beautiful  problems  of  mechanical 
construction  which  are  presented  to  us  by  the  skeleton  as  a  whole. 
Certain  problems  of  this  class  are  by  no  means  neglected  by  writers 
on  anatomy,  and  many  have  been/ handed  down  from  Borelli,  and 
even  from  older  writers.  For  instance,  it  is  an  old  tradition  of 
anatomical  teaching  to  point  out  in  the  human  body  examples  of 
the  three  orders  of  levers  * ;  again,  the  principle  that  the  limb-bones 
tend  to  be  shortened  in  order  to  support  the  weight  of  a  very  heavy 
animal  is  well  understood  by  comparative  anatomists,  in  accordance 
with  Euler's  law,  that  the  weight  which  a  column  liable  to  flexure 
is  capable  of  supporting  varies  inversely  as  the  square  of  its  length ; 
and  again,  the  statical  equilibrium  of  the  body,  in  relation  foi 
instance  to  the  erect  posture  of  man,  has  long  been  a  favourite  theme 
of  the  philosophical  anatomist.  But  the  general  method,  based 
upon  that  of  graphic  statics,  to  which  we  have  been  introduced  in 
our  study  of  a  bone,  has  not,  so  far  as  J  know,  been  apphed  to  the 
general  fabric  of  the  skeleton.  Yet  it  is  plain  that  each  bone  plays 
a  part  in  relation  to  the  whole  body,  analogous  to  that  which  a  little 
trabecula,  or  a  little  group  of  trabeculae,  plays  within  the  bone 
itself:  that  is  to  say,  in  the  normal  distribution  of  forces  in  the 
body  the  bones  tend  to  follow  the  lines  of  stress,  and  especially 
the  pressure-Hnes.  To  demonstrate  this  in  a  comprehensive  way 
would  doubtless  be  difficult ;  for  we  should  be  dealing  with  a  frame- 
work of  very  great  complexity,  and  should  have  to  take  account  of 

*  E.g.  (1)  the  head,  nodding  backwards  and  forwards  on  a  fulcrum,  represented 
by  the  atlas  vertebra,  lying  between  the  weight  and  the  power;  (2)  the  foot,  raising 
on  tip-toe  the  weight  of  the  body  against  the  fulcrum  of  the  ground,  where  the 
weight  is  between  the  fulcrum  and  the  power,  the  latter  being"  represented  by  the 
tendo  Achillis;  (3)  the  arm,  lifting  a  weight  in  the  hand,  with  the  power  (i.e.  the 
biceps  muscle)  between  the  fulcrum  and  the  weight.  (The  second  case,  by  the  way, 
has  been  much  disputed;  cf.  Hay  craft  in  Scha,{eT'B  Textbook  of  Physiology,  1900, 
p.  251.)  Cf.  (int.  al.)  G.  H.  Meyer,  Statik  u.  Mechanik  der  menschlicken  Knochen- 
geriiste,  1873,  pp.  13-25. 


XVI]      COMPARATIVE  ANATOMY  OF  BRIDGES       989 

a  great  variety  of  conditions*.  This  framework  is  complicated  as 
we  see  it  in  the  skeleton,  where  (as  we  have  said)  it  is  only,  or  chiefly, 
the  struts  of  the  whole  fabric  which  are  represented ;  but  to  under- 
stand the  mechanical  structure  in  detail,  we  should  have  to  follow 
out  the  still  more  complex  arrangement  of  the  ties,  as  represented 
by  the  muscles  and  ligaments,  and  we  should  also  require  much 
detailed  information  as  to  the  weights  of  the  various  parts  and  as 
to  the  other  forces  concerned.  Without  these  latter  data  we  can 
only  treat  the  question  in  a  preUminary  and  imperfect  way.  But, 
to  take  once  again  a  small  and  simplified  part  of  a  big  problem, 
let  us  think  of  a  quadruped  (for  instance,  a  horse)  in  a  standing 
posture,  and  see  whether  the  methods  and  terminology  of  the 
engineer  may  not  help  us,  as  they  did  in  regard  to  the  minute 
structure  of  the  single  bone.  And  let  us  note  in  passing  that  the 
"standing  posture,"  whether  on  two  legs  or  on  four,  is  no  very 
common  thing;  but  is  (so  to  speak),  with  all  its  correlated  anatomy, 
a  privilege  of  the  few. 

Standing  four-square  upon  its  fore-legs  and  hind-legs,  with  the 
weight  of  the  body  suspended  between,  the  quadruped  at  once 
suggests  to  us  the  analogy  of  a  bridge,  carried  by  its  two  piers. 
And  if  it  occurs  to  us,  as  naturalists,  that  we  never  look  at  a 
standing  quadruped  without  contemplating  a  bridge,  so,  conversely, 
a  similar  idea  has  occurred  to  the  engineer ;  for  Professor  Fidler, . 
in  this  Treatise  on  Bridge-Construction,  deals  with  the  chief  descrip- 
tive part  of  his  subject  under  the  heading  of  "The  Comparative 
Anatomy  of  Bridges"]"."  The  designation  is  most  just,  for  in 
studying  the  various  types  of  bridge  we  are  studying  a  series  of 
well-planned  skeletons  %]    and   (at  the  cost  of  a  little  pedantry) 

*  Our  problem  is  analogous  to  Thomas  Young's  problem  of  the  best  disposition 
of  the  timbers  in  a  wooden  ship  {Phil.  Trans.  1814,  p.  303).  He  was  not  long  of 
finding  that  the  forces  which  act  upon  the  fabric  are  very  numerous  and  very 
variable,  and  that  the  best  mode  of  resisting  them,  or  best  structural  arrangement 
for  ultimate  strength,  becomes  an  immensely  complicated  problem. 

f  By  a  bolder  metaphor  Fontenelle  said  of  Newton  that  he  had  "fait  I'anatomie 
de  la  lumiere." 

X  In  like  manner.  Clerk  Maxwell  could  not  help  employing  the  term  "skeleton" 
in  defining  the  mathematical  conception  of  a  "frame,"  constituted  by  points  and 
their  interconnecting  lines :  in  studying  the  equilibrium  of  which,  we  consider  its 
different  points  as  rnutually  acting  on  each  other  with  forces  whose  directions  are 
those  of  the  lines  joining  each  pair  of  points.  Hence  (says  Maxwell),  "in  order  to 
exhibit  the  mechanical  action  of  the  frame  in  the  most  elementary  manner,  we  may 


990    ON  FOKM  AND  MECHANICAL  EFFICIENCY  [ch. 

we  might  go  even  .further,  and  study  (after  the  fashion  of  the 
anatomist)  the  "osteology"  and  "desmology"  of  the  structure,  that 
is  to  say  the  bones  which  are  represented  by  "struts,"  and  the 
Hgaments,  etc.,  which  are  represented  by  "ties."  Furthermore 
after  the  methods  of  the  comparative  anatomist,  we  may  classify 
the  families,  genera  and  species  of  bridges  according  to  their  dis- 
tinctive mechanical  features,  which  correspond  to  certain  definite 
conditions  and  functions. 


Fig.  468.  Skeleton  of  an  American  bison.  (An  unusually  weU-mounted  skeleton, 
of  American  workmanship,  now  in  the  Anatomical  Museum  of  Edinburgh 
University.) 

In  more  ways  than  one,  the  quadrupedal  bridge  is  a  remarkable 
one;  and  perhaps  its  most  remarkable  peculiarity  is  that  it  is  a 
jointed  and  flexible  bridge,  remaining  in  equilibrium  under  con- 
siderable and  sometimes  great  modifications  of  its  curvature,  such 
as  we  see,  for  instance,  when  a  cat  humps  or  flattens  her  back. 
The  fact  thsit  flexibility  is  an  essential  feature  in  the  quadrupedal 

draw  it  as  a  skeleton,  in  which  the  different  points  are  joined  by  straight  lines, 
and  we  may  indicate  by  numbers  attached  to  these  lines  the  tensions  or  com- 
pressions in  the  corresponding  pieces  of  the  frame"  {Trans.  R.S.E.  xxvi,  p.  1, 
1870).  It  follows  that  the  diagram  so  constructed  represents  a  "diagram  of 
forces,"  in  this  limited  sense  that  it  is  geometrical  as  regards  the  position  and 
direction  of  the  forces,  but  arithmetical  as  regards  their  magnitude.  It  is  to  just 
such  a  diagram  that  the  animal's  skeleton  tends  to  approximate. 


XVI]      COMPARATIVE  ANATOMY  OF  BRIDGES       991 

bridge,  while  it  is  the  last  thing  which  an  engineer  desires  and 
the  first  which  he  seeks  to  provide  against,  will  impose  certain 
important  limiting  conditions  upon  the  design  of  the  skeletal  fabric. 
But  let  us  begin  by  considering  the  quadruped  at  rest,  when  he 
stands  upright  and  motionless  upon  his  feet,  and  when  his  legs 
exercise  no  function  save  only  to  carry  the  weight  of  the  whole 
body.  So  far  as  that  function  is  concerned,  we  might  now  perhaps 
compare  the  horse's  legs  with  the  tall  and  slender  piers  of  some 
railway  bridge;  but  it  is  obvious  that  these  jointed  legs  are  ill- 
adapted  to  receive  the  horizontal  thrust  of  any  arch  that  may  be 
placed  atop  of  them.  Hence  it  follows  that  the  curved  backbone 
of  the  horse,  which  appears  to  cross  like  an  arch  the  span  between 
his  shoulders  and  his  flanks,  cannot  be  regarded  as  an  arch,  in  the 


Fig.  469.     a,  tied  arch;  b,  bowstring  girder. 


engineer's  sense  of  the  word.  It  resembles  an  arch  in  form,  but 
not  in  function,  for  it  cannot  act  as  an  arch  unless  it  be  held  back 
at  each  end  (as  every  arch  is  held  back)  by  ahutinents  capable  of 
resisting  the  horizontal  thrust;  and  these  necessary  abutments  are 
not  present  in  the  structure.  But  in  various  ways  the  engineer 
can  modify  his  superstructure  so  as  to  supply  the  place  of  these 
external  reactions,  which  in  the  simple  arch  are  obviously  indis- 
pensable. Thus,  for  example,  we  may  begin  by  inserting  a  straight 
steel  tie,  AB  (Fig.  469),  uniting  the  ends  of  the  curved  rib  AaB\ 
and  this  tie  will  supply  the  place  of  the  external  reactions,  converting 
the  structure  into  a  "tied  arch,"  such  as  we  may  see  in  the  roofs 
of  many  railway  stations.  Or  we  may  go  on  to  fill  in  the  space 
between  arch  and  tie  by  a  "web-system,"  converting  it  into  what 
the  engineer  describes  as  a  "parabolic  bowstring  girder"  (Fig.  4696). 
In  either  case,  the  structure  becomes  an . independent  "detached 


992    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

girder,"  supported  at  each  end  but  not  otherwise  fixed,  and  con- 
sisting essentially  of  an  upper  compression-member,  AaB,  and  a 
lower  tension-member,  AB.  But  again,  in  the  skeleton  of  the 
quadruped,  the  necessary  tie,  AB,  of  the  simple  bow-girder  is  not  to 
be  found ;  and  it  follows  that  these  comparatively  simple  types  of 
bridge  do  not  correspond  to,  nor  do  they  help  us  to  understand, 
the  type  of  bridge  which  Nature  has  designed  in  the  skeleton  of  the 
quadruped.  Nevertheless  if  we  try  to  look,  as  an  engineer  would 
look,  at  the  actual  design  of  the  animal  skeleton  and  the  actual 
distribution  of  its  load,  we  find  that  the  one  is  most  admirably 
adapted  to  the  other,  according  to  the  strict  principles  of  engineering 
construction.  The  structure  is  not  an  arch,  nor  a  tied  arch,  nor 
a  bowstring  girder:  but  it  is  strictly  and  beautifully  comparable 
to  the  main  girder  of  a  double-armed  cantilever  bridge. 


Fig.  470.     A  two-armed  cantilever  of  the  Forth  Bridge.     Thick  lines,  com- 
pression-members (bones);   thin  lines,  tension -members  (ligaments). 

Obviously,  in  our  quadrupedal  bridge,  the  superstructure  does 
not  terminate  (as  it  did  in  our  former  diagram)  at  the  two  points 
of  support,  but  it  extends  beyond  them,  carrying  the  head  at  one 
end  and  sometimes  a  heavy  tail  at  the  other,  upon  projecting  arms 
or  "cantilevers." 

In  a  typical  cantilever  bridge,  such  as  the  Forth  Bridge  (Fig.  470), 
a  certain  simplification  is  introduced.  For  each  pier  carries,  in  this 
case,  its  own  double-armed  cantilever,  linked  by  a  short  connecting 
girder  to  the  next,  but  so  jointed  to  it  that  no  weight  is  transmitted 
from  one  cantilever  to  another.  The  bridge  in  short  is  cut  into 
separate  sections,  practically  independent  of  one  another;  at  the 
joints  a  certain  amount  of  bending  is  not  precluded,  but  shearing 
strain  is  evaded;  and  each  pier  carries  only  its  own  load.  By 
this  arrangement  the  engineer  finds  that  design  and  construction 
are  alike  simplified  and  faciUtated.     In  the  horse  or  the  ox,  it  is 


XVI]      COMPARATIVE  ANATOMY  OF  BRIDGES       993 

obvious  that  the  two  piers  of  the  bridge,  that  is  to  say  the  fore-legs 
and  the  hind-legs,  do  not  bear  (as  they  do  in  the  Forth  Bridge) 
separate  and  independent  loads,  but  the  whole  system  forms  a 
continuous  structure.  In  this  case,  the  calculation  of  the  loads 
will  be  a  little  more  difficult  and  the  corresponding  design  of  the 
structure  a  Httle  more  complicated.  We  shall  accordingly  simplify 
our  problem  very  considerably  if,  to  begin  with,  we  look  upon  the 
quadrupedal  skeleton  as  constituted  of  two  separate  systems,  that 
is  to  say  of  two  balanced  cantilevers,  one  supported  on  the  fore-legs 
and  the  other  on  the  hind;  and  we  may  deal  afterwards  with  the 
fact  that  these  two  cantilevers  are  not  independent,  but  are  bound 
up  in  one  common  field  of  force  and  plan  of  construction. 

In  both  horse  and  ox  it  is  plain  that  the  two  cantilever  systems 
into  which  we  may  thus  analyse  the  quadrupedal  bridge  are  unequal 
in  magnitude  and  importance.  The  fore-part  of  the  animal  is  much 
bulkier  than  its  hind-quarters,  and  the  fact  that  the  fore-legs  carry, 
as  they  so  evidently  do,  a  greater  weight  than  the  hind-legs  has 
long  been  known  and  is  easily  proved ;  we  have  only  to  walk  a  horse 
on  to  a  weigh-bridge,  weigh  fijst  his  fore-legs  and  then  his  hind-legs, 
to  discover  that  what  we  may  call  his  front  half  weighs  a  good  deal 
more  than  what  is  carried  on  his  hind  feet,  say  about  three-fifths 
of  the  whole  weight  of  the  animal. 

The  great  (or  anterior)  cantilever  then,  in  the  horse,  is  constituted 
by  the  heavy  head  and  still  heavier  neck  on  one-  side  of  that  pier 
which  is  represented  by  the  fore-legs,  and  by  the  dorsal  vertebrae 
carrying  a  large  part  of  the  weight  of  the  trunk  upon  the  other 
side;  and  this  weight  is  so  balanced  over  the  fore-legs  that  the 
cantilever,  while  "anchored"  to  the  other  parts  of  the  structure, 
transmits  but  little  of  its  weight  to  the  hind-legs,  and  the  amount 
so  transmitted  will  vary  with  the  attitude  of  tne  head  and  with 
the  position  of  any  artificial  load*.  Under  certain  conditions,  as 
when  the  head  is  thrust  well  forward,  it  is  evident  that  the  hind-legs 
will  be  actually  relieved  of  a  portion  of  the  comparatively  small 
load  which  is  their  normal  share. 

*  When  the  jockey,  crouches  over  the  neck  of  his  race-horse,  and  when  Tod 
Sloan  introduced  the  "American  seat,"  the  avowed  object  in  both  cases  is  to  relieve 
the  hind-legs  of  weight,  and  so  leave  them  free  for  the  work  of  propulsion.  On 
the  share  taken  by  the  hind-limbs  in  this  latter  duty,  and  other  matters,  cf. 
Stillman,  The.  Tlnrne,  in  Motion,  1882,  p.  69. 


994    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

But  here  we  pass  from  the, statical  problem  to  the  dynamical, 
from  the  horse  at  rest  to  the  horse  in  motion,  from  the  observed 
fact  that  weight  lies  mainly  over  the  fore-legs  to  the  question  of 
what  advantage  is  gained  by  such  a  distribution  of  the  load.  Taking 
the  hind-legs  as  the  main  propulsive  agency,  as  we  may  now  safely 
do,  the  moment  of  propulsion  is  about  the  hind-hooves;  then  (as  we 
see  in  Fig.  471)  we  may  take  the  weight,   W  =  A  sin  a,  and  the 

W     H 

propulsive    force,  f=  A  cos  a,   and    -y  =  y,   WL  =  fH  being  the 

balanced  condition.  From  the  statical  point  of  view  the  load  must 
balance  over  the  fore-legs;  from  the  dynamical  point  of  view  it 
might  well  lie  even  farther  forward.     And  when  the  jockey  crouches 


Fig.  471. 

over  the  horse's  neck,  and  when  Tod  Sloan  introduced  the  "  American 
seat,"  both  shew  a  remarkable,  though  perhaps  unconscious,  insight 
into  the  dynamical  proposition. 

Our  next  problem  is  to  discover,  in  a  rough  and  approximate 
way,  some  of  the  structural  details  which  the  balanced  load  upon 
the  double  cantilever  will  impress  upon  the  fabric. 


Working  by  the  methods  of  graphic  statics,  the  engineer's  task  is, 
in  theory,  one  of  great  simplicity.  He  begins  by  drawing  in  outline 
the  structure  which  he  desires  to  erect;  he  calculates  the  stresses 
and  bending-moments  necessitated  by  the  dimensions  and  load  on 
the  structure;  he  draws  a  new  diagram  representing  these  forces, 
and  he  designs  and  builds  his  fabric  on  the  lines  of  this  statical 
diagram.  He  does,  in  short,  precisely  what  we  have  seen  nature 
doing  in  the  case  of  the  bone.     For  if  we  had  begun,  as  it  were. 


xvi]  THE  SKELETON  AS  A  BRIDGE  995 

by  blocking  out  the  femur  roughly,  and  considering  its  position 
and  dimensions,  its  means  of  support  and  the  load  which  it  has  to 
bear,  we  could  have  proceeded  at  once  to  draw  the  system  of 
stress-lines  which  must  occupy  that  field  of  force :   and  to  precisely 


B 

Fig.  472.     A,  Span  of  proposed  bridge.     B,  Stress  diagram,  or  diagram 
of  bending-moments*. 

those  stress-lines  has  Nature  kept  in  the  building  of  the  bone,  down 
to  the  minute  arrangement  of  its  trabecular 

The  essential  function  of  a  bridge  is  to  stretch  across  a  certain 
span,  and  carry  a  certain  definite  load;   and  this  being  so,  the  chief 


XfJwTX 


Fig.  473.     The  bridge  construnted,  as  a  parabolic  girder. 

problem  in  the  designing  of  a  bridge  is  to  provide  due  resistance 
to  the  "bending-moments"  which  result  from  the  load.  These 
bending-moments  will  vary  from  point  to  point  along  the  girder, 

*  This  and  the  following  diagrams  are  borrowed  and  adapted  from  Professor 
Fidler's  Bridge  Construction.  We  may  reflect  with  advantage  on  Clerk  Maxwell's 
saying  that  "the  use  of  diagrams  is  a  particular  instance  of  that  method  of  symbols 
which  is  so  powerful  an  aid  in  the  advancement  of  science" ;  and  on  his  explanation 
that  "a  diagram  differs  from  a  picture  in  this  respect  that  in  a  diagram  no  attempt 
is  made  to  represent  those  factors  of  the  actual  material  system  which,  are  not  the 
special  objects  of  our  study." 


996    ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

and  taking  the  simplest  case  of  a  uniform  load,  whether  supported 
at  one  or  both  ends,  they  will  be  represented  by  points  on  a  parabola. 
If  the  girder  be  of  uniform  depth  and  section,  that  is  to  say  if  its 
two  flanges,  respectively  under  tension  and  compression,  be  equal 
and  parallel  to  one  another,  then  the  stress  upon  these  flanges  will 
vary  as  the  bending-moments,  and  will  accordingly  be  very  severe 
in  the  middle  and  will  dwindle  towards  the  ends.  But  if  we  make 
the  dejpih  of  the  girder  everjrwhere  proportional  to  the  bending- 
moments,  that  is  to  say  if  we  copy  in  the  girder  the  outlines  of  the 
bending-moment  diagram,  then  our  design  will  automatically  meet 
the  circumstances  of  the  case,  for  the  horizontal  stress  in  each  flange 
will  now  be  uniform  throughout  the  length  of  the  girder.     In  short. 


/ 

w 

Fig.  474. 

in  Professor  Fidler's  words,  "Every  diagram  of  moments  represents 
the  outline  of  a  framed  structure  which  will  carry  the  given  load 
with  a  uniform  horizontal  stress  in  the  principal  members." 

In  the  above  diagrams  (Fig.  474,  a,  b)  (which  are  taken  from 
the  original  ones  of  Culmann),  we  see  at  once  that  the  loaded  beam 
or  bracket  {a)  has  a  "danger-point"  close  to  its  fixed  base,  that  is 
to  say  at  the  .point  remotest  from  its  load.  But  in  the  parabolic 
bracket  (6)  there  is  no  danger-point  at  all,  for  the  dimensions  of 
the  structure  are  made  to  increase  jpari  passu  with  the  bending- 
moments:  stress  and  resistance  vary  together.  Again  in  Fig.  475, 
we  have  a  simple  span  (A),  with  its  stress  diagram  (B);  and  in 
(C)  we  have  the  corresponding  parabolic  girder,  whose  stresses 
are  now  uniform  throughout.  In  fact  we  see  that,  by  a  process  of 
conversion,  the  stress  diagram  in  each  case  becomes  the  structural 


XVI 


OF  RECIPROCAL  DIAGRAMS 


997 


diagram  in  the  other*.  Now  all  this  is  but  the  modern  rendering 
of  one  of  Galileo's  most  famous  propositions.  In  the  Dialogue  which 
we  have  already  quoted  more  tlfan  oncef,  Sagredo  says  "It  would 
be  a  fine  thing  if  one  could  discover  the  proper  shape  to  give  a  solid 
in  order  to  make  it  equally  resistant  at  every  point,  in  which  case 
a  load  placed  at  the  middle  would  not  produce  fracture  more  easily 


I 


Fig.  475. 

than  if  placed  at  any  other  point  J."  And  Gahleo  (in  the  person 
of  Salviati)  first  puts  the  problem  into  its  more  gen'eral  form;  and 
then  shews  us  how,  by  giving  a  parabolic  outline  to  our  beam,  we 
have  its  simple  and  comprehensive  solution.  It  was  such  teaching 
as  this  that  led  R.  A.  Millikan  to  say  that  "we  owe  our  present-day 
civilisation  to  Galileo." 


*  The  method  of  constructying  reciprocal  diagrams,  of  which  one  should  represent 
the  outlines  of  a  frame  and  the  other  the  system  of  forces  necessary  to  keep  it 
in  equilibrium,  was  first  indicated  in  Culmann's  Graphische  Statik;  it  was  greatly 
developed  soon  afterwards  by  Macquorn  Rankine  (Phil.  Mag.  Feb,  1864,  and 
Applied  Mechanics,  passim),  to  whom  the  application  of  the  principle  to  engineering 
practice  is  mainly  due.  See  also  Fleeming  Jenkin,  On  the  practical  application 
of  reciprocal  figures  to  the  calculation  of  strains  in  framework,  Trans.  R.S.E. 
XXV,  pp.  441^48,  1869;  and  Clerk  Maxwell,  ibid,  xxvi,  p.  9,  1870,  and  Phil.  Mag. 
April  1864. 

t  Dialogues  concerning  Two  New  Sciences  (1638);   Crew  and  Salvio's  translation 
p.  140  seq. 

X  As  in  the  great  case  of  the  Eiffel  Tower,  supra,  p.  29. 


998    ON  FOKM  AND  MECHANICAL  EFFICIENCY  [ch. 

In  the  case  of  our  cantilever  bridge,  we  shew  the  primitive  girder 
in  Fig.  475,  A,  with  its  bending-moment  diagram  (B);  and  it  is 
evident  that,  if  we  turn  this  diagram  upside  down,  it  will  still  be 
illustrative,  just  as  before,,  of  the  bending-moments  from  point 
to  point:  for  as  yet  it  is  merely  a  diagram,  or  graph,  of  relative 
magnitudes. 

To  either  of  these  two  stress  diagrams,  direct  or  inverted,  we 
may  fit  the  design  of  the  construction,  as  in  Figs.  475,  C  and  476. 


Fig.  476. 

Now  in  different  animals  the  amount  and  distribution  of  the 
load  differ  so  greatly  that  we  can  expect  no  single  diagram,  drawn 
from  the  comparative  anatomy  of  bridges,  to  apply  equally  well  to 
all  the  cases  met  with  in  the  comparative  anatomy  of  quadrupeds; 
but  nevertheless  we  have  already  gained  an  insight  into  the  general 
principles  of  "structural  design"  in  the  quadrupedal  bridge. 

.  In  our  last  diagram  the  upper  member  of  the  cantilever  is  under 
tension;  it  is  represented  in  the  quadruped  by  the  ligamentum  nuchae 
on  the  one  side  of  the  cantilever,  and  by  the  supraspinous  ligaments 
of  the  dorsal  vertebrae  on  the  other.  The  compression-member 
is  similarly  represented,  on  both  sides  of  the  cantilever,  by  the 
vertebral  column,  or  rather  by  the  bodies  of  the  vertebrae;  while 
the  web,  or  "filling,"  of  the  girders,  that  is  to  say  the  upright  or 
sloping  members  which  extend  from  one  flange  to  the  other,  is 
represented  on  the  one  hand  by  the  spines  of  the  vertebrae,  and  on 
the  other  hand  by  the  obli(|ue  interspinous  ligaments  and  muscles — 
that  is  to  say,  by  compression -members  and  tension-members 
inclined  in  opposite  directions  to  one  another.  The  high  spines  over 
the  quadruped's  withers  are  no  other  than  the  high  sisruts  which 
rise  over  the  supporting  piers  in  the  parabolic  girder,  and  correspond 
to  the  position  of  the  maximal  bending-moments.  The  fact  that 
these  tall  vertebrae  of  the  withers  usually  slope  backwards,  some- 


XVI]  OF  RECIPROCAL  DIAGRAMS  999 

times  steeply,  in  a  quadruped,  is  easily  and  obviously  explained*. 
For  each  vertebra  tends  to  act  as  a  "hinged  lever,"  and  its  spine, 
acted  on  by  the  tensions  transmitted  by  the  ligaments  on  either  side, 
takes  up  its  position  as  the  diagonal  of  the  parallelogram  of  forces 
to  which  it  is  exposed. 

It  happens  that  in  these  comparatively  simple  types  of  cantilever 
bridge  the  whole  of  the  parabolic  curvature  is  transferred  to  one 
or  other  of  the  principal  members,  either  the  tension-member  or 
the  compression-member  as  the  case  may  be.  But  it  is  of  course 
equally  permissible  to  have  both  members  curved,  in  opposite 
directions.  This,  though  not  exactly  the  case  in  the  Forth  Bridge, 
is  approximately  so;  for  here  the  main  compression-member  is 
curved  or  arched,  and  the  main  tension-member  slopes  downwards 
on  either  side  from  its  maximal  height  above  the  piers.  In  short, 
the  Forth  Bridge  (Fig.  470)  is  a  nearer  approach  than  either  of 
the  other  bridges  which  we  have  illustrated  to  the  plan  of  the 
quadrupedal  skeleton;  for  the  main  compression-member  almost 
exactly  recalls  the  form  of  the  backbone,  while  the  main  tension- 
meraber,  though  not  so  closely  similar  to  the  supraspinous  and 
nuchal  ligaments,  corresponds  to  the  plan  of  these  in  a  somewhat 
simplified  form. 

We  may  now  pass  without  difficulty  from  the  two-armed  canti- 
lever supported  on  a  single  pier,  as  it  is  in  each  separate  section  of  the 
Forth  Bridge,  or  as  we  have  imagined  it  to  be  in  the  fore-quarters 
of  a  horse,  to  the  condition  which  actually  exists  in  a  quadruped, 
when  a  two-armed  cantilever  has  its  load  distributed  over  two 
separate  piers.  This  is  not  precisely  what  an  engineer  calls  a 
"continuous"  girder,  for  that  term  is  applied  to  a  girder  which, 
as  a  continuous  structure,  has  three  supports  and  crosses  two  or  more 
spans,  while  here  there  is  only  one.     But  nevertheless,  this  girder 

*  The  form  and  direction  of  the  vertebral  spines  have  been  frequently  and 
elaborately  described;  cf.  (e.g.)  H.  Gottlieb,  Die  Anticlinie  der  Wirbelsaule  der 
Saugethiere,  Morphol.  Jahrb.  lxix,  pp.  179-220,  1915,  and  many  works  quoted 
therein.  According  to  Morita,  Ueber  die  Ursachen  der  Richtung  und  Gestalt  der 
thoracalen  Dornfortsatze  der  8augethierwirbelsaule  {ibi  cit.  p.  201),  various  changes 
take  place  in  the  direction  or  inclination  of  these  processes  in  rabbits,  after  section 
of  the  interspinous  ligaments  and  muscles.  These  changes  seem  to  be  very  much 
what  we  should  expect,  on  simple  mechanical  grounds.  See  also  0.  Fischer, 
Theoretische  Grundlagen  fur  eine  Mechanik  der  lebenden  Korper,  Leipzig,  1906, 
pp.  X,  372. 


1000  ON  FOR]\J  AND  MECHANICAL  EFFICIENCY  [ch. 

is  effectively  continuous  from  the  head  to  the  tip  of  the  tail;  and 
at  each  point  of  support  (A  and  5)  it  is  subjected  to  the  negative 
bending-moment  due  to  the  overhanging  load  on  each  of  the 
projecting  cantilever  arms  AH  and  BT.  The  diagram  of  bending- 
moments  will  (according  to  the  ordinary  conventions)  lie  below  the 
base  line  (because  the  moments  are  negative),  and  must  take  some 
such  forfn  as  that  shewn  in  the  diagram :  for  the  girder  must  suffer 
its  greatest  bending  stress  not  at  the  centre,  but  at  the  two  points 
of  support  A  and  5,  where  the  moments  are  measured  by  the 
vertical  ordinates.  It  is  plain  that  this  figure  only  differs,  from 
a  representation  of  two  independent  two-armed  cantilevers  in  the 
fact  that  there  is  no  point  midway  in  the  span  where  the  bending- 
moment  vanishes,  but  only  a  region  between  the  two  piers  in  which 
it  tends'  to  diminish. 


H  A  B  T 

Fig.  477.    Two-armed  cantilever  and  its  stress  diagram. 

The  diagram  effects  a  graphic  summation  of  the  positive  and 
negative  moments,  but  its  form  may  assume  various  modific^ttions 
according  to  the  method  of  graphic  summation  which  we  choose 
to  adopt;  and  it  is  obvious  also  that  the  form  of  the  diagram 
rriay  assume  many  modifications  of  detail  according  to  the  actual 
distribution  of  the  load.  In  all  cases  the  essential  points  to  be 
observed  are  these:  firstly  that  the  girder  which  is  to  resist  the 
bending-moments  induced  by  the  load  must  possess  its  two  principal 
members — an  upper  tension-member  or  tie,  represented  by  ligament 
(whose  tension  doubtless  varies  along  its  length),  and  a  lower 
compression-member  represented  by  bone:  these  members  being 
united  by  a  web  represented  by  the  vertebral  spines  with  their 
interspinous  ligaments,  and  being  placed  one  above  the  other  in 
the  order  named  because  the  moments  are  negative;  secondly  we 
observe  that  the  depth  of  the  web,  or  distance  apart  of  the  principal 


xvij  OF  RECIPROCAL  DIAGRAMS  1001 

members — that  is  to  say  the  height  of  the  vertebral  spines — must 
be  proportional  to  the  bending-moment  at  each  point  along  the 
length  of  the  girder. 

In  the  case  of  an  animal  carrying  most  oi  his  weight  upon  his 
fore-legs,  as  the  horse  or  the  ox  do,  the  bending-moment  diagram 
will  be  unsymmetrical,  after  the  fashion  of  Fig.  478,  the  precise  form 
depending  on  the  distribution  of  weights  and  distances. 


7a/7  Head 

Fig.  478.     Stress -diagram  of  horse's  backbone. 

On  the  other  hand  the  Dinosaur,  with  his  light  head  and  enormous 
tail  would  give  us  a  moment-diagram  with  the  opposite  kind  of 
asymmetry,  the  greatest  bending  stress  being  now  found  over  the 
haunclies,  at  B  (Fig.  479).  A  glance  at  the  skeleton  of  Diplodocus 
will  shew  us  the  high  vertebral  spines  over  the  loins,  in  precise 
correspondence  with  the  requirements  of  this  diagram:  just  as  in 
the  horse,  under  the  opposite  conditions  of  load,  the  highest  vertebral 
spines  are  those  of  the  withers,  that  is  to  say  those  of  the  posterior 
cervical  and  anterior  dorsal  vertebrae. 

We  have  now  not  only  dealt  with  the  general  resemblance,  both 
in  structure  and  in  function,  of  the  quadrupedal  backbone  with  its 
associated  ligaments  to  a  double-armed  cantilever  girder,  but  we 
have  begun  to  see  how  the  characters  of  the  vertebral  system  must 
differ  in  different  quadrupeds,  according  to  the  conditions  imposed 
by  the  varying  distribution  of  the  load :  and  in  particular  how  the 
height  of  the  vertebral  spines  which  constitute  the  web  will  be 
in  a  definite  relation,  as  regards  magnitude  and  position,  to  the 
bending-moments  induced  thereby.  We  should  require  much  de- 
tailed information  as  to  the  actual  weights  of  the  several  parts  of 
the  body  before  we  could  follow  out  quantitatively  the  mechanical 
efficiency  of  each  type  of  skeleton;  but  in  an  approximate  way 
what  we  have  already  learnt  will  enable  us  to  trace  many  interesting 
correspondences  between  structure  and  function  in  this  particular 
part  of  comparative  anatomy.  We  must,  however,  be  careful  to 
note  that  the  great  cantilever  system  is  not  of  necessity  constituted 


1002  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

by  the  vertebral  column  and  its  ligaments  alone,  but  that  the  pelvis, 
firmly  united  as  it  is  to  the  sacral  vertebrae,  and  stretching  back- 
wards far  beyond  the  acetabulum,  becomes  an  intrinsic  part  of  the 
system ;  and  heljnng  (as  it  does)  to  carry  the  load  of  the  abdominal 
viscera,  it  constitutes  a  great  portion  of  the  posterior  cantilever  arm, 
or  even  its  chief  portion  in  cases  where  the  size  and  weight  of  the 
tail  are  insignificant,  as  is  the  case  in  the  majority  of  terrestrial 
mammals. 


Tail  Head 

Fig.  479.     Stress- diagram  of  backbone  of  Dinosaur. 

We  may  also  note  here,  that  just  as  a  bridge  is  often  a  " combined" 
or  composite  structure,  exhibiting  a  combination  of  principles  in 
its  construction,  so  in  the  quadruped  we  have,  as  it  were,  another 
girder  supported  by  the  same  piers  to  carry  the  viscera;  and  con- 
sisting of  an  inverted  parabolic  girder,  whose  compression-member 
is  again  consti'tuted  by  the  backbone,  its  tension-member  by  the 
line  of  the  sternum  and  the  abdominal  muscles,  while  the  ribs  and 
intercostal  muscles  play  the  part  of  the  web  or  filling. 

A  very  few  instances  must  suffice  to  illustrate  the  chief  variations 
in  the  load,  and  therefore  in  the  bending-moment  diagram,  and 
therefore  also  in  the  plan  of  construction,  of  various  quadrupeds. 
But  let  us  begin  by  setting  forth,  in  a  few  cases,  the  actual  weights 
which  are  borne  by  the  fore-limbs  and  the  hind-limbs,  in  our 
quadrupedal  bridge*. 


On 

On 

%on 

%on 

Gross  weight 

fore-feet 

hind-feet 

fore-feet 

hind -feet 

ton    cwt. 

cwt. 

cwt. 

Camel  (Bactrian) 

-     14-25 

9-25 

4-5 

67-3 

32-7 

Llama 

-      2-75 

1-75 

0-875 

66-7 

33-3 

Elephant  (Indian) 

1     15-75 

20-5 

14-75 

58-2 

41-8 

Horse 

-      8-25 

4-75 

3-5 

57-6 

42-4 

Horse  (large  Clydes- 

-    15-5 

8-5 

7-0 

54-8 

45-2 

dale) 

*  I  owe  the  first  four  of  these  determinations  to  the  kindness  of  Sir  P.  Chalmers 
Mitchell,  who  had  them  made  for  me  at  the  Zoological  Society's  Gardens;  while 
the  great  Clydesdale  carthorse  was  weighed  for  me  by  a  friend  in  Dundee. 


XVI]        IN  THE  SKELETON  OF  QUADRUPEDS       1003 

It  will  be  observed  that  in  all  these  animals  the  load  upon  the 
fore-feet  preponderates  considerably  over  that  upon  the  hind,  the 
preponderance  being  rather  greater  in  the  elephant  than  in  the  horse, 
and  markedly  greater  in  the  camel  and  the  llama  than  in  the  other 
two.  But  while  these  weights  are  helpful  and  suggestive,  it  is 
obvious  that  they  do  not  go  nearly  far  enough  to  give  us  a  full 
insight  into  the  constructional  diagram  to  which  the  animals  are 
conformed.  For  such  a  purpose  we  should  require  to  weigh  the 
total  load,  not  in  two  portions  but  in  many;  and  we  should  also 
have  to  take  close  account  of  the  general  form  of  the  animal,  of 
the  relation  between  that  form  and  the  distribution  of  the  load, 
and  of  the  actual  directions  of  each  bone  and  ligament  by  which 
the  forces  of  compression  and  tension  >were  transmitted.  All  this 
lies  beyond  us  for  the  present;   but  nevertheless  we  may  consider. 


Tail 


Head 


Fig.  480.     Stress- diagram  of  Titanotheriuni. 


very  briefly,  the  principal  cases  involved  in  our  enquiry,  of  which 
the  above  animals  form  a  partial  and  preliminary  illustration. 

(1)  Wherever  we  have  a  heavily  loaded  anterior  cantilever  arm, 
that  is  to  say  whenever  the  head  and  neck  represent  a  considerable 
fraction  of  the  whole  weight  of  the  body,  we  tend  to  have  large 
bending-moments  over  the  fore-legs,  and  correspondingly  high  spines 
over  the  vertebrae  of  the  withers.  This  is  the  case  in  the  great 
majority  of  four-footed  terrestrial  animals,  the  chief  exceptions 
being  found  in  animals  with  comparatively  small  heads  but  large 
and  heavy  tails,  such  as  the  anteaters  or  the  Dinosaurian  reptiles, 
and  also  (very  naturally)  in  animals  such  as  the  crocodile,  where 
the  "  bridge  "  can  scarcely  be  said  to  be  developed,  for  the  long  heavy 
body  sags  down  to  rest  upon  the  ground.  The  case  is  sufficiently 
exemplified  by  the  horse,  and  still  more  notably  by  the  stag,  the  ox, 
or  the  pig.     It  is  illustrated  in  the  skeleton  of  a  bison  (Fig.  468),  or 


1004  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

in  the  accompanying  diagram  of  the  conditions  in  the  great  extinct 
Titanotherium. 

(2)  In  the  elephant  and  the  camel  we  have  similar  conditions, 
but  slightly  modified.  In  both  cases,  and  especially  in  the  latter, 
the  weight  on  the  fore-quarters  is  relatively  krge ;  and  in  both  cases 
the  bending-moments  are  all  the  larger,  by  reason  of  the  length 
and  forward  extension  of  the  camel's  neck  and  the  forward  position 
of  the  heavy  tusks  of  the  elephant.  In  both  cases  the  dorsal  spines 
are  large,  but  they  do  not  strike  us  as  exceptionally  so;  but  in 
both  cases,  and  especially  in  the  elephant,  they  slope  backwards 
in  a  marked  degree.  Each  spine,  as  already  explained,  must  in  all 
cases  assume  the  position  of  the  diagonal  in  the  parallelogram  of 
forces  defined  by  the  tensions  acting  on  it  at  its  extremity;  for  it 
constitutes  a  "hinged  lever,"  by  which  the  bending-moments  on 
either  side  are  automatically  balanced ;  and  it  is  plain  that  the  more 
the  spine  slopes  backwards  the  more  it  indicates  a  relatively  large 
strain  thrown  upon  the  great  ligament  of  the  neck,  and  a  relief 
of  strain  upon  the  more  directly  acting,  but  weaker,  Hgaments  of 
the  back  and  loins.  In  both  cases,  the  bending-moments .  would 
seem  to  be  more  evenly  distributed  over  the  region  of  the  back  than, 
for  instance,  in  the  stag,  with  its  light  hind-quarters  and  heavy  load 
of  antlers:  and  in  bo^h  cases  the  high  "girder"  is  considerably 
prolonged,  by  an  extension  of  the  tall  spines  backwards  in  the 
direction  of  the  loins.  When  we  come  to  such  a  case  as  the  mam- 
moth, with  its  immensely  heavy  and  immensely  elongated  tusks, 
we  perceive  at  once  that  the  bending-moments  over  the  fore-legs 
are  now  very  severe;  and  we  see  also  that  the  dorsal  spines  in  this 
region  are. much  more  conspicuously  elevated  than  in  the  ordinary 
elephant. 

(3)  In  the  case  of  the  giraffe  we  have,  without  doubt,  a  very 
heavy  load  upon  the  fore-legs,  though  no  weighings  are  at  hand 
to  define  the  ratio;  but  as  far  as  possible  this  disproportionate 
load  would  seem  to  be  relieved  by  help  of  a  downward  as  well  as 
backward  thrust,  through  the  sloping  back  to  the  unusually  low 
hind-quarters.  The  dorsal  spines  of  the  vertebrae  are  very  high 
and  strong,  and  the  whole  girder-system  very  perfectly  formed.  The 
elevated  rather  than  protruding  position  of  the  head  lessens  the 
anterior  bending-moment  as  far  as  possible,  but  it  leads  to  a  strong 


XVI]       IN  THE  SKELETON  OF  QUADRUPEDS       1005 

compressional  stress  transmitted  almost  directly  downwards  through 
the  neck:  in  correlation  with  which  we  observe  that  the  bodies  of 
the  cervical  vertebrae  are  exceptionally  large  and  strong,  and  steadily 
increase  in  size  and  strength  from  the  head  downwards. 

(4)  In  the  kangaroo,  the  fore-limbs  are  entirely  relieved  of  theii* 
load,  and  accordingly  the  tall  spines  over  the  withers,  which  were  so 
conspicuous  in  all  heavy-headed  quadrupeds,  have  now  completely 
vanished.  The  creature  has  become  bipedal,  and  body  and  tail 
form  the  extremities  of  a  single  balanced  cantilever,  whose  maximal 
bending-moments  are  marked  by  strong,  high  lumbar  and  sacral 
vertebrae,  and  by  iliac  bones  of  peculiar  form,  of  exceptional  strength 
and  nearly  upright  position. 

Precisely  the  same  condition  is  illustrated  in  the  Iguanodon,  and 
better  still  by  reason  of  the  great  bulk  of  the  creature  and  of  the 
heavy  load  which  falls  to  be  supported  by  the  great  cantilever  and 
by  the  hind-legs  which  form  its  piers.  The  long  heavy  body  and 
neck  require  a  balance-weight  (as  in  the  kangaroo)  in  the  form*  of 
a  long  heavy  tail;  and  the  double-armed  cantilever,  so  constituted, 
shews  a  beautiful  parabolic  curvature  in  the  graded  heights  of  the 
whole  series  of  vertebral  spines,  which  rise  to  a  maximum  over  the 
haunches  and  die  away  slowly  towards  the  neck  and  towards  the 
tip  of  the  tail. 

(5)  In  the  case  of  some  of  the  great  American  fossil  reptiles  such 
as  Diplodocus,  it  has  always  been  a  more  or  less  disputed  question 
whether  or  not  they  assumed,  like  Iguanodon,  an  erect,  bipedal 
attitude.  In  all  of  them  we  see  an  elongated  pelvis,  and,  in  still 
more  marked  degree,  we  see  elevated  spinous  processes  of  the 
vertebrae  over  the  hind-limbs ;  in  all  of  them  we  have  a  long  heavy 
tail,  and  in  most  of  them  we  have  a  marked  reduction  in  size  and 
weight  both  of  the  fore-limb  and  of  the  head  itself.  The  great  size 
of  these  animals  is  not  of  itself  a  proof  against  the  erect  attitude; 
because  it  might  well  have  been  accompanied  by  an  aquatic  or 
partially  submerged  habitat,  and  the  crushing  stress  of  the  creature's 
huge  bulk  proportionately  relieved.  But  we  must  consider  each 
such  case  in  the  whole  light  of  its  own  evidence ;  and  it  is  easy  to 
see  that,  just  as  the  quadrupedal  mammal  may  carry  the  greater 
part  but  not  all  of  its  w^eight  upon  its  fore-limbs,  so  a  heavy-tailed 
reptile  may  carry  the  greater  part  upon  its  hind-limbs,  without 


1006  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

this  process  going  so  far  as  to  relieve  its  fore-limbs  of  all  weight 
whatsoever.  This  would  seem  to  be  the  case  in  such  a  form  as 
Diplodocus,  ■  and  also,  in  Stegosaurus,  whose  restoration  by  Marsh 
is  doubtless  substantially  correct*.  The  fore-limbs,  though  com- 
paratively small,  are  obviously  fashioned  for  support,  but  the  weight 
which  they  have  to  tarry  is  far  less  than  that  which  the  hind-limbs 
bear.  The  head  is  small  and  the  neck  short,  while  on  the  other  hand 
the  hind-quarters  and  the  tail  are  big  and  massive.  The  backbone 
bends  into  a  great  double-armed  cantilever,  culminating  over  the 
pelvis  and  the  hind-limbs,  and  here  furnished  with  its  highest  and 
strongest  spines  to  separate  the  tension-member  from  the  com- 


Fig.  481.     Diagram  of  Stegosaurus.  * 

pression-member  of  the  girder.  The  fore-legs  form  a  secondary 
supporting  pier  to  this  great  continuous  cantilever,  the  greater  part 
of  whose  weight  is  poised  upon  the  hind-limbs  alone. 

(6)  In  the  slender  body  of  a  weasel,  neither  head  nor  tail  is  such 
as  to  form  an  efficient  cantilever;  and  though  the  lithe  body  is 
arched  in  active  exercise,  our  parallel  of  the  bridge  no  longer  works 
well.  What  else  to  compare  it  with  is  far  from  clear;  but  the 
mechanism  has  some  resemblance  (perhaps)  to  an  elastic  spring. 
Animals  of  this  habit  of  body  are  all  small;  their  bodily  weight 
is  a  light  burden,  and  gravity  becomes  an  ineffectual  force, 

*  This  pose  of  Diplodocus,  and  of  other  Sauropodous  reptiles,  has  been  much 
discussed.  Cf.  {int.  al.)  0.  Abel,  Ahh.  k.  k.  zool.  hot.  Ges.  Wien,  v,  1909-10  (60  pp.); 
Tornier,  SB.  Ges.  Naturf.  Fr.  Berlin,  1909,  pp.  193-209;  0.  P.  Hay,  Amer.  Nat. 
Oct.  1908;  Tr.  Wash.  Acad.  Sci.  xlii,  pp.  1-25,  1910;  Holland,  Amer.  Nat.  May 
1910,  pp.  259-283;  Matthew,  ibid.  pp.  547-560;  C.  W.  Gilmore  [Restoration  of 
Stegosaurus),  Pr.  U.S.  Nat.  Miiseum,  1915. 


XVI]  IN  AQUATIC  ANIMALS  1007 

(7)  An  abnormal  and  very  curious  case  is  that  of  the  sloth,  which 
hangs  by  hooked  hands  and  feet,  head  downwards,  from  high 
branches  in  the  Brazilian  forest.  The  vertebrae  are  unusually 
numerous,  they  are  all  much  alike  one  to  another,  and  (as  we  might 
well  suppose)  the  whole  pensile  chain  of  vertebrae  hangs  in  what 
closely  approximates  to  a  catenary  curve*. 

(8)  We  find  a  highly  important  corollary  in  the  case  of  aquatic 
animals.  For  here  the  eifect  of  gravity  is  neutralised;  we  have 
neither  piers  nor  cantilevers;  and  we  find  accordingly  in  all  aquatic 
mammals  of  whatsoever  group — whales,  seals  or  sea-cows — that 
the  high  arched  vertebral  spines  over  the  withers,  or  corresponding 
structures  over  the  hind-limbs,  have  both  entirely  disappeared. 

But  in  the  whale  or  dolphin  (and  not  less  so  in  the  aquatic  bird), 
stiffness  must  be  ensured  in  order  to  enable  the  muscles  to  act  against 
the  resistance  of  the  water  in  the  act  of  swimming;  and  accordingly 
Nature  must  provide  against  bending-moments  irrespective  of 
gravity.  In  the  dolphin,  at  any  rate  as  regards  its  tail-end,  the 
conditions  will  be  not  very  different  from  those  of  a  column  or 
beam  with  fixed  ends,  in  which,  under  deflection,  there  will  be  two 
points  of  contrary  flexure,  as  at  C,  D,  in  Fig.  482. 

Here,  between  C  and  D  we  have  a  varying  bending-moment, 
represented  by  a  continuous  curve  with  its  maximal  elevation  mid- 
way between  the  points  of  inflection. 
And  correspondingly,  in  our  dolphin, 
we  have  a  continuous  series  of  high 
dorsal  spines,  rising  to  a  maximum 
about  the   middle  of  the  animal's 

body,  and  falling  to  nil  at  some  distance  from  the  end  of  the  tail.  It 
is  their  business  (as  usual)  to  keep  the  tension-member,  represented  by 
the  strong  supraspinous  hgaments,  wide  apart  from  the  compression- 
member,  which  is  as  usual  represented  by  the  backbone  itself.  But 
in  our  diagram  we  see  that  on  the  farther  side  of  C  and  D  we  have  a 
negative  curve  of  bending-moments,  or  bending-moments  in  a  contrary 
direction.     Without  enquiring  how  these  stresses  are  precisely  met 

♦  A  hmvy  cord,  or  a  cord  carrying  equal  weights  for  equal  distances  along  its 
line,  hangs  in  a  catenary:  imagine  it  frozen  and  inverted,  and  we  have  an  arch, 
carrying  Ihe  same  sort  of  load,  and  under  compression  only.  On  the  other  hand, 
a  flexible  cable  (itself  of  neghgible  weight),  carrying  a  uniform  load  along  the  line 
of  its  horizontal  projection,  hangs  in  the  form  of  a  parabola. 


1008  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

towards  the  dolphin's  head  (where  the  coalesced  cervical  vertebrae 
suggest  themselves  as  a  partial  explanation),  we  see  at  once  that 
towards  the  tail  they  are  met  by  the  strong  series  of  chevron-bones, 
which  in  the  caudal  region,  where  tall  dorsal  spines  are  no  longer 
needed,  take  their  place  below  the  vertebrae,  in  precise  correspondence 
with  the  bending-moment  diagram.  In  many  cases  other  than  these 
aquatic  ones,  when  we  have  to  deal  with  animals  with  long  and 
heavy  tails  (like  the  Iguanodon  and  the  kangaroo  of  which  we  have 
already  spoken),  w^e  are  apt  to  meet  with  similar,  though  usually 
shorter  chevron-bones;  and  in  all  these  cases  we  may  see  without 
difficulty  that  a  negative  bending-moment  in  the  vertical  direction 
has  to  be  resisted  or  controlled. 

In  the  dolphin  we  may  find  an  illustration  of  the  fact  that  not 
only  is  it  necessary  to  provide  for  rigidity  in  the  vertical  direction 
but  often  also  in  the  horizontal,  where  a  tendency  to  bending  must  be 
resisted  on  either  side.  This  function  is  effected  in  part  by  the  ribs 
with  their  associated  muscles,  but  they  extend  but  a  little  way  and 

their  efficacy  for  this  purpose  can 
be  but  small.  We  have,  however, 
behind  the  region  of  the  ribs  and 
on  either  side  of  the  backbone 
a  strong  series  of  elongated  and 
flattened  transverse  processes, 
forming  a  web  for  the  support 
of  a  tension-member  in  the  usual 
form  of  Hgament,  and  so  playing 
a  part  precisely  analogous  to  that 
performed  by  the  dorsal  spines  in 
the  same  animal.  In  an  ordinary 
fish,  such  as  a  cod  or  a  haddock, 
we  see  precisely  the  same  thing: 
the  backbone  is  stiffened  by  the  indispensable  help  of  its  three  series 
of  ligament-connected  processes,  the  dorsal  and  the  two  transverse 
series ;  but  there  are  no  such  stiffeners  in  the  eel.  When  we  come  to 
the  region  of  the  tail,  where  rigidity  gives  place  to  lateral  flexibility, 
the  three  stiffeners  give  place  to  two — the  dorsal  and  haemal  s|)ines 
of  the  caudal  vertebrae.  And  here  we  see  that  the  three  series  of 
processes,  or  struts,  tend  (when  all  three  are  present)  to  be  arranged 


Fig.  483.     a,  dorsal  and  b,  caudal 
vertebrae  of  haddock. 


XVI]  ON  STRENGTH  AND  FLEXIBILITY  1009 

well-nigh  at  equal  angles,  of  120°,  with  one  another,  giving  the 
greatest  and  most  uniform  strength  of  which  such  a  system  is 
capable.  On  the  other  hand,  in  a  flat  fish,  such  as  a  plaice,  where 
from  the  natural  mode  of  progression  it  is  necessary  that  the  back- 
bone should  be  flexible  in  one  direction  while  stiffened  in  another, 
we  find  the  whole  outline  of  the  fish  comparable  to  that  of  a  double 
bowstring  girder,  the  compression-member  being  (as  usual)  the 
backbone  itself,  the  tension-member  on  either  side  being  constituted 
by  the  interspinous  ligaments  and  muscles,  while  the  web  or  filUng 
is  very  beautifully  represented  by  the  long  and  evenly  graded  neural 
and  haema)  spines,  which  spring  symmetrically  up  and  down  from 
each  individual  vertebra. 

In  the  skeleton  of  the  flat  fishes,  the  web  of  the  otherwise  perfect 
parabolic  girder  has  to  be  cut  away  and  encroached  on  to  make  room' 
for  the  viscera.  When  the  body  is  long  and  the  vertebrae  many, 
as  in  the  sole,  the  space  required  is  small  compared  with  the  length 
of  the  girder,  and  the  strength  of  the  latter  is  not  much  impaired. 
In  the  shorter,  rounder  kinds  with  fewer  vertebrae,  like  the  turbot, 
the  visceral  cavity  is  large  compared  with  the  length  of  the  fish, 
and  its  presence  would  seem  to  weaken  the  girder  very  seriously. 
But  Nature  repairs  the  breach  by  framing  in  the  hinder  part  of  the 
space  with  a  strong  curved  bracket  or  angle-iron,  which  takes  the 
place  very  efl5.ciently  of  the  bony  struts  which  have  been  cut  away. 

The  main  result  at  which  we  have  now  arrived,  in  regard  to  the 
construction  of  the  vertebral  column  and  its  associated  parts,  is 
that  we  may  look  upon  it  as  a  certain  type  of  girder,  whose  depth 
is  everywhere  very  nearly  proportional  to  the  height  of  the  corre- 
sponding ordinate  in  the  diagram  of  moments:  just  as  it  is  in  a  girder 
designed  by  a  modern  engineer.  In  short,  after  the  nineteenth  or 
twentieth  century  engineer  has  done  his  best  in  framing  the  design 
of  a  big  cantilever,  he  may  find  that  some  of  his  best  ideas  had,  so 
to  speak,  been  anticipated  ages  again  the  fabric  of  the  great  saurians 
and  the  larger  mammals. 

But  it  is  possible  that  the  modern  engineer  might  4)e  disposed  to 
criticise  the  skeleton  girder  at  two  or  three  points ;  and  in  particular 
he  might  think  the  girder,  as  we  see  it  for  instance  in  Diplodocus  or 
Stegosaurus,  not  deep  enough  for  carrying  the  animal's  enormous 


1010  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

weight  of  some  twenty  tons.  If  we  adopt  a  much  greater  depth 
(or  ratio  of  depth  to  length)  as  in  the  modern  cantilever,  we  shall 
greatly  increase  the  strength  of  the  structure ;  but  at  the  same  time 
we  should  greatly  increase  its  rigidity,  and  this  is  precisely  what,  in 
the  circumstances  of  the  case,  it  would  seem  that  Nature  is  bound 
to  avoid.  We  need  not  suppose  that  the  great  saurian  was  by  any 
means  active  and  limber;  but  a  certain  amount  of  activity  and 
flexibility  he  was  bound  to  have,  and  in  a  thousand  ways  he  would 
find  the  need  of  a  backbone  that  should  hQ  flexible  as  well  as  strong. 
Now  this  opens  up  a  new  aspect  of  the  matter  and  is  the  beginning 
of  a  long,  long  story,  for  in  every  direction  this  double  requirement 
of  strength  and  flexibility  imposes  new  conditions  upon  the  design. 
To  represent  all  the  porrelated  quantities  we  should  have  to  construct 
not  only  a  diagram  of  moments  but  also  a  diagram  of  elastic 
deflection  and  its  so-called  ''curvature";  and  the  engineer  would 
want  to  know  something  more  about  the  material  of  the  ligamentous 
tension-member — its  flexibility,  its  modulus  of  elasticity  in  direct 
tension,  its  elastic  limit,  and  its  safe  working  stress. 

In  various  ways  our  structural  problem  is  beset  by  "limiting 
conditions."  Not  only  must  rigidity  be  associated  with  flexibility, 
but  also  stability  must  be  ensured  in  various  positions  and  attitudes ; 
and  the  primary  function  of  support  or  weight-carrying  must  be 
combined  with  the  provision  of  points  d'appui  iot  the  muscles  con- 
cerned in  locomotion.  We  canr^ot  hope  to  arrive  at  a  numerical 
or  quantitative  solution  of  this  complicate  problem,  but  we  have 
found  it  possible  to  trace  it  out  in  part  towards  a  qualitative  solution. 
And  speaking  broadly  we  may  certainly  say  that  in  each  case  the 
problem  has  been  solved  by  Nature  herself,  very  much  as  she  solves 
the  difficult  problems  of  minimal  areas  in  a  system  of  soap-bubbles ; 
so  that  each  animal  is  fitted  with  a  backbone  adapted  to  his  own 
individual  needs,  or  (in  other  words)  corresponding  to  the  mean 
resultant  of  the  many  stresses  to  which  as  a  mechanical  system  it 
is  exposed. 

The  mechanical  construction  of  a  bird  is  a  more  elaborate  aff'air 
than  a  quadruped's,  inasmuch  as  it  has  a  double  part  to  play,  the 
bird's  whole  weight  being  borne  now  by  its  legs  and  now  by  its  wings. 
As  it  stands  on  the  ground  our  bird  is  a  balanced  cantilever,  carried 


XVI 


ON  STRENGTH  AND  FLEXIBILITY  1011 


on  two  legs  as  on  a  pier,  the  cantilever  being  constituted  by  the 
pelvic  bones,  drawn  out  fore  and  aft  and  firmly  welded  to  a  long 
stretch  of  vertebral  column.  The  centre  of  gravity  is  kept  in  a  line 
passing  through  the  acetabulum^,  and  the  long  toes  help  to  preserve 
an  unstable  but  well-ad j  usted  equilibrium .  One  arm  of  the  cantilever 
carries  head,  neck  and  wings,  the  other,  the  shorter  arm,  carries  the 
abdomen;  but  the  whole  weight  of  the  viscera  hangs  in  the  abdomen 
as  in  a  bag,  and  on  the  other  hand  head  and  neck  are  kept  small  and 
light,  and  their  purchase  on  the  fulcrum  is  under  constant  modifica- 
tion and  control.     A  stork  or  a  heron  is  continually  balancing  itself; 


Fig.  484.     Pelvis  oi  Apteryx.     The  line  AB  is  vertical,  or  nearly  so, 
in  the  standing  posture  of  the  bird. 


as  the  beak  is  thrust  forward  a  leg  stretches  back,  as  the  bird  walks 
along  its  whole  body  sways  in  keeping.  No  less  elegant  is  the 
perfect  balance  of  the  same  birds  at  rest — the  heron  standing  on 
one  leg,  even  on  a  tree  top,  the  flamingo  also  on  one  long  leg, 
with  its  neck  close  coiled  and  its  head  tucked  amongst  the 
feathers. 

The  approximately  parabolic  form  of  the  great  pelvic  cantilever 
is  best  seen  in  the  ostrich  and  other  running  birds,  but  more 
commonly  the  strength  of  the  cantilever  is  got  in  other  ways. 
Usually,  as  in  the  fowl,  it  consists  of  a  thin  shell  of  bone  curved 
over  like  the  bonnet  of  a  motor  car  and  stiffened,  or  "cambered,"  by 
ridges  converging  on  the  acetabulum.  A  doubled  sheet  of  paper, 
cut  roughly  to  the  shape  of  the  pelvis  and  then  pinched  up  into  folds 


1012  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch 

on  either  side,  as  in  Fig.  485,  will  serve  as  a  model  of  the  skeletal 
cantilever  and  shew  how  its  limp  surface  is  stiffened  by  the 
folds. 

Save  in  the  ostrich,  and  a  few  other  flightless  birds,  the  breast-bone 
or  sternum  is  a  broad,  flat  bone,  produced  into  a  deep,  descending 
ridge  or  "keel,"  Very  .firmly  fixed  to  the  sternum  on  either  side 
is  a  short  strong  bone,  the  coracoid;  attached  to  it  again,  and 
bending  backwards  over  the  ribs,  is  the  scapula;  and  at  the  junction 
of  scapula  and  coracoid  is  the  socket,  or  glenoid  cavity,  for  the 
wing.  The  clavicles,  fused  into  a  "merry-thought,"  run  from  near 
the  glenoid  cavity  to  the  front  end  of  the  keel;  in  strong-flying 
birds  they  are  stout  and  curved,  and  a  continuous  curve  sweeps 


Fig.  485.     Rough  paper  model  of  a  fowl's  pelvis. 

round  from  scapula  to  sternal  keel.  The  keel  is  commonly  explained 
as  necessary  to  give  space  enough  for  the  attachment  of  the  muscles 
of  flight,  but  this  explanation  is  inadequate,  even  untrue;  for  one 
thing,  the  great  pectoral  muscle  springs  from  the  edge,  not  from 
the  broad  surface  of  the  keel.  The  keel  is  essentially  a  flange,  and, 
as  in  a  piece  of  T  iron,  adds  immensely  to  the  strength  and  stiffness 
of  the  construction*;  that  it  tends  to  give  the  fibres  of  the  muscle 
more  stretch  and  play,  and  a  straighter  pull  on  the  arm-bone  to 
which  they  run,  is  a  secondary  advantage.  Strong  as  they  are,  these 
bones  are  exquisitely  light  and  thin.  A  great  frigate-bird,  with  a  7-foot 
span  of  wings,  weighs  a  little  over  a  couple  of  pounds,  and  all  its 
bones  weigh  about  four  ounces.  The  bones  weigh  less  than  the 
feathers  |. 

♦  T  irons,  if  I  am  not  mistaken,  were  among  the  many  inventions  of  Robert 
Stephenson,  in  his  construction  of  the  Menai  tubular  bridge  a  hundred  years  ago. 
t  Cf.  R.  C.  Murphy,  Natural  History,  Oct.  1939. 


XVI]  ON  STRENGTH  AND  FLEXIBILITY  1013 

While  the  bird  stands/on  the  ground  its  backbone  is,  as  in  ourselves, 
its  skeletal  axis,  and  it,  including  the  great  cantilever  associated  with 


Fig.  485.  a,  Sternum  and  shoulder-girdle  of  a  skua  gull,  b,  section  of  do.,  through 
the  hne  AB.  St,  sternum;  Ca,  its  carina  or  keel;  Co,  coracoid  bone;  Sc, 
scapula;  F,  merry-thought  or  furcula. 

it,  carries,  and  transmit^  to  the  legs,  the  whole  weight  of  the  body. 
But  as  soon  as  a  bird  spreads  its  wings  and  rests  upon  the  air,  legs, 


Fig.  487.     A  flying  bird.     Sternum,  shoulder -girdle  and  wings  combine  to 
support  the  body;  and  all  the  rest  lies  as  a  dead  weight  thereon. 

backbone,  cantilever  and  all  become  merely  so  much  weight  to  be 
carried;  and  the  whole  rests,  as  on  a  floor,  on  the  strong,  stiff 
platform  made  of  sternum  and  shoulder-girdle,  which  the  wings  (so 


1014  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

to  speak)  take  hold  of  and  support,  and  which  is  now,  mechanically 
speaking,  the  axis  of  the  body.  The  bird  has  two  points  of  suspension, 
which  it  uses  alternately:  the  one  through  the  two  acetabula,  the 
other  through  the  glenoid  cavities  and  the  outstretched  wings. 
Glenoid  cavity  and  acetabulum  are  but  a  little  way  apart,  and  the 
bird  swings  its  weight  over  from  one  to  the  other  easily  and  smoothly. 
At  first  sight  it  seems  a  curious  feature  of  the  bird's  skeleton  that 
breast-bone,  shoulder-girdle,  wings  and  all  are  but  very  slightly 
attached  to  the  rest  of  the  body,  and  to  what  we  look  on,  usually, 
as  its  main  axis  of  support ;  the  only  skeletal  attachment  is  by  the 
framework  of  the  ribs,  and  these  are  sUght  and  slender.  The  fact 
is  that  the  two  skeletal  axes,  the  backbone  and  the  breast-bone,  have 
their  separate  and  independent  roles,  and  each  is  but  loosely  connected 
with  the  other. 


Fig.  488.     Diagram  of  a  continuous  girder. 


The  curvature  of  the  bird's  neck  is  very  beautiful:  one  curve 
leads  on  to  another;  and  indeed  the  bird's  whole  axial  skeleton, 
from  head  to  tail,  is  one  even  and  continuous  curve.  Where  a 
bridge  crosses  the  gap  between  two  piers,  it  sags  as  the  load  passes 
over;  where  successive  girders  cross  successive  gaps,  each  sags  in 
its  turn  under  the  travelHng  load.  But  suppose  one  continuous 
girder  to  cross  two  gaps;  it  bends  in  a  more  complicated  way,  and 
one  half  tends  to  bend  up  while  the  other  is  sagging  down.  We 
cannot  analyse  the  whole  field  of  force  to  which  the  bird  is  subject, 
but  we  reahse  that  it  is  a  continuous  field,  in  which  what  the  engineer 
calls  a  "continuous  girder"  has  its  great  part  to  play.  The 
continuous  girder  is  apt  to  sag  and  bend  and  sway  in  an  erratic 
fashion  unless  its  ends  be  firm  and  secure,  and  the  bird's  head  must, 
of  necessity,  be  under  some  analogous  control;  the  semicircular 
canals  are  the  potent  factors  in  equilibrmm,  and  the  bird  "keeps 


XVI]  ON  STRENGTH  AND  FLEXIBILITY  1015 

a  level  head  "  by  their  help  and  guidance.  Then,  between  the  level 
of  the  skull  and  the  level  of  the  great  pelvic  cantilever  a  continuous 
field  of  force  governs  and  defines  the  S-shaped  curvature.  Man's 
vertebral  column  shews,  mutatis  mutandis,  the  same  phenomenon 
of  continuous  but  alternating  curvature.  The  dorsal  region  is,  of 
necessity,  concave  towards  the  cavity  of  the  chest,  and  as  a  simple 
consequence  the  cervical  and  lumbar  regions  curve  the  other  way. 
The  typically  aquatic  birds,  such  as  swim  under  water  as 
penguins  and  divers  do,  have  characteristic  features  and  adaptations 
of  their  own.  Just  as  the  cantilever  girder  becomes  obsolete  in  the 
aquatic  mammal  so  does  it  tend  to  weaken  and  disappear  in  the 
aquatic  bird.  There  is  a  marked  contrast  between  the  high-arched 
strongly  built  pelvis  in  the  ostrich  or  the  hen,  and  the  long,  thin, 
comparatively  straight  and  apparently  weakly  bone  which  represents 
it  in  a  diver,  a  grebe  or  a  penguin.  Wings  large  enough  foi;  air 
would  be  an  obstruction  under  water,  and  small  wings  are  enough ; 
for  they  have  to  produce  thrust  only,  not  Uft,  and  the  former  is 
but  a  small  fraction  of  the  latter  load.  The  feet  also  are  now  mainly 
concerned  with  the  same  forward  thrust,  and  we  begin  to  see  how 
the  long  narrow  pelvis  gives  just  the  point  d^appui  which  that 
thrust  requires. 

The  woodcock,  as  ornithologists  are  aware,  shews  us  an  osteological  paradox, 
which  is  commonly  described  by  saying  that  this  bird's  ear  is  in  front  of  its 
eye !  If  we  hold  a  woodcock's  skull  level,  beak  and  all,  this  indeed  seems  to 
be  the  case,  but  no  woodcock  does  so.  Standing  or  flying,  the  woodcock 
holds  its  beak  pointing  downwards,  and  its  skull  is  then  level,  like  that  of 
other  birds ;  in  other  words,  its  beak  is  not  in  a  line  with  the  basi-cranial  axis, 
as  a  guillemot's  is,  but  bends  sharply  downwards.  When  the  axis  of  the  skull 
is  horizontal,  the  beak  points  downwards  at  an  angle  of  nearly  60°,  and  the 
auditory  aperture  is  then  as  much  behind  the  eye  as  in  other  birds. 

There  is  a  certain  other  principle  much  to  the  fore  in  the  con- 
struction of  the  skeleton,  well  known  to  the  designer  of  a  hydroplane 
or  "flying  boat,"  and  not  wholly  neglected  by  the  bridge-building 
engineer;  it  is  the  principle  of  non-rigid,  flexible  or  elastic  stability*. 
A  homely  comparison  between  a  basket  and  a  tin-can  tells  us  in 
a  moment  what  it  means,  and  shews  us  some  at  least  of  its  peculiar 
advantages.    This  method  of  construction  helps  to  distribute  the  load, 

*  India-rubber  has  great  elastic  stability.  It  is  not  compressible,  but  is  almost  as 
incbmpressible  as  water  itself,  as  J.  D.  Forbes  discovered  a  century  ago. 


1016  ON  FORM  AND  MECHANICAL  EFFICIENCY  [en. 

bridges  over  points  or  areas  where  pressure  might  be  unduly  con- 
centrated or  confined,  adapts  itself  to  a  sudden  impact  or  concentrated 
stress,  helps  to  lessen  or  to  guard  against  shock,  and  imparts  to  the 
whole  structure  a  quality  which  we  may  call,  for  short,  resiliency. 

The  engineer  finds  it  easiest  of  attainment  when  his  principal 
members  are  in  tension;  hence  elastic  movement  and  resilience  are 
apt  to  be  conspicuous  in  a  suspension-bridge.  One  way  and  another, 
resihence  shews  to  perfection  in  a  bird.  The  S-shaped  curve  of  the 
neck  carrying  the  Hght  weight  of  the  head,  the  zig-zag  flexures  of 
the  legs  bearing  the  balanced  burden  of  the  body,  the  supple  basket 


Fig.  489..   A  woodcock's  skull,  in  (or  nearly  in)  the  natural  attitude.     A,  B,  the 
baMS  cranii;  E,  auditory  meatus;  0,  orbit;  Q,  quadrate  bone. 

of  the  ribs,  each  rib  in  two  halves  one  flexed  on  the  other,  all  these 
are  such  as  to  make  the  whole  framework  act  like  an  elastic  spring, 
absorbing  every  shock  as  the  bird  lights  on  or  rises  from  the  ground. 
Bird,  beast  and  man  exhibit  this  resilience,  each  in  its  degree; 
a  springy  step  is  part  of  the  joy  of  youth,  and  its  loss  is  one  of  the 
first  infirmities  of  age. 

Nature's  engineering  is  marvellous  in  our  eyes,  and  our  finest 
work  is  narrow  in  scope  and  clumsy  in  execution  compared  to  her 
construction  and  design.  But  following  her  example,  wittingly  or 
unwittingly,  our  own  problems  evolve  and  our  ambitions  enlarge 
towards   the   conception   of  an   "organised   structure."     In   such 


XVI]  ON  THE  SKELETON  AS  A  WHOLE  1017 

triumphs  of  modern  mechanism  as  a  torpedo,  a  racing  aeroplane, 
a  high-speed  rail  way- train,  the  whole  construction  is  knit  together 
in  a  new  way.  It  finds  its  streamlined  outline  in  what  seems  to 
be  a  simple  and  natural  way;  it  is  solid  and  robust,  it  is  graceful 
as  well  as  strong;  it  is  no  longer  a  bundle  of  parts,  it  has  become 
an  organic  whole :  its  likeness,  even  its  outward  likeness,  to  a  living 
organism  has  becorne  patent  and  clear. 

Throughout  this  short  discussion  of  the  principles  of  construction, 
we  see  the  same  general  principles  at  work  in  the  skeleton  as  a  whole 
as  we  recognised  in  the  plan  and  construction  of  an  individual 
bone.  That  is  to  say,  we  see  a  tendency  for  material  to  be  laid 
down  just  in  the  lines  of  stress,  and  so  to  evade  thereby  the 
distortions  and  disruptions  due  to  shear.  In  these  phenomena 
there  lies  a  definite  law  of  growth,  whatever  its  ultimate  expression 
or  explanation  may  come  to  be.  Let  us  not  press  either  argument 
or  hypothesis  too  far:  but  be  content  to  see  that  skeletal  form,  as 
brought  about  by  growth,  is  to  a  very  large  extent  determined  by 
mechanical  considerations,  and  tends  to  manifest  itself  as  a  diagram, 
or  reflected  image,  of  mechanical  stress.  If  we  fail,  owing  to  the 
immense  complexity  of  the  case,  to  unravel  all  the  mathematical 
principles  involved  in  the  construction  of  the  skeleton,  we  yet  gain 
something,  and  not  a  little,  by  applying  this  method  to  the  famihar 
objects  of  anatomical  study:  ohvia  conspicimus,  nubem  pellente 
tnathesi*. 

Before  we  leave  this  subject  of  mechanical  adaptation,  let  us 
dwell  once  more  for  a  moment  upon  the  considerations  which  arise 
from  our  conception  of  a  field  of  force,  or  field  of  stress,  in  which 
tension  and  compression  (for  instance)  are  inevitably  combined,  and 
are  met  by  the  materials  naturally  fitted  to  resist  them.  It  has 
been  remarked  over  and  over  again  how  harmoniously  the  whole 
organism  hangs  together,  and  how  throughout  its  fabric  one  part 
is  related  and  fitted  to  another  in  strictly  functional  correlation. 
But  this  conception,  though  never  denied,  is  sometimes  apt  to  be 
forgotten  in  the  course  of  that  process  of  more  and  more  minute 

*  The  motto  was  Macquorn  Rankine's,  \vl  1857;  cf.  Trans.  R.S.K.  xxvi,  p.  715, 
1872. 


1018  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

dnalysis  by  which,  for  simpUcity's  sake,  we  seek  to  unravel  the 
intricacies  of  a  complex  organism. 

As  we  analyse  a  thing  into  its  parts  or  into  its  properties,  we  tend 
to  magnify  these,  to  exaggerate  their  apparent  independence,  and 
to  hide  from  ourselves  (at  least  for  a  time)  the  essential  integrity 
and  individuality  of  the  composite  wh^le.  We  divide  the  body  into 
its  organs,  the  skeleton  into  its  bones,  as  in  very  much  the  same 
fashion  we  make  a  subjective  analysis  of  the  mind,  according  to 
the  teachings  of  psychology,  into  component  factors :  but  we  know 
very  well  that  judgment  and  knowledge,  courage  or  gentleness, 
love  or  fear,  have  no  separate  existence,  but  are  somehow  mere 
manifestations,  or  imaginary  coefficients,  of  a  most  complex  integral. 
And  likewise,  as  biologists,  we  may  go  so  far  as  to  say  that  even 
the  bones  themselves  are  only  in  a  limited  and  even  a  deceptive 
sense,  separate  and  individual  things.  The  skeleton  begins  as  a 
continuum,  and  a  continuum  it  remains  all  life  long.  The  things 
that  link  bone  with  bone,  cartilage,  ligaments,  membranes,  are 
fashioned  out  of  the  same  primordial  tissue,  and  come  into  being 
pari  passu  with  the  bones  themselves.  The  entire  fabric  has  its 
soft  parts  and  its  hard,  its  rigid  and  its  flexible  parts;  but  until  we 
disrupt  and  dismember  its  bony,  gristly  and  fibrous  parts  one  from 
another,  it  exists  simply  as  a  "skeleton,"  as  one  integral  and 
individual  whole. 

A  bridge  was  once  upon  a  time  a  loose  heap  of  pillars  and  rods 
and  rivets  of  steel.  But  the  identity  of  these  is  lost,  just  as  if  they 
were  fused  into  a  solid  mass,  when  once  the  bridge  is  built;  their 
separate  functions  are  only  to  be  recognised  and  analysed  in  so  far 
as  we  can  analyse  the  stresses,  the  tensions  and  the  pressures,  which 
affect  this  part  of  the  structure  or  that;  and  these  forces  are  not 
themselves  separate  entities,  but  are  the  resultants  of  an  analysis 
of  the  whole  field  of  force.  Moreover  wJien  the  bridge  is  broken  it 
is  no  longer  a  bridge,  and  all  its  strength  is  gone.  So  is  it  precisely 
with  the  skeleton.  In  it  is  reflected  a  field  of  force:  and  keeping 
pace,  as  it  were,  in  action  and  interaction  with  this  field  of  force,  the 
whole  skeleton  and  every  part  thereof,  down  to  the  minute  intrinsic 
structure  of  the  bones  themselves,  is  related  in  form  and  in  position 
to  the  lines  of  force,  to  the  resistances  it  has  to  encounter;  for  by 
one  of  the  mysteries  of  biology,  resistance  begets  resistance,  and 


XVI]  THE  PROBLEM  OF  PHYLOGENY  1019 

where  pressure  falls  there  growth  springs  up  in  strength  to  meet  it. 
And,  pursuing  the  same  train  of  thought,  we  see  that  all  this  is  true 
not  of  the  skeleton  alone  but  of  the  whole  fabric  of  the  body.  Muscle 
and  bone,  for  instance,  are  inseparably  associated  and  connected; 
they  are  moulded  one  with  another ;  they  come  into  being  together, 
and  act  and  react  together*.  We  may  study  them  apart,  but  it 
is  as  a  concession  to  our  weakness  and  to  the  narrow  outlook  of 
our  minds.  We  see,  dimly  perhaps  but  yet  with  all  the  assurance 
of  conviction,  that  between  muscle  and  bone  there  can  be  no  change 
in  the  one  but  it  is  correlated  with  changes  in  the  other;  that 
through  and  through  they  are  linked  in  indissoluble  association; 
that  they  are  only  separate  entities  m  this  limited  and  suboi'dinate 
sense,  that  they  are  parts  of  a  whole  which,  when  it  loses  its  com- 
posite integrity,  ceases  to  exist. 

The  biologist,  as  well  as  the  philosopher,  learns  to  recognise  that 
the  whole  is  not  merely  the  sum  of  its  parts.  It  is  this,  and  much 
more  than  this.  For  it  is  not  a  bundle  of  parts  but  an  organisation 
of  parts,  of  parts  in  their  mutual  arrangement,  fitting  one  with 
another,  in  what  Aristotle  calls  "a  single  and  indivisible  principle 
of  unity  " ;  and  this  is  no  merely  metaphysical  conception,  but  is  in 
biology  t])e  fundamental  truth  which  lies  at  the  basis  of  Geoffroy's 
(or  Goethe's)  law  of  "compensation,"  or  "balancement  of  growth," 

Nevertheless  Darwin  found  no  difficulty  in  believing  that  "natural 
selection  will  tend  in  the  long  run  to  reduce  any  part  of  the  organisa- 
tion, as  soon  as,  through  changed  habits,  it  becomes  superfluous: 
without  by  any  means  causing  some  oth^r  part  to  be  largely  developed 
in  a  corresponding  degree.  And  conversely,  that  natural  selection 
may  perfectly  well  succeed  in  largely  developing  an  organ  without 
requiring  as  a  necessary  compensation  the  reduction  of  some  ad- 
joining partf  "  This  view  has  been  developed  into  a  doctrine  of 
the  "independence  of  single  characters"  (not  to  be  confused  with 
the  germinal  "unit  characters"  of  Mendelism),  especially  by  the 
palaeontologists.  Thus  Osborn  asserts  a  "principle  of  hereditary 
correlation,"  combined  with  a  "principle  of  hereditary  separability ^ 

*  John  Hunter  was  seldom  wrong;  but  I  cannot  believe  that  he  was  right  when 
he  said  {Scientific  Works,  ed.  Owen,  i,  p.  371),  "The  bones,  in  a  mechanical  view, 
appear  to  be  the  first  that  are  to  be  considered.  We  can  study  their  shape, 
connections,  number,  uses,  etc.,  without  considering  any  other  part  of  the  body.'' 

t  Origin  of  Species,  6th  ed.  p.  118. 


1020  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

whereby  the  body  is  a  colony,  a  mosaic,  of  single  individual  and 
separable  characters*."  I  cannot  think  that  there  is  more  than 
a  very  small  element  of  truth  in  this  doctrine.  As  Kant  said,  "die 
Ursache  der  Art  der  Existenz  bei  jedem  Theile  eines  lebenden 
Korpers  ist  im  Ganzen  enthalten.'"  And,  according  to  the 'trend  or 
aspect  of  our  thought,  we  may  look  upon  the  coordinated  parts,, 
now  as  related  and  fitted  to  the  end  or  function  o/the  whole,  and  now 
as  related  to  or  resulting  from  the  physical  causes  inherent  in  the 
entire  system  of  forces  to  which  the  whole  has  been  exposed,  and 
under  whose  influence  it  has  come  into  being  f. 

In  John  Hunter's  day  the  anatomist  studied  every  bone  of  the 
skeleton  in  its  own  place,  in  order  to  discover  its  useful  purpose  and 
understand  its  mechanical  perfection.  The  morphologist  of  a  hun- 
dred years  later  preferred  to  study  an  isolated  bone  from  many 
animals,  collar-bones  or  shoulder-blades  by  themselves,  apart  from 
the  field  of  force  in  which  their  work  was  done,  in  the  search  for 
signs  of  blood-relationship  and  common  ancestry.  Truth  lies  both 
ways;  immediate  use  and  old  inheritance  are  blended  in  Nature's 
handiwork  as  in  our  own.  In  the  marble  columns  and  architraves 
of  a  Greek  temple  we  still  trace  the  timbers  of  its  wooden  prototype, 
and  see  beyond  these  the  tree-trunks  of  a  primeval  sacred  grove; 
roof  and  eaves  of  a  pagoda  recall  the  sagging  mats  which  roofed  an 

*  Amer.  Naturalist,  April,  1915,  p.  198,  etc.     Cf.  infra,  p.  1036. 

t  Drie.sch  saw  in  "Entelechy"  that  something  which  differentiates  the  whole 
from  the  sum  of  its  parts  in  the  case  of  the  organism:  "The  organism,  we  know, 
is  a  system  the  single  constituents  of  which  are  inorganic  in  themselves;  only  the 
whole  constituted  by  them  in  their  typical  order  or  arrangement  owes  its  specificity 
to  'Entelechy'"  {Gijford  Lectures,  1908,  p.  229):  and  I  think  it  could  be  shewn 
that  many  other  philosophers  have  said  precisely  the  same  thing.  So  far  as  the 
argument  goes,  I  fail  to  see  how  this  Entelechy  is  shewn  to  be  peculiarly  or 
specifically  related  to  the  living  organism.  The  conception  (at  the  bottom  of 
General  Smuts's  'Holism'')  that  the  whole  is  always  something  very  different  from 
its  parts  is  a  very  ancient  doctrine.  The  reader  will  perhaps  remember  how, 
in  another  vein,  the  theme  is  treated  by  Martinus  Scriblerus  (Huxley  quoted  it 
once,  for  his  own  ends):  "In  every  Jack  there  is  a  meat-roasting  Quality,  which 
neither  resides  in  the  fly,  nor  in  the  weight,  nor  in  any  particular  wheel  of  the  Jack, 
but  is  the  result  of  the  whole  composition;  etc.,  etc."  Indeed  it  was  at  that  very 
time,  in  the  early  eighteenth  century,  that  the  terms  organism  and  organisation  were 
coming  into  use,  to  connote  that  harmonious  combination  of  parts  "qui  conspirent 
toutes  ensembles  a  produire  cet  efFet  general  que  nous  nommons  la  vie"  (Buffon). 
Cf.  Ch.  Robin,  Recherches  sur  Torigine  et  le  sens  des  termes  organisme  et  organisation, 
Jl.  de  VAnat.  lx,  pp.  l-.'i.^,  1880. 


XVI 


THE  PROBLEM  OF  PHYLOGENY 


1021 


earlier  edifice;  Anglo-Saxon  land-tenure  influences  the  planning  of 
our  streets,  and  the  cliff-dwelling  and  the  cave-dwelling  linger  on 
in  the  construction  of  our  homes!  So  we  see  enduring  traces  of 
the  past  in  the  living  organism — landmarks  which  have  lasted  on 
through  altered  functions  and  altered  needs ;  and  yet  at  every  stage 
new  needs  are  met  and  new  functions  effectively  performed. 

When  we  consider  (for  instance)  the  several  bones  in  a  fish's 
shoulder-girdle — clavicle,  supra-clavicle,  post-clavicle,  post-temporal 
and  so  on — and  recognise  these  in  this  fish  or  that  under  countless 
minor  transformations,  we  have  something  which  is  not  only  wide- 


Fig.  490.     Skeleton  of  moonfish,  Vomer  sp.     From  L.  Agassiz. 


spread  but  is  rooted  in  antiquity,  and  whose  full  significance  seems 
beyond  our  reach.  But  take  the  skeleton  of  some  particular  fish,  a 
moonfish  or  a  John  Dory  will  do  very  well,  and  look  at  its  shoulder- 
girdle  from  the  mechanical  point  of  view.  It  is  a  deal  more  than  is 
needed  for  the  support  of  the  small,  weak  pectoral  fin;  but  another 
function,  and  its  perfect  adaptation  for  that  function,  are  not  hard  to 
see.  The  flattened  body  of  the  fish  is  built  (as  we  have  seen  also  in  the 
plaice)  on  the  plan  of  a  parabolic  girder;  but  out  of  this  girder  a 
great  gap  has  had  to  be  cut,  to  hold  the  viscera.  The  great  shoulder- 
girdle  serves  to  strengthen  and  complete  the  girder,  to  bind  its 
upper  and  lower  members  together,  and  to  compensate  for  the  part 


1022  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

which  has  been  taken  away^^  It  fulfils  this  function  by  various 
means;  by  the  way  in  which  the  two  sides  of  the  girdle  are  conjoined 
into  a  single  arch ;  by  its  strong  attachment  to  the  head,  and  again 
to  the  pelvis,  and  through  the  latter  to  the  chain  of  ossicles  which 
bound  or  constitute  the  abdominal  border  of  the  fish ;  and  a  large 
part  of  the  stress  upon  the  shoulder-girdle  proper  is  taken  up,  or 
relieved,  by  the  strong  post-clavicular  bones,  which  form  a  supple- 
mentary arch  running  downwards  from  the  clavicle  (just  where  it 
begins  to  incline  forward),  straight  to  the  ventral  border,  to  be 
firmly  attached  there  to  the  ventral  ossicles.  Similarly  we  notice 
at  the  hinder  border  of  the  abdominal  cavity,  a  strong  curved  bone 
running  from  the  anterior  part  of  the  ventral  fin  to  a  solid  attach- 
ment with  the  vertebral  column,  stiffening  the  ventral  part,  and 
helping  the  shoulder-girdle  to  restore  full  strength  to  the  girder 
after  it  had  been  reduced,  so  to  speak,  to  the  brink  of  inevitable 
collapse.  The  skull  itself  is  not  only  streamlined  with  the  rest  of  the 
body,  but  is  an  intrinsic  part  of  the  whole  engineering  construction. 
The  lines  of  stress  run  simply  and  clearly  through  the  skeleton,  and 
a  bone  can  no  longer  teach  us  its  full  and  proper  lesson  after  we  have 
taken  it  apart.  To  look  on  the  hereditary  or  evolutionary  factor 
as  tlie  guiding  principle  in  morphology  is  to  give  to  that  science 
a  one-sided  and  fallacious  simplicity*. 

It  would  seem  to  me  that  the  mechanical  principles  and  phenomena 
which  we  have  dealt  with  in  this  chapter  are  of  no  small  importance 
to  the  morphologist,  all  the  more  when  he  is  inclined  to  direct  his 
study  of  the  skeleton  exclusively  to  the  problem  of  phylogeny ;  and 
especially  when,  according  to  the  methods  of  modern  comparative 
morphology,  he  is  apt  to  take  the  skeleton  to  pieces,  and  to  draw 
from  the  comparison  of  a  series  of  scapulae,  humeri,  or  individual 
vertebrae,  conclusions  as  to  the  descent  and  relationship  of  the 
animals  to  which  they  belong. 

It  would,  I  dare  say,  be  an  exaggeration  to  see  in  every  bone 
nothing  more  than  a  resultant  of  immediate  and  direct  physical  or 
mechanical  conditions ;  for  to  do  so  would  be  to  deny  the  existence, 

*  The  extreme  evolutionary,  or  phylogenetic,  aspect  of  morphology  was  being 
questioned  even  forty  years  ago.  "Where  we  once  thought  we  detected  relation- 
ships we  now  know  we  were  often  being  misled,  and  the  old-time  supposition 
that  mere  community  of  structure  is  necessarily  an  index  of  community  of  origin 
has  gone  to  the  wall"  (G.  B.  Howes,  in  Nature,  Jan.  10,  1901). 


XVI]  THE  PROBLEM  OF  PHYLOGENY  1023 

in  this  connection,  of  a  principle  of  heredity.  And  though  I  have 
tried  throughout  this  book  to  lay  emphasis  on  the  direct  action  of 
causes  other  than  heredity,  in  short  to  circumscribe  the  employment 
of  the  latter  as  a  working  hypothesis  in  morphology,  there  can  still 
be  no  question  whatsoever  but  that  heredity  is  a  vastly  important 
as  well  as  a  mysterious  thing ;  it  is  one  of  the  great  factors  in  biology, 
however  we  may  attempt  to  figure  to  ourselves,  or  howsoever  we 
may  fail  even  to  imagine,  its  underlying  physical  explanation.  But 
I  maintain  that  it  is  no  less  an  exaggeration  if  we  tend  to  neglect 
these  direct  physical  and  mechanical  modes  of  causation  altogether, 
and  to  see  in  the  characters  of  a  bone  merely  the  results  of  variation 
and  of  heredity,  and  to  trust,  in  consequence,  to  those  characters 
as  a  sure  and  certain  and  unquestioned  guide  to  affinity  and  phylo- 
geny.  Comparative  anatomy  has  its  physiological  side,  which  filled 
men's  minds  in  John  Hunter's  day,  and  in  Owen's  day;  it  has  its 
classificatory  and  phylogenetic  aspect,  which  all  but  filled  men's 
minds  in  the  early  days  of  Darwinism;  and  we  can  lose  sight  of 
neither  aspect  without  risk  of  error  and  misconception. 

It  is  certain  that  the  question  of  phylogeny,  always  difficult, 
becomes  especially  so  in  cases  where  a  great  change  of  physical  or 
mechanical  conditions  has  come  about,  and  where  accordingly  the 
former  physical  and  physiological  constraints  are  altered  or  removed. 
The  great  depths  of  the  sea  differ  from  other  habitations  of  the 
living,  not  least  in  their  eternal  quietude.  The  fishes  which  dwell 
therein  are  quaint  and  strange;  their  huge  heads,  prodigious  jaws, 
and  long  tails  and  tentacles  are,  as  it  were,  gross  exaggerations  of 
the  common  and  conventional  forms.  We  look  in  vain  for  any 
purposeful  cause  or  physiological  explanation  of  these  enormities; 
and  are  left  under  a  vague  impression  that  life  has  been  going  on 
in  the  security  of  all  but  perfect  equilibrium,  and  that  the  resulting 
forms,  liberated  from  many  ordinary  constraints,  have  grown  with 
unusual  freedom*. 

To  discuss  these  questions  at  length  would  be  to  enter  on  a 
discussion  of  Lamarck's  philosophy  of  biology,  and  of  many  other 
things  besides.  But  let  us  take  one  single  illustration.  The  affinities 
of  the  whales  constitute,  as  will  be  readily  admitted,  a  very  hard 
problem  in  phylogenetic  classification.     We  know  now  that  the 

*  Cf.  supra,  p.  423. 


1024  ON  FORM  AND  MECHANICAL  EFFICIENCY  [ch. 

extinct  Zeuglodons  are  related  to  the  old  Creodont  carnivores,  and 
thereby  (though  distantly)  to  the  seals*;  and  it  is  supposed,  but  it 
is  by  no  means  so  certain,  that  in  turn  they  are  to  be  considered 
as  representing,  or  as  allied  to,  the  ancestors  of  the  modern  toothed 
whales  t.  The  proof  of  any  such  a  contention  becomes,  to  my 
mind,  extraordinarily  difficult  and  complicated ;  and  the  arguments 
commonly  used  in  such  cases  may  be  said  (in  Bacon's  phrase)  to 
allure,  rather  than  to  extort  assent.  Though  the  Zeuglodons  were 
aquatic  animals,  we  do  not  know,  and  we  have  no  right  to  suppose 
or  to  assume,  that  they  swam  after  the  fashion  of  a  whale  (any 
more  than  the  seal  does),  that  they  dived  like  a  whale,  or  leaped 
like  a  whale.  But  the  fact  that  the  whale  does  these  things,  and 
the  way  in  which  he  does  them,  is  reflected  in  many  parts  of  his 
skeleton — perhaps  more  or  less  in  all:  so  much  so  that  the  lines 
of  stress  which  these  actions  impose  are  the  very  plan  and  working- 
diagram  of  great  part  of  his  structure.  That  the  Zeuglodon  has 
a  scapula  Hke  that  of  a  whale  is  to  my  mind  no  necessary  argument 
that  he  is  akin  by  blood-relationship  to  a  whale:  that  his  dorsal 
vertebrae  are  very  different  from  a  whale's  is  no  conclusive  argument 
that  such  blood-relationship  is  lacking.  The  former  fact  goes  a  long 
way  to  prove  that  he  used  his  flippers  very  much  as  a  whale  does; 
the  latter  goes  still  farther  to  prove  that  his  general  movements 
and  equilibrium  in  the  water  were  totally  different.  The  whale 
may  be  descended  from  the  Carnivora,  or  might  for  that  matter, 
as  an  older  school  of  naturalists  believed,  be  descended  fropa  the 
Ungulates;  but  whether  or  no,  we  need  not  expect  to  find  in  him 
the  scapula,  the  pelvis  or  the  vertebral  column  of  the  lion  or  of  the 
cow,  for  it  would  be  physically  impossible  that  he  could  live  the  life 
he  does  with  any  one  of  them.  In  short,  when  we  hope  to  find  the 
missing  links  between  a  whale  and  his  terrestrial  ancestors,  it  must 
be  not  by  means  of  conclusions  drawn  from  a  scapula,  an  axis,  or 

*  See  {int.  al.)  my  paper  On  the  affinities  oi Zeuglodon  in  Studies  from  the  Museum 
of  University  College,  Dundee,  1889. 

t  "There  can  be  no  doubt  that  Fraas  is  correct  in  regarding  this  type  (Procetus) 
as  an  annectant  form  between  the  Zeuglodonts  and  the  Creodonta,  but.  although 
the  origin  of  the  Zeuglodonts  is  thus  made  clear,  it  still  seems  to  be  by  no  means 
so  certain  as  that  author  beheves,  that  they  may  not  themselves  be  the  ancestral 
forms  of  the  Odontoceti"  (Andrews,  Tertiary  Vertebrata  of  the  Fayum,  1906, 
p.  235). 


i 


XVI]  THE  PROBLEM  OF  PHYLOGENY  1025 

even  from  a  tooth,  but  by  the  discovery  of  forms  so  intermediate 
in  their  general  structure  as  to  indicate  an  organisation  and,  ipso 
facto,  a  mode  of  life,  intermediate  between  the  terrestrial  and  the 
Cetacean  form.  There  is  no  vahd  syllogism  to  the  effect  that  A 
has  a  flat  curved  scapula  like  a  seal's,  and  B  has  a  fl.a,t  curved 
scapula  like  a  seal's :  and  therefore  A  and  B  are  related  to  the  seals 
and  to  each  other;  it  is  merely  a  flagrant  case  of  an  "undistributed 
middle."  But  there  is  validity  in  an  argument  that  B  shews  in 
its  general  structure,  extending  over  this  bone  and  that  bone, 
resemblances  both  to  A  and  to  the  seals :  and  that  therefore  he  may 
be  presumed  to  be  related  to  both,  in  his  hereditary  habits  of  life 
and  in  actual  kinship  by  blood.  It  is  cognate  to  this  argument  that 
(as  every  palaeontologist  knows)  we  find  clues  to  affinity  more 
easily,  that  is  to  say  with  less  confusion  and  perplexity,  in  certain 
structures  than  in  others.  The  deep-seated  rhythms  of  growth 
which,  as  I  venture  to  think,  are  the  chief  basis  of  morphological 
heredity,  bring  about  similarities  of  form  which  endure  in  the 
absence  of  conflicting  forces ;  but  a  new  system  of  forces,  introduced 
by  altered  environment  and  habits,  impinging  on  those  particular 
parts  of  the  fabric  which  he  within  this  particular  field  of  force,  will 
assuredly  not  be  long  of  manifesting  itself  in  notable  and  inevitable 
modifications  of  form.  And  if  this  be  really  so,  it  will  further 
imply  that  modifications  of  form  will  tend  to  manifest  themselves, 
not  so  much  in  small  and  isolated  phenomena,  in  this  part  of  the 
fabric  or  in  that,  in  a  scapula  for  instance  or  a  humerus :  but  rather 
in  some  slow,  general,  and  more  or  less  uniform  or  graded  modifica- 
tion, spread  over  a  number  of  correlated  parts,  and  at  times 
extending  over  the  whole,  or  over  great  portions,  of  the  body. 
Whether  any  such  general  tendency  to  widespread  and  correlated 
transformation  exists,  we  shall  attempt  to  discuss  in  the  following 
chapter. 


CHAPTER  XVII 

ON  THE  THEORY  OF  TRANSFORMATIONS,  OR  THE 
COMPARISON  OF  RELATED  FORMS 

In  the  foregoing  chapters  of  this  book  we  have  attempted  to  study 

the  inter-relations  of  growth  and  form,  and  the  part  which  the 

physical  forces  play  in  this  complex  interaction;    and,  as  part  of 

the  same  enquiry,  we  have  tried  in  comparatively  simple  cases 

to   use  mathematical  methods  and  mathematical  terminology  to 

describe  and  define  the  forms  of  organisms.     We  have  learned  in  so 

doing  that  our  own  study  of  organic  form,  which  we  call  by  Goethe's 

name  of  Morphology,  is  but  a  portion  of  that  wider  Science  of  Form 

wtjch  deals  with  the  forms  assumed  by  matter  under  all  aspects 

and  conditions,  and,  in  a  still  wider  sense,  with  forms  which  are 

theoretically  im^aginable. 

The  study  of  form  may  be  descriptive  merely,  or  it  may  become 

analytical.     We  begin  by  describing  the  shape  of  an  object  in  the 

simple  words  of  common  speech :  we  end  by  defining  it  in  the  precise 

language  of  mathematics;   aild  the  one  method  tends  to  follow  the 

other  in  strict  scientific  order  and  historical  continuity.     Thus,  for 

instance,  the  form  of  the  earth,  of  a  raindrop  or  a  rainbow,  the 

shape  of  the  hanging  chain,  or  the  path  of  a  stone  thrown  up  into 

the  air,  may  all  be  described,  however  inadequately,  in  common 

words ;   but  when  we  have  learned  to  comprehend  and  to  define  the 

sphere,  the  catenary,  or  the  parabola,  we  have  made  a  wonderful 

ai^d  perhaps  a  manifold  advance.     The  mathematical  definition  of 

a  "form"  has  a  quahty  of  precision  which  was^ quite  lacking  in  our 

earlier  stage  of  mere  description;   it  is  expressed  in  few  words  or 

in  still  briefer  symbols,  and  these  words  or  symbols  are  so  pregnant 

with  meaning  that  thought  itself  is  economised;    we  are  brought 

by  means  of  it  in  touch  with  Galileo's  aphorism  (as  old  as  Plato,  as 

old  as  Pythagoras,  as  old  perhaps  as  the  wisdom  of  the  Egyptians), 

that  "the  Book  of  Nature  is  written  in  characters  of  Geometry*." 

♦  Cf.  Plutarch,  Symp.  viii,  2,  on  the  meaning  of  Plato's  aphorism  ("if  it  actually 
was  Plato's"):   ttwj  HXdrufv  iXeye  t6v  d^bv  ad  yeuifxeTpe'tv. 


CH.  XVII]     COMPARISON  OF  RELATED  FORMS         1027 

We  are  apt  to  think  of  mathematical  definitions  as  too  strict  and 
rigid  for  common  use,  but  their  rigour  is  combined  with  all  but 
endless  freedom.  The  precise  definition  of  an  ellipse  introduces  us 
to  all  the  elHpses  in  the  world;  the  definition  of  a  "conic  section" 
enlarges  our  concept,  and  a  "curve  of  higher  order"  all  the  toore 
extends  our  range  of  freedom*.  By  means  of  these  large  limitations, 
by  this  controlled  and  regulated  freedom,  we  reach  through  mathe- 
matical analysis  to  mathematical  synthesis.  We  discover  homologies 
or  identities  which  were  not  obvious  before,  and  which  our  descrip- 
tions obscured  rather  than  revealed :  as  for  instance,  when  we  learn 
that,  however  we  hold  our  chain,  or  however  we  fixe  our  bullet,  the 
contour  of  the  one  or  the  path  of  the  other  is  always  mathematically 
homologous. 

Once  more,  and  this  is  the  greatest  gain  of  all,  we  pass  quickly  and 
easily  from  the  mathematical  concept  of  form  in  its  statical  aspect  to 
form  in  its  dynanncal  relations:  we  rise  from  the  conception  of 
form  to  an  understanding  of  the  forces  which  gave  rise  to  it;  and 
in  the  representation  of  form  and  in  the  comparison  of  kindred 
forms,  we  see  in  the  one  case  a  diagram  of  forces  in  equihbrium, 
and  in  the  other  case  we  discern  the  magnitude  and  the  direction 
of  the  forces  which  have  sufficed  to  convert  the  one  form  into  the 
other.  Here,  since  a  change  of  material  form  is  only  effected  by 
the  movement  of  matter  |,  we  have  once  again  the  support  of  the 
schoolman's  and  the  philosopher's  axiom,  Ignorato  motu,  ignoratur 
Natural' 


*  So  said  Gustav  Theodor  Fechner,  the  author  of  Fechner's  Law,  a  hundred 
years  ago.  (Ueber  die  mathematische  Behandlung  organischer  Gestalten  und 
Processe,  Berichte  d.  k.  sacks.  Gesellsch.,  Maih.-phys.  CI.,  Leipzig,  1849,  pp.  50-64.) 
Fechner's  treatment  is  more  purely  mathematical  and  less  physical  in  its  scope  and 
bearing  than  ours,  and  his  paper  is  but  a  short  one,  but  the  conclusions  to  which 
he  is  led  differ  Uttle  from  our  own.  Let  me  quote  a  single  sentence  which,  together 
with  its  context,  runs  precisely  on  Aie  lines  which  we  have  followed  in  this  book : 
"So  ist  also  die  mathematische  Bestimmbarkeit  im  Gebiete  des  Organischen  ganz 
eben  so  gut  vorhanden  als  in  dem  des  Unorganischen,  und  in  letzterem  eben  solchen 
oder  aquivalenten  Beschrankungen  unterworfen  als  in  ersterem ;  und  nur  sofern  die 
unorganischen  Formen  und  das  unorganische  Geschehen  sich  einer  einfacheren 
Gesetzlichkeit  mehr  nahern  als  die  organischen,  kann  die  Approximation  im 
unorganischen  Gebiet  leichter  und  weiter  getrieben  werden  als  im  organischen. 
Dies  ware  der  ganze,  sonach  rein  relative,  Unterschied."  Here,  in  a  nutshell,  is 
the  gist  of  the  whole  matter. 

t  "We  can  move  matter,  that  is  all  we  can  do  to  it"  (Oliver  Lodge). 


1028      THE  THEORY  OF  TRANSFORMATIONS        [ch. 

There  is  yet  another  way — we  learn  it  of  Henri  Poincare — to  regard 
the  function  of  mathematics,  and  to  reahse  why  its  laws  and  its 
methods  are  hound  to  underlie  all  parts  of  physical  science.  Every 
natural  phenomenon,  however  simple,  is  really  composite,  and  every 
visible  action  and  effect  is  a  summation  of  countless  subordinate 
actions.  Here  mathematics  shews  her  pecuUar  power,  to  combine 
and  to  generahse.  The  concept  of  an  average,  the  equation  to 
a  curve,  the  description  of  a  froth  or  cellular  tissue,  all  come  within 
the  scope  of  mathematics  for  no  other  reason  than  that  they  are 
summations  of  more  elementary  principles  or  phenomena.  Growth 
and  Form  are  throughout  of  this  composite  nature;  therefore  the 
laws  of  mathematics  are  bound  to  underlie  them,  and  her  methods 
to  be  peculiarly  fitted  to  interpret  them. 

In  the  morphology  of  living  things  the  use  of  mathematical 
methods  and  symbols  has  made  slow  progress ;  and  there  are  various 
reasons  for  this  failure  to  employ  a  method  whose  advantages  are 
so  obvious  in  the  investigation  of  other  physical  forms.  To  begin 
with,  there  would  seem  to  be  a  psychological  reason,  lying  in  the 
fact  that  the  student  of  living  things  is  by  nature  and  training  an 
observer  of  concrete  objects  and  phenomena  and  the  habit  of  mind 
which  he  possesses  and  cultivates  is  alien  to  that  of  the  theoretical 
mathematician.  But  this  is  by  no  means  the  only  reason;  for  in 
the  kindred  subject  of  mineralogy,  for  instance,  crystals  were  still 
treated  in  the  days  of  Linnaeus  as  wholly  within  the  province  of 
the  naturalist,  and  were  described  by  him  after  the  simple  methods 
in  use  for  animals  and  plants:  but  as  soon  as  Haiiy  shewed  the 
application  of  mathematics  to  the  description  and  classification  of 
crystals,  his  methods  were  immediately  adopted  and  a  new  science 
came  into  being. 

A  large  part  of  the  neglect  and  suspicion  of  mathematical  methods 
in  organic  morphology  is  due  (as  we  have  partly  seen  in  our  opening 
chapter)  to  an  ingrained  and  deep-seated  belief  that  even  when  we 
seem  to  discern  a  regular  mathematical  figure  in  an  organism,  the 
sphere,  the  hexagon,  or  the  spiral  which  we  so  recognise  merely 
resembles,  but  is  never  entirely  explained  by,  its  mathematical 
analogue ;  in  short,  that  the  details  in  which  the  figure  differs  from 
its  mathematical  prototype  are  more  important  and  more  interesting 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1029 

than  the  features  in  which  it  agrees;  and  even  that  the  pecuUar 
aesthetic  pleasure  with  which  we  regard  a  living  thing  is  spmehow 
bound  up  with  the  departure  from  mathematical  regularity  which 
it  manifests  as  a  peculiar  attribute  of  life.  This  view  seems  to  me 
to  involve  a  misapprehension.  There  is  no  such  essential  difference 
between  these  phenomena  of  organic  form  and  those  which  are 
manifested  in  portions  of  inanimate  matter*.  The  mathematician 
knows  better  than  we  do  the  value  of  an  approximate  resultf.  The 
child's  skipping-rope  is  but  an  approximation  to  Huygens's  catenary 
curve — but  in  the  catenary  curve  lies  the  whole  gist  of  the  matter. 
We  may  be  dismayed  too  easily  by  contingencies  which  are  nothing 
short  of  irrelevant  compared  to  the  main  issue ;  there  is  a  jyrinciple ' 
of  negligibility.  Someone  has  said  that  if  Tycho  Brahe's  instruments 
had  been  ten  times  as  exact  there  would  have  been  no  Kepler,  no 
Newton,  and  no  astronomy. 

If  no  chain  hangs  in  a  perfect  catenary  and  no  raindrop  is  a  perfect 
sphere,  this  is  for  the  reason  that  forces  and  resistances  other  than 
the  main  one  are  inevitably  at  work.  The  same  is  true  of  organic 
form,  but  it  is  for  the  mathematician  to  unravel  the  conflicting 
forces  which  are  at  work  together.  And  this  process  of  investigation 
may  lead  us  on  step  by  step  to  new  phenomena,  as  it  has  done 
in  physics,  where  sometimes  a  knowledge  of  form  leads  us  to  the 
interpretation  of  forces,  and  at  other  times  a  knowledge  of  the  forces 
at  work  guides  us  towards  a  better  insight  into  form.  After  the 
fundamental  advance  had  been  made  which  taught  us  that  the  world 

*  M,  Bergson  repudiates,  with  peculiar  confidence,  the  appHcation  of  mathe- 
matics to  biology;  cf.  Creative  Evolution,  p.  21,  "Calculation  touches,  at  most, 
certain  phenomena  of  organic  destruction.  Organic  creation,  on  the  contrary, 
the  evolutionary  phenomena  which  properly  constitute  life,  we  cannot  in  any  way 
subject  to  a  mathematical  treatment."  Bergson  thus  follows  Bichat:  "C'est 
peu  connaitre  les  fonctions  animales  que  de  vouloir  les  soumettre  au  moindre 
calcul,  parceque  leur  instabUite  est  extreme.  Les  phenomenes  restent  toujours 
les  memes,  et  c'est  ce  qui  nous  importe;  mais  leurs  variations,  en  plus  ou  en  moins, 
sont  sans  nombre"  (La  Vie  et  la  Mort,  p.  257). 

t  When  we  make  a  'first  approximation'  to  the  solution  of  a  physical  problem, 
we  usually  mean  that  we  are  solving  one  part  while  neglecting  others.  Geometry 
deals  with  pure  forms  (such  as  a  straight  line),  defined  by  a  single  law;  but  these 
are  few  compared  with  the  mixed  forms,  like  the  surface  of  a  polyhedron,  or  a 
segment  of  a  sphere,  or  any  ordinary  mechanical  construction  or  any  ordinary 
physical  phenomenon.  It  is  only  in  a  purely  mathematical  treatment  of  physics 
that  the  "single  law"  can  be  dealt  with  alone,  and  the  approximate  solution 
dispensed  with  accordingly. 


1030       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

was  round,  Newton  shewed  that  the  forces  at  work  upon  it  must 
lead  to  its  being  imperfectly  spherical,  and  in  the  course  of  time  its 
oblate  spheroidal  shape  was  actually  verified.  But  now,  in  turn, 
it  has  been  shewn  that  its  form  is  still  more  complicated,  and  the 
next  step  is  to  seek  for  the  forces  that  have  deformed  the  oblate 
spheroid.  As  Newton  somewhere  says,  "Nature  dehghts  in  trans- 
formations." 

The  organic  forms  which  we  can  define  more  or  less  precisely 
in  mathematical  terms,  and  afterwards  proceed  to  explain  and  to 
account  for  in  terms  of  force,  are  of  many  kinds,  as  we  have  seen; 
but  nevertheless  they  are  few  in  number  compared  with  Nature's 
all  but  infinite  variety.  The  reason  for  this  is  not  far  to  seek. 
The  living  organism  represents,  or  occupies,  a  field  of  force  which 
is  never  simple,  and  which  as  a  rule  is  of  immense  complexity.  And 
just  as  in  the  very  simplest  of  actual  cases  we  meet  with  a  departure 
from  such  symmetry  as  could  only  exist  under  conditions  of  ideal 
simplicity,  so  do  we  pass  quickly  to  cases  where  the  interference  of 
numerous,  though  still  perhaps  very  simple,  causes  leads  to  a  recultant 
complexity  far  beyond  our  powers  of  analysis.  Nor  must  we  foi*get 
that  the  biologist  is  much  more  exacting  in  his  requirement^,  as 
regards  form,  than  the  physicist;  for  the  latter  is  usually  content 
with  either  an  ideal  or  a  general  description  of  form,  while  the 
student  of  living  things  must  needs  be  specific.  Material  things, 
be  they  living  or  dead,  shew  us  but  a  shadow  of  mathematical 
perfection*.  The  physicist  or  mathematician  can  give  us  perfectly 
satisfying  expressions  for  the  form  of  a  wave,  or  even  of  a  heap 
of  sand ;  but  we  never  ask  him  to  define  the  form  of  any  particular 
wave  of  the  sea,  nor  the  actual  form  of  any  mountain-peak  or  hill. 

In  this  there  lies  a  certain  justification  for  a  saying  of  Minot's,  of  the 
greater  part  of  which,  nevertheless,  I  am  heartily  inclined  to  disapprove. 
"We  biologists,"  he  says,  "cannot  deplore  too  frequently  or  too  emphatically 
the  great  mathematical  delusion  by  which  men  often  of  great  if  limited  ability 
have  been  misled  into  becoming  advocates  of  an  erroneous  conception  of 
accuracy.  The  delusion  is  that  no  science  is  accurate  imtil  its  results  can  be 
expressed  mathematically.  The  error  comes  from  the  assumption  that 
mathematics  can  express  complex  relations.  Unfortunately  mathematics 
have  a  very  limited  scope,  and  are  based  upon  a  few  extremely  rudimentary 

*  Cf.  Haton  de  la  GoupiUiere,  op.  cit.:  "On  a  souvent  I'occasion  de  saisir  dans 
la  nature  un  reflet  des  formes  rigoureuses  qu'etudie  la  geometrie," 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1031 

experiences,  which  we  make  as  very  little  children  and  of  which  no  adult  has 
any  recollection.  The  fact  that  from  this  basis  men  of  genius  have  evolved 
wonderful  methods  of  dealing  with  numerical  relations  should  not  blind  us 
to  another  fact,  namely,  that  the  observational  basis  of  mathematics  is, 
psychologically  speaking,  very  minute  compared  with  the  observational  basis 
of  even  a  single  minor  branch  of  biology. . . .  While  therefore  here  and  there 
the  mathematical  methods  may  aid  us,  we  need  a  kind  and  degree  of  accuracy 
of  which  mathematics  is  absolutely  incapable. . .  .  With  human  minds  constituted 
as  they  actually  are,  we  cannot  anticipate  that  there  will  ever  be  a  mathe- 
matical expression  for  any  organ  or  even  a  single  cell,  although  formulae  will 
continue  to  be  useful  for  dealing  now  and  then  with  isolated  details..." 
{op.  cit.  p.  19,  1911).  It  were  easy  to  discuss  and  criticise  these  sweeping 
assertions,  which  perhaps  had  their  origin  and  parentage  in  an  obiter  dictum 
of  Huxley's,  to  the  effect  that  "Mathematics  is  that  study  which  knows  nothing 
of  observation,  nothing  of  experiment,  nothing  of  induction,  nothing  of 
causation"  {cit.  Cajori,  Hist,  of  Elem.  Mathematics,  p.  283).  But  Gauss, 
"rex  mathematicorum,"  called  mathematics  "a  science  of  the  eye";  and 
Sylvester  assures  us  that  "most,  if  not  all,  of  the  great  ideas  of  modem 
mathematics  have  had  their  origin  in  observation"  {Brit.  Ass.  Address,  1869, 
and  Laws  of  Verse,  p.  120,  1870. 

Reaumur  said  the  same  thing  two  hundred  years  ago  {Mem.  i,  p.  49,  1734). 
Maupertuis,  he  said,  was  both  naturalist  and  mathematician;  and  all  his 
mathematics  "n'ont  en  rien  affaibli  son  gout  pour  les  insectes,  personne 
peut-etre  n'a  plus  d' amour  pour  eux."  He  goes  on  to  say:  "L' esprit 
d'observation  qu'on  regarde  comme  le '  caractere  d' esprit  essentiel  aux 
naturalistes,  est  egalement  necessaire  pour  faire  des  progres  en  quelque  science 
que  ce  soit.  C'est  I'esprit  d'observation  qui  fait  appercevoir  ce  qui  a 
echappe  aux  autres,  qui  fait  saisir  des  rapports  qui  sont  entre  des  choses 
qui  semblent  differentes,  ou  qui  fait  trouver  les  differences  qui  sont  entre 
celles  qui  paroissent  semblables.  On  ne  resoud  les  problemes  les  plus  epineux 
de  Geometric  qu'apres  avoir  S9u  observer  des  rapports,  qui  ne  se  decouvrent 
qu'a  un  esprit  penetrant,  et  extremement  attentif.  Ce  sont  des  observations 
qui  mettent  en  etat  de  resoudre  les  problemes  de  physique  comme  ceux 
d'histoire  naturelle,  car  I'histoire  natureUe  a  ses  problemes  a  resoudre,  et 
elle  n'en  a  meme  que  trop  qui  ne  sont  pas  resolus."  It  is  in  a  deeper  sense 
than  this,  however,  that  the  modem  physicist  looks  on  mathematics  as  an 
"empirical"  science,  and  no  longer  a  matter  of  pure  intuition,  or  "reine 
Anschauung."  Cf.  Max  Bom,  on  Some  philosophical  aspects  of  modem 
physics,  Proc.  R.S.E.  Lvn,  pp.  1-18,  1936.. 

For  one  reason  or  another  there  are  very  many  organic  forms  which 
we  cannot  describe,  still  less  define,  in  mathematical  terms:  just  as 
there  are  problems  even  in  physical  science  beyond  the  mathematics 
of  our  age.  We  never  even  seek  for  a  formula  to  define  this  fish  or 
that,  or  this  or  that  vertebrate  skull.  But  we  may  already  use 
mathematical  language  to  describe,  even  to  define  in  general  terms, 


1032       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

the  shape  of  a  snail-shell,  the  twist  of  a  horn,  the  outline  of  a  leaf, 
the  texture  of  a  bone,  the  fabric  of  a  skeleton,  the  stream-lines  of 
fish  or  bird,  the  fairy  lace-work  of  an  insect's  wing.  Even  to  do 
this  we  must  learn  from  the  mathematician  to  ehminate  and  to 
discard;  to  keep  the 'type  in  mind  and  leave  the  single  case,  with 
all  its  accidents,  alone ;  and  to  find  in  this  sacrifice  of  what  matters 
little  and  conservation  of  what  matters  much  one  of  the  pecuUar 
excellences  of  the  method  of  mathematics*. 

In  a  very  large  part  of  morphology,  our  essential  task  lies  in  the 
comparison  of  related  forms  rather  than  in  the  precise  definition 
of  each;  and  the  deformation  of  a  complicated  figure  may  be  a 
phenomenon  easy  of  comprehension,  though  the  figure  itself  have 
to  be  left  unanalysed  and  undefined.  This  process  of  comparison, 
of  recognising  in  one  form  a  definite  permutation  or  deformation  of 
another,  apart  altogether  from  a  precise  and  adequate  understanding 
of  the  original  "type"  or  standard  of  comparison,  Hes  within  the 
immediate  province  of  mathematics,  and  finds  its  solution  in  the 
elementary  use  of  a  certain  method  of  the  mathematician.  This 
method  is  the  Method  of  Coordinates,  on  which  is  based  the  Theory 
of  Transformations  f. 

I  imagine  that  when  Descartes  conceived  the  method  of  co- 
ordinates, as  a  generalisation  from  the  proportional  diagrams  of  the 
artist  and  the  architect,  and  long  before  the  immense  possibilities 
of  this  analysis  could  be  foreseen,  he  had  in  mind  a  very  simple 
purpose ;  it  was  perhaps  no  more  than  to  find  a  way  of  translating 
the  form  of  a  curve  (as  well  as  the  position  of  a  point)  into  numbers 
and  into  words.  This  is  precisely  what  we  do,  by  the  method  of 
coordinates,  every  time  we  study  a  statistical  curve ;  and  conversely, 
we  translate  numbers  into  form  whenever  we  "plot  a  curve,"  to 
illustrate  a  table  of  mortality,  a  rate  of  growth,  or  the  daily  variation 
of  temperature  or  barometric  pressure.     In  precisely  the  same  way 

*  Cf.  W.  H.  Young,  The  mathematical  method  and  its  limitations,  Congresso 
dei  Matematici,  Bologna,  1928. 

f  The  mathematical  Theory  of  Transformations  is  part  of  the  Theory  of  Groups, 
of  great  importance  in  modern  mathematics.  A  distinction  is  drawn  between 
Substitution-groups  and  Transformation-groups,  the  former  being  discontinuous, 
the  latter  continuous — in  such  a  way  that  within  one  and  the  same  group  each 
transformation  is  infinitely  little  different  from  another.  The  distinction  among 
biologists  between  a  mutation  and  a  variation  is  curiously  analogous. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1033 

it  is  possible  to  inscribe  in  a  net  of  rectangular  coordinates  the 
outline,  for  instance,  of  a  fish,  and  so  to  translate  it  into  a  table  of 
numbers,  from  which  again  we  may  at  pleasure  reconstruct  the 
curve. 

But  it  is  the  next  step  in  the  employment  of  coordinates  which 
is  of  special  interest  and  use  to  the  morpnologist ;  and  this  step 
consists  in  the  alteration,  or  deformation,  of  our  system  of  coordinates, 
and  in  the  study  of  the  corresponding  transformation  of  the  curve 
or  figure  inscribed  in  the  coordinate  network. 

Let  us  inscribe  in  a  system  of  Cartesian  coordinates  the  outUne 
of  an  organism,  however  complicated,  or  a  part  thereof:  such  as 
a  fish,  a  crab,  or  a  mammalian  skull.  We  may  now  treat  this 
complicated  figure,  in  general  terms,  as  a  function  of  x,  y.  If  we 
submit  our  rectangular  system  to  deformation  on  simple  and 
recognised  lines,  altering,  for  instance,  the  direction  of  the  axes, 
the  ratio  of  xjy,  or  substituting  for  x  and  y  some  more  complicated 
expressions,  then  we  obtain  a  new  system  of  coordinates,  whose 
deformation  from  the  original  type  the  inscribed  figure  will  precisely 
follow/  In  other  words,  we  obtain  a  new  figure  which  represents 
the  old  figure  under  a  more  or  less  homogeneous  strain,  and  is 
a  function  of  the  new  coordinates  in  precisely  the  same  way  as  the 
old  figure  was  of  the  original  coordinates  x  and  y. 

The  problem  is  closely  akin  to  that  of  the  cartographer  who 
transfers  identical  data  to  one  projection  or  another*;  and  whose 
object  is  to  secure  (if  it  be  possible)  a  complete  Correspondence, 
in  each  small  unit  of  area,  between  the  one  representation  and  the 
other.  The  morphologist  will  not  seek  to  draw  his  organic  forms 
in  a  new  and  artificial  projection;  but,  in  the  converse  aspect  of 
the  problem,  he  will  enquire  whether  two  different  but  more  or  less 
obviously  related  forms  can  be  so  analysed  and  interpreted  that 
each  may  be  shewn  to  be  a  transformed  representation  of  the  other. 
This  once  demonstrated,  it  will  be  a  comparatively  easy  task  (in  all 
probability)  to  postulate  the  direction  and  magnitude  of  the  force 
capable  of  effecting  the  required  transformation.  Again,  if  such 
a  simple  alteration  of  the  system  of  forces  can  be  proved  adequate 
to  meet  the  case,  we  may  find  ourselves  able  to  dispense  with  many 

*  Cf.  (e.g.)  Tissot,  Memoire  sur  la  representation  des  surfaces,  et  les  projections 
des  cartes  geographiques,  Paris,  1881. 


1034       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

widely  current  and  more  complicated  hypotheses  of  biological 
causation.  For  it  is  a  maxim  in  physics  that  an  effect  ought  not 
to  be  ascribed  to  the  joint  operation  of  many  causes  if  few  are 
adequate  to  the  production  of  it :  Frustra  fit  per  plura,  quod  fieri 
potest  per  pauciora. 

We  might  suppose  that  by  the  combined  action  of  appropriate 
forces  any  material  form  could  be  transformed  into  any  other:  just 
as  out  of  a  "shapeless"  mass  of  clay  the  potter  or  the  sculptor 
models  his  artistic  product;  or  just  as  we  attribute  to  Nature  herself 
the  power  to  effect  the  gradual  and  successive  transformation  of 
the  simple  germ  into  the  complex  organism.  But  we  need  not 
let  these  considerations  deter  us  from  our  method  of  comparison 
of  related  forms.  We  shall  strictly  limit  ourselves  to  cases  where 
the  transformation  necessary  to  effect  a  comparison  shall  be  of 
a  simple  kind,  and  where  the  transformed,  as  well  as  the  original, 
coordinates  shall  constitute  an  harm<5ni6us  and  more  or  less  sym- 
metrical system.  We  should  fall  into  deserved  and  inevitable 
confusion  if,  whether  by  the  mathematical  or  any  other  method, 
we  attempted  to  compare  organisms  separated  far  apart  in  Nature 
and  in  zoological  classification.  We  are  limited,  both  by  our  method 
,  and  by  the  whole  nature  of  the  case,  to  the  comparison  of  organisms 
such  as  are  manifestly  related  to  one  another  and  belong  to  the  same 
zoological  class.  For  it  is  a  grave  sophism,  in  natural  history  as  in 
logic,  to  make  a  transition  into  another  kind*. 

Our  enquiry  lies,  in  short,  just  within  the  limits  which  Aristotle 
himself  laid  down  when,  in  defining  a  "genus,"  he  shewed  that 
(apart  from  those  superficial  characters,  such  as  colour,  which  he 
called  "accidents")  the  essential  differences  between  one  "species" 
and  another  are  merely  differences  of  proportion,  of  relative  mag- 
nitude, or  (as  he  phrased  it)  of  "excess  and  defect."  "Save  only 
for  a  difference  in  the  way  of  excess  or  defect,  the  parts  are  identical 
in  the  case  of  such  animals  as  are  of  one  and  the  saiiie  genus ;  and 
by  '  genus '  I  mean,  for  instance,  Bird  or  Fish."  And  again :  "  Within 
the  limits  of  the  same  genus,  as  a  general  rule,  most  of  the  parts 
exhibit  differences ...  in  the  way  of  multitude  or  fewness,  magnitude 

*  The  saying  heterogenea  comparari  non  possunt  is  discussed  by  Coleridge  in  his 
Aids  to  Reflexion. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1035 

or  parvitude,  in  short,  in  the  way  of  excess  or  defect.  For  'the 
more'  and  'the  less'  may  be  represented  as  'excess'  and  *  defect'*." 
It  is  precisely  this  difference  of  relative  magnitudes,  this  Aristotelian 
"excess  and  defect"  in  the  case  of  form,  which  our  coordinate 
method  is  especially  adapted  to  analyse,  and  to  reveal  and  demon- 
strate as  the  main  cause  of  what  (again  in  the  Aristotelian  sense) 
we  term  "specific"  differences. 

The  appUcability  of  our  method  to  particular  cases  will  depend 
upon,  or  be  further  Hmited  by,  certain  practical  considerations  or 
qualifications.  Of  these  the  chief,  and  indeed  the  essential,  con- 
dition is,  that  the  form  of  the  entire  structure  under  investigation 
should  be  found  to  vary  in  a  more  or  less  uniform  manner,  after  the 
fashion  of  an  approximately  homogeneous  and  isotropic  body.  But 
an  imperfect  isotropy,  provided  always  that  some  "principle  of 
continuity"  run  through  its  variations,  will  not  seriously  interfere 
with  our  method ;  it  will  only  cause  our  transformed  coordinates 
to  be  somewhat  less  regular  and  harmonious  than  are  those,  for 
instance,  by  which  the  physicist  depicts  the  motions  of  a  perfect 
fluid,  or  a  theoretic  field  of  force  in  a  uniform  medium. 

Again,  it  is  essential  that  our  structure  vary  in  its  entirety,  or 
at  least  that  "  independent  variants  "  should  be  relatively  few.  That 
independent  variations  occur,  that  localised  centres  of  diminished 
or  exaggerated  growth  will  now  and  then  be  found,  is  not  only 
probable  but  manifest;  and  they  may  even  be  so  pronounced  as  to 
appear  to  constitute  new  formations  altogether.  Such  independent 
variants  as  these  Aristotle  himself  clearly  recognised:  "It  happens 
further  that  some  have  parts  which  others  have  not;  for  instance, 
some  [birds]  have  spurs  and  others  not,  some  have  crests,  or  combs, 
and  others  not;  but,  as  a  general  rule,  most  parts  and  those  that 
go  to  make  up  the  bulk  of  the  body  are  either  identical  with  one 
another,  or  differ  from  one  another  in  the  way  of  contrast  and  of 
excess  and  defect.  For  '  the  more '  and  '  the  less '  may  be  represented 
as  'excess'  or  ' defect '|." 

If,  in  the  evolution  of  a  fish,  for  instance,  it  be  the  case  that  its 

*  Historia  Animalium  i,  1. 

•j;  Aristotle's  argument  is  even  more  subtle  and  far-reaching;  for  the  diflferences 
of  which  he  speaks  are  not  merely  those  between  one  bird  and  another,  but  between 
them  all  and  the  very  type  itself,  or  Platonic  "idea"  of  a  bird. 


1036       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

several  and  constituent  parts — head,  body  and  tail,  or  this  fin 
and  that  fin — represent  so  many  independent  variants,  then  our 
coordinate  system  will  at  once  become  too  complex  to  be  intelligible ; 
we  shall  be  making  not  one  comparison  but  several  separate  com- 
parisons, and  our  general  method  will  be  found  inapplicable.  Now 
precisely  this  independent  variability  of  parts  and  organs — here, 
there,  and  everywhere  within  the  organism — would  appear  to  be 
implicit  in  our  ordinary  accepted  notions  regarding  variation;  and, 
unless  I  am  greatly  mistaken,  it  is  precisely  on  such  a  conception  of 
the  easy,  frequent,  and  normally  independent  variability  of  parts  that 
our  conception  of  the  process  of  natural  selection  is  fundamentally 
based.  For  the  morphologist,  when  comparing  one  organism  with 
another,  describes  the  differences  between  them  point  by  point,  and 
"character"  by  "character*."  If  he  is  from  time  to  time  con- 
strained to  admit  the  existence  of  "correlation"  between  characters 
(as  a  hundred  years  ago  Cuvier  first  shewed  the  way),  yet  all  the 
while  he  recognises  this  fact  of  correlation  somewhat  vaguely,  as 
a  phenomenon  due  to  causes  which,  except  in  rare  instances,  he  can 
hardly  hope  to  trace ;  and  he  falls  readily  into  the  habit  of  thinking 
and  talking  of  evolution  as  though  it  had  proceeded  on  the  lines  of 
his  own  descriptions,  point  by  point,  and  character  by  character  f. 

With  the  "characters"  of  Mendelian  genetics  there  is  no  fault 
to  be  found;  tall  and  short,  rough  and  smooth,  plain  or  coloured 
are  opposite  tendencies  or  contrasting  qualities,  in  plain  logical 
contradistinction.  But  when  the  morphologist  compares  one  animal 
with  another,  point  by  point  or  character  by  character,  these  are 
too  often  the  mere  outcome  of  artificial  dissection  and  analysis. 
Rather  is  the  living  body  one  integral  and  indivisible  whole,  in 

*  Cf.  supra,  p.  1020. 

t  Cf.  H.  F.  Osbom,  On  the  origin  of  single  characters,  as  observed  in  fossil 
and  living  animals  and  plants,  Amer.  Nat.  xlix,  pp.  193-239,  1915  (and  other 
papers);  ibid.  p.  194,  "Each  individual  is  composed  of  a  vast  number  of  somewhat 
similar  new  or  old  characters,  each  character  has  its  independent  and  separate 
history,  each  character  is  in  a  certain  stage  of  evolution,  each  character  is  correlated 

with  the  other  characters  of  the  individual The  real  problem  has  always  been 

that  of  the  origin  and  development  of  characters.  Since  the  Origin  of  Species 
appeared,  the  terms  variation  and  variability  have  always  referred  to  single 
characters;  if  a  species  is  said  to  be  variable,  we  mean  that  a  considerable 
number  of  the  single  characters  or  groups  of  characters  of  which  it  is  composed  are 
variable,"  etc. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1037 

wEich  we  cannot  find,  when  we  come  to  look  for  it,  any  strict 
dividing  line  even  between  the  head  and  the  body,  the  muscle  and 
the  tendon,  the  sinew  and  the  bone.  Characters  which  we  have 
differentiated  insist  on  integrating  themselves  again;  and  aspects 
of  the  organism  are  seen  to  be  conjoined  which  only  our  mental 
analysis  had  put  asunder.  The  coordinate  diagram  throws  into 
rehef  the  integral  solidarity  of  the  organism,  and  enables  us  to  see 
how  simple  a  certain  kind  of  correlation  is  which  had  been  apt  to 
seem  a  subtle  and  a  complex  thing. 

But  if,  on  the  other  hand,  diverse  and  dissimilar  fishes  can  be 
referred  as  a  whole  to  identical  functions  of  very  different  coordinate 
systems,  this  fact  will  of  itself  constitute  a  proof  that  variation  has 
proceeded  on  definite  and  orderly  line?,  that  a  comprehensive  "law 
of  growth"  has  pervaded  the  whole  structure  in  its  integrity,  and 
that  some  more  or  less  simple  and  recognisable  system  of  forces 
has  been  in  control.  It  will  not  only  shew  how  real  and  deep-seated 
is  the  phenomenon  of  "correlation,"  in  regard  to  form,  but  it  will 
also  demonstrate  the  fact  that  a  correlation  which  had  seemed  too 
complex  for  analysis  or  comprehension  is,  in  many  cases,  capable  of 
very  simple  graphic  expression.  This,  after  many  trials,  I  believe  to 
be  in  general  the  case,  bearing  always  in  mind  that  the  occurrence  of 
independent  or  localised  variations  must  sometimes  be  considered. 

We  are  deahng  in  this  chapter  with  the  forms  of  related  organisms,  in  order 
to  shew  that  the  differences  between  them  are  as  a  general  rule  simple  and 
symmetrical,  and  just  such  as  might  have  been  brought  about  by  a  slight  and 
simple  change  in  the  system  of  forces  to  which  the  living  and  growing  organism 
was  exposed.  Mathematically  speaking,  the  phenomenon  is  identical  with  one 
met  with  by  the  geologist,  when  he  finds  a  bed  of  fossils  squeezed  flat  or  other- 
wise symmetrically  deformed  by  the  pressures  to  which  they,  and  the  strata 
which  contain  them,  have  been  subjected.  In  the  first  step  towards  fossilisation, 
when  the  body  of  a  fish  or  shellfish  is  silted  over  and  buried,  we  may  take  it 
that  the  wet  sand  or  mud  exercises,  approximately,  a  hydrostatic  pressure — 
that  is  to  say  a  pressure  which  is  uniform  in  all  directions,  and  by  which  the 
form  of  the  buried  object  will  not  be  appreciably  changed.  As  the  strata 
consolidate  and  accumulate,  the  fosail  Organisms  which  they  contain  will 
tend  to  be  flattened  by  the  vast  superincumbent  load,  just  as  the  stratum 
which  contains  them  will  also  be  compressed  and  will  have  its  molecular 
arrangement  more  or  less  modified*.  But  the  deformation  due  to  direct 
vertical  pressure  in  a  horizontal  stratum  is  not  nearly  so  striking  as  are  the 
deformations  produced  by  the  oblique  or  shearing  stresses  to  which  inclined 

*  Cf.  Sorby,  Quart.  Journ.  Geol.  Soc.  (Proc),  1879,  p.  88. 


1038       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

and  folded  strata  have  been  exposed,  and  by  which  their  various  "dislocations  " 
have  been  brought  about.  And  especially  in  mountain  regions,  where  "these 
dislocations  are  especially  numerous  and  complicated,  the  contained  fossils 
are  apt  to  be  so  curiously  and  yet  so  symmetrically  deformed  (usually  by  a 
simple  shear)  that  they  may  easily  be  interpreted  as  so  many  distinct  and 
separate  "species*."  A  great  number  of  described  species,  and  here  and 
there  a  new  genus  (as  the  genus  Ellipsolithes  for  an  obliquely  deformed 
Goniatite  or  Nautilus),  are  said  to  rest  on  no  other  foundation  f. 

If  we  begin  by  drawing  a  net  of  rectangular  equidistant  coordinates 
(about  tlie  axes  x  and  y),  we  may  alter  or  deform;  this  network  in 
various  ways,  several  of  which  are  very  simple  indeed.  Thus  (1)  we 
may  alter  the  dimensions  of  our  system,  extending  it  along  one  or 


/ 

/^ 

N 

\ 

/ 

/ 

\ 

\ 

1 

\ 

V 

y 

/ 

Fig.  491. 


Fig.  492. 


other  axis,  and  so  converting  each  little  square  into  a  corresponding 
and  proportionate  oblong  (Figs.  491,  492).  It  follows  that  any 
figure  which  we  may  have  inscribed  in  the  original  net,  and  which 
we  transfer  to  the  new,  will  thereby  be  deformed  in  strict  proportion 
to  the  deformation  of  the  entire  configuration,  being  still  defined 
by  corresponding  points  in  the  network  and  being  throughout  in 
conformity  with  the  original  figure.     For  instance,  a  circle  inscribed 

*  Cf.  Ale.  D'Orbigny,  Cmirs  dim.  de  PaUontologie,  etc.,  i,  pp.  144-148,  1849; 
see  also  Daniel  Sharpe,  On  slaty  cleavage,  Q.J.G.S.  m,  p.  74,  1847. 

t  Thus  Ammonites  erugatus,  when  compressed,  has  been  described  &a  A .  planorbis : 
cf.  J.  F.  Blake,  Phil.  Mag.  (5),  vi,  p.  260,  1878.  Wettstein  has  shewn  that^several 
species  of  the  fish-genus  Lepidopus  have  been  based  on  specimens  artificially 
deformed  in  various  ways:  Ueber  die  Fischfauna  des  Tertiaren  Glarnerschiefers, 
Abh.  Schw.  Palaeont.  Gesellsch.  xiii,  1886  (see  especially  pp.  23-38,  pi.  i).  The 
whole  subject,  interesting  as  it  is,  has  been  httle  studied ;  both  Blake  and  Wettstein 
deal  with  it  mathematically. 


XVII]    THE  COMPARISON  OF  RELATED   FORMS    1039 

in  the  original  "Cartesian"  net  will  now,  after  extension  in  the 
^/-direction,  be  found  elongated  into  an  ellipse.  In  elementary 
mathematical  language,  fo;-  the  original  x  and  y  we  have  substituted 
iCj  and  c?/i ,  and  the  equation  to  our  original  circle,  x^  ^  ip'  =  a^, 
becomes  that  of  the  ellipse,  x^  +  c^yx  ^  ^^• 

If  I  draw  the  cannon-bone  of  an  ox  (Fig.  493,  A),  for  instance, 
within  a  system  of  rectangular  coordinates,  and  then  transfer  the 
same  drawing,  point  for  point,  to  a  system  in  which  for  the  x  of 
the  original  diagram  we  substitute  x'  =  2x/3,  we  obtain  a  drawing 
(B)  which  is  a  very  close  approximation  to  the  cannon-bone  of  the 
sheep.     In  other  words,  the  main  (and  perhaps  the  only)  difference 


r 

< 

7 

X 

/ 1 

— 

.  \ 

\ 

U 

si 

ft 

Fi 

?.4 

94. 

A  B  C 

Fig.  493. 

between  the  two  bones  is  simply  that  that  of  the  sheep  is  elongated 
along  the  vertical  axis  as  compared  with  that  of  the  ox,  in  the  pro- 
portion of  3/2.  And  similarly,  the  long  slender  cannon-bone  of  the 
giraife  (C)  is  referable  to  the  same  identical  t3rpe,  subject  to  a  reduction 
of  breadth,  or  increase  of  length,  corresponding  to  x"  =  x/3. 

(2)  The  second  type  is  that  where  extension  is  not  equal  or 
uniform  at  all  distances  from  the  origin:  but  grows  greater  or  less, 
as,  for  instance,  when  we  stretch  sl  tapering  elastic  band.  In  such 
cases,  as  I  have  represented  it  in  Fig.  494,  the  ordinate  increases 
logarithmically,  and  for  y  we  substitute  e^.  It  is  obvious  that  this 
logarithmic  extension  may  involve  both  abscissae  and  ordinates, 
X  becoming  e^  while  y  becomes  e^.  The  circle  in  our  original  figure 
is  now  deformed  into  some  such  shape  as  that  of  Fig.  495.     This 


1040       THE  THEORY  OF  TRAN^SFORMATIONS        [ch. 

method  of  deformation  is  a  common  one,  and  will  often  be  of  use 
to  us  in  our  comparison  of  organic  forms. 


Fig.  495. 


Fig.  496. 

(3)  Our  third  type  is  the  "simple  shear,"  where  the  rectangular 
coordinates  become  "obHque,"  their  axes  being  inclined  to  one 
another  at  a  certain  angle  oj.  Our  original  rectangle  now  becomes 
such  a  figure  as  that  of  Fig.  496.  The  system  may  now  be  described 
in  terms  of  the  oblique  axes  X,  Y;  or  may  be  directly  referred 
to  new  rectangular  coordinates  f,  rj  by  the  simple  transposition 
X  =  i  —  7)  cot  oj,  y  =  r]  cosec  co. 


Fig.  497. 

(4)  Yet  another  important  class  of  deformations  may  be  repre- 
sented by  the  use  of  radial  coordinates,  in  which  one  set  of  lines  are 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1041 

represented  as  radiating  from  a  point  or  "focus,"  while  the  other 
set  are  transformed  into  circular  arcs  cutting  the  radii  orthogonally. 
These  radial  coordinates  are  especially  apphcable  to  cases  where 
there  exists  (either  within  or  without  the  figure)  some  part  which 
is  supposed  to  suffer  no  deformation ;  a  simple  illustration  is  afforded 
by  the  diagrams  which  illustrate  the  flexure  of  a  beam  (Fig.  497). 
In  biology  these  coordinaites  will  be  especially  applicable  in  cases 
where  the  growing  structure  includes  a  "node,"  or  point  where 
growth  is  absent  or  at  a  minimum;  and  about  which  node  the  rate 
of  growth  may  be  assumed  to  increase  symmetrically.     Precisely 


Fig.  498. 

such  a  case  is  furnished  us  in  a  leaf  of  an  ordinary  dicotyledon. 
The  leaf  of  a  typical  monocotyledon — such  as  a  grass  or  a  hyacinth, 
for  instance — grows  continuously  from  its  base,  and  exhibits  no 
node  or  "point  of  arrest."  Its  sides  taper  off  gradually  from  its 
broad  base  to  its  slender  tip,  according  to  some  law  of  decrement 
specific  to  the  plant;  and  any  alteration  in  the  relative  velocities 
of  longitudinal  and  transverse  growth  will  merely  make  the  leaf 
a  little  broader  or  narrower,  and  will  effect  no  other  conspicuous 
alteration  in  its  contour.  But  if  there  once  come  into  existence 
a  node,  or  "locus  of  no  growth,"  about  which  we  may  assume 
growth — which  in  the  hyacinth  leaf  was  longitudinal  and  trans- 
verse— to  take  place  radially  and  transversely  to  the  radii,  then  we 


1042       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

shall  soon  see  the  sloping  sides  of  the  hyacinth  leaf  give  place  to 
a  more  typical  and  "leaf -like"  shape.  If  we  alter  the  ratio  between 
the  radial  and  tangential  velocities  of  growth — in  other  words,  if  we 
increase  the  angles  between  corresponding^radii — we  pass  successively 
through  the  various  configurations  which  the  botanist  describes  as 
the  lanceolate,  the  ovate,  and  the  cordiform  leaf.  These  successive 
changes  may  to  some  extent,  and  in  appropriate  cases,  be  traced  as 
the  individua,l  leaf  grows  to  maturity ;  but  as  a  much  more  general 
rule,  the  balance  of  forces,  the  ratio  between  radial  and  tangential 
velocities  of  growth,  remains  so  nicely  and  constantly  balanced  that 
the  leaf  increases  in  size  without  conspicuous  modification  of  form. 
It  is  rather  what  we  may  call  a  long-period  variation,  a  tendency  for 
the  relative  velocities  to  alter  from  one  generation  to  another,  whose 
result  is  brought  into  view  by  this  method  of  illustration. 

There  are  various  corollaries  to  this  method  of  describing  the  form 
of  a  leaf  which  may  be  here  alluded  to.     For  instance,  the  so-called 
A  unsymmetrical  leaf*  of  a  begonia, 

in  which  one  side  of  the  leaf  may  be 
merely  ovate  while  the  other  has  a 
cordate  outhne,  is  seen  to  be  really 
a  case  of  unequal,  and  not  truly 
asymmetrical,  growth  on  either  side 
of  the  midrib.  There  is  nothing 
more  mysterious  in  its  conformation 
than,  for  instance,  in  that  of  a  forked 
twig  in  which  one  limb  of  the  fork 
has  grown  longer  than  the  other. 
The  case  of  the  begonia  leaf  is  of 
sufficient  interest  to  deserve  illus- 
tration, and  in  Fig.  499  I  have 
outlined  a  leaf  of  the  large  Begonia 
daedalea.  On  the  smaller  left-hand 
Fig.  499  Begonia  daedalea.  ^^^^  ^f  ^he  leaf  I  have  taken  at 
random  three  points  a,  6,  c,  and  have  measured  the  angles,  AOa,  etc., 

*  Cf.  Sir  Thomas  Browne,  in  The  Garden  of  Cyrus:  "But  why  ofttimes  one 
side  of  the  leaf  is  unequal!  unto  the  other,  |is  in  Hazell  and  Oaks,  why  on  either 
side  the  master  vein  the  lesser  and  derivative  channels  stand  not  directly  opposite, 
not  at  equall  angles,  respectively  unto  the  adverse  side,  but  those  of  one  side  do 
often  exceed  the  other,  as  the  Wallnut  and  many  more,  deserves  another  enquiry." 


xvii]    THE  COMPARISON  OF  RELATED  FORMS     1043 

which  the  radii  from  the  hilus  of  the  leaf  to  these  points  make  with 
the  median  axis.  On  the  other  side  of  the  leaf  I  have  marked  the 
points  a\  h\  c',  such  that  the  radii  drawn  to  this  margin  of  the  leaf 
are  equal  to  the  former,  Oa'  to  Oa,  etc.  Now  if  the  two  sides  of  the 
leaf  are  mathematically  similar  to  one  another,  it  is  obvious  that 
the  respective  angles  should  be  in  continued  proportion,  i.e.  as  AOa 
is  to  AOa',  so  should  AOh  be  to  AOh'.  This  proves  to  be  very 
nearly  the  case.  For  I  have  measured  the  three  angles  on  one  side, 
and  one  on  the  other,  and  have  then  compared,  as  follows,  the 
calculated  with  the  observed  values  of  the  other  two : 

AOa  AOb  AOc  AOa'  AOh'  AOc' 

Observed  values         12°  28-5°  88°  —  —  157° 

Calculated    „  —  —  —  21-5°  5M°  — 

Observed      „  —  —  —  20  52  — 

The  agreement  is  very  close,  and  what  discrepancy  there  is  may 
be  amply  accounted  for,  firstly,  by  the  slight  irregularity  of  the 
sinuous  margin  of  the  leaf;  and  secondly,  by  the  fact  that  the  true 
axis  or  midrib  of  the  leaf  is  not  straight  but  shghtly  curved,  and 
therefore  that  it  is  curvilinear  and  not  rectihnear  triangles  which 
we  ought  to  have  measured.  When  we  understand  these  few  points 
regarding  the  peripheral  curvature  of  the  leaf,  it  is  easy  to  see  that 
its  principal  veins  approximate  closely  to  a  beautiful  system  of 
isogonal  coordinates.  It  is  also  obvious  that  we  can  easily  pass, 
by  a  process  of  shearing,  from  those  cases  where  the  principal  veins 
start  from  the  base  of  the  leaf  to  those  where  they  arise  successively 
from  the  midrib,  as  they  do  in  most  dicotyledons. 

It  may  sometimes  happen  that  the  node*,  or  "point  of  arrest," 
is  at  the  upper  instead  of  the  lower  end  of  the  leaf-blade;  and 
occasionally  there  is  a  node  at  both  ends.  In  the  former  case, 
as  we  have  it  in  the  daisy,  the  form  of  the  leaf  will  be,  as  it  were, 
inverted,  the  broad,  more  or  less  heart-shaped,  outhne  appearing 
at  the  upper  end,  while  below  the  leaf  tapers  gradually  downwards 
to  an  ill-defined  base.  In  the  latter  case,  as  in  Dionaea,  we  obtain 
a  leaf  equally  expanded,  and  similarly  ovate  or  cordate,  at  both 
ends.  We  may  notice,  lastly,  that  the  shape  of  a  solid  fruit,  such 
as  an  apple  or  a  cherry,  is  a  solid  of  revolution,  developed  from 
similar  curves  and  to  be  explained  on  the  same  principle.     In  the 

*  "Node,"  in  the  botanical,  not  the  mathematical,  sense. 


1044       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

cherry  we  have  a  "point  of  arrest"  at  the  base  of  the  berry,  where 
it  joins  its  peduncle,  and  about  this  point  the  fruit  (in  imaginary 
section)  swells  out  into  a  cordate  outhne ;  while  in  the  apple  we  have 
two  such  well-marked  points  of  arrest,  above  and  below,  and  about 
both  of  them  the  same  conformation  tends  to  arise.  The  bean  and 
the  human  kidney  owe  their  "reniform"  shape  to  precisely  the 
same  phenomenon,  namely,  to  the  existence  of  a  node  or  "hilus," 
about  which  the  forces  of  growth  are  radially  and  symmetrically 
arranged.  When  the  seed  is  small  and  the  pod  roomy,  the  seed 
may  grow  round,  or  nearly  so,  like  a  pea ;  but  it  is  flattened  and 
bean-shaped,  or  elliptical  like  a  kidney-bean,  when  compressed 
within  a  narrow  and  elongated  pod.  If  the  original  seed  have  any 
simple  pattern,  of  the  nature  for  instance  of  meridians  or  parallels 
of  latitude,  it  is  easy  to  see  how  these  will  suffer  a  conformal  trans- 
formation, corresponding  to  the  deformation  of  the  sphere*. 

We  might  go  farther,  and  farther  than  we  have  room  for  here, 
to  illustrate  the  shapes  of  leaves  by  means  of  radial  coordinates, 
and  even  to  attempt  to  define  them  by  polar  equations.  In*  a 
former  chapter  we  learned  to  look  upon  the  curve  of  sines  as  an 
easy,  gradual  and  natural  transition — perhaps  the  simplest  and  most 
natural  of  all — from  minimum  to  corresponding  maximum,  and  so 
on  alternately  and  continuously;  and  we  found  the  same  curve 
going  round  like  the  hands  of  a  clock,  when  plotted  on  radial  co- 
ordinates and  (so  to  speak)  prevented  from  leaving  its  place.  Either 
way  it  represents  a  "simple  harmonic  motion."  Now  we  have  just 
seen  an  ordinary  dicotyledonous  leaf  to  have  a  "point  of  arrest," 
or  zero-growth  in  a  certain  direction,  while  in  the  opposite  direction 
towards  the  tip  it  has  grown  with  a  maximum  velocity.  This 
progress  from  zero  to  maximum  suggests  one-half  of  the  sine-curve ; 
in  other  words,  if  we  look  on  the  outline  of  the  leaf  as  a  vector- 
diagram  of  its  own  growth,  at  rates  varying  from  zero  to  zero  in 
a  complete  circuit  of  360°,  this  suggests,  as  a  possible  and  very 
simple  case,  the  plotting  of  r  =  sin  ^/2.  Doing  so,  we  obtain  a 
curve  (Fig.  500)  closely  resembling  what  the  botanists  call  a  reniform 
(or  kidney-shaped)  leaf,  that  is  to  say,  with  a  cordate  outline  at  the 
base  formed  of  two  "auricles,"  one  on  either  side,  and  then  rounded 

*  Vide  supra,  p.  524. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1045 

off  with  no  projecting  apex*.  The  ground-ivy  and  the  dog-violet 
(Fig.  501)  illustrate  such  a  leaf;  and  sometimes,  as  in  the  violet, 
the  veins  -of  the  leaf  show  similar  curves  congruent  with  the  outer 
edge.  Moreover  the  violet  is  a  good  example  of  how  the  reniform 
leaf  may  be  drawn  out  more  and  more  into  an  acute  and  ovate 
form. 

From  sin^/2  we  may  proceed  to  any  other  given  fraction  of  ^, 
and  plot,  for  instance,  r  =  sin  5^/3,  as  in  Fig.  502 ;  which  now  no 
longer  represents  a  single  leaf  but  has  become  a  diagram  of  the 


Fig.  501.     Violet  leaf. 


Fig.  500.     Curve  resembling  the  out- 
line of  a  reniform  leaf:  r=:8in^/2. 

five  petals  of  a  pentamerous  flower.  Abbot  Guido  Grandi,  a  Pisan 
mathematician  of  the  early  eighteenth  century,  drew  such  a  curve 
and  pointed  out  its  botanical  analogies ;  and  we  still  call  the  curves 
of  this  family  "Grandi's  curvesj*." 

The  gamopetalous  corolla  is  easily  transferred  to  polar  coordinates, 
in  which  the  radius  vector  riow  consists  of  two  parts,  the  one  a 
constant,  the  other  expressing  the  amplitude  (or  half-amplitude)  of 
the  sine-curve ;  we  may  write  the  formula  r  =  a  +  b  cos  nO.  In 
Fig.  503  w  ^  5 ;  in  this  figure,  if  the  radius  of  the  outermost  circle 
be  taken  as  unity,  the  outer  of  the  two  sinuous  curves  has  a:b  as 

*  Fig.  500  illustrates  the  whole  leaf,  but  only  shows  one-half  of  the  sine-curve. 
The  rest  is  got  by  reflecting  the  moiety  already  drawn  in  the  horizontal  axis 
(^=7r/2). 

t  Dom.  Guido  Grandus,  Flores  geometrici  ex  rhodonearum  et  cloeliarum  curvarum 
descriptione  resultantes . . . ,  Florentiae,  1728.  Cf.  Alfred  Lartigue,  Biodynamique 
generale,  Paris,  1930 — a  curious  but  eccentric  book. 


1046       THE  THEORY  OF  TRANSFORMATIONS        [ch. 
9:1,  and  the  inner  curve  as  3 : 1 ;  while  the  five  petals  become  separate 

when  a  =  b,  and  the  formula  reduces  to  r  =  cos^  —  . 

z 

In  Fig.  504  we  have  what  looks  like  a  first  approximation  to  a 

horse-chestnut  leaf.     It  consists  of  so  many  separate  leaflets,  akin 

to  the  five  petals  in  Fig.  503;    but  these  are  now  inscribed  in  (or 

have  a  locus  m)  the  cordate  or  reniform  outline  of  Fig.  500.     The 

new  curve  is,  in  short,  a  composite  one;   and  its  general  formula  is 

r  =  sin  6 12. sin  nd.    The  small  size  of  the  two  leaflets  adjacent  to 

the  petiole  is  characteristic  of  the  curve,  and  helps  to  explain  the 

development  of  "stipules." 


Fig.  502.  Grandi's  curves  based  on 
r  =  sin5^,  and  illustrating  the  five 
petals  of  a  simple  flower. 


Fig.  503.  Diagram  illustrating  a  corolla 
of  five  petals,  or  of  five  lobes,  are 
based  on  the  equation  r  =  a  +  b  cos  6. 


In  this  last  case  we  have  combined  one  curve  with  another,  and 
the  doing  so  opens  out  a  new  range  of  possibilities.  On  the  outhne 
of  the  simple  leaf,  whether  ovate,  lanceolate  or  cordate,  we  may 
superpose  secondary  sine-curves  of  lesser  period  and  varying  ampli- 
tude, after  the  fashion  of  a  Fourier  series ;  and  the  results  will  vary 
from  a  mere  crenate  outline  to  the  digitate  lobes  of  an  ivy-leaf,  or 
to  separate  leaflets  such  as  we  have  just  studied  in  the  horse-chestnut. 
Or  again,  we  may  inscribe  the  separate  petals  of  Fig.  505  within  a 
spiral  curve,  equable  or  equiangular  as  the  case  may  be ;  and  then, 
continuing  the  series  on  and  on,  we  shall  obtain  a  figure  resembling 
the  clustered  leaves  of  a  stonecrop,  or  the  petals  of  a  water-lily  or 
other  polypetalous  flower. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1047 

Most  of  the  transformations  which  we  have  hitherto  considered 
(other  than  that  of  the  simple  shear)  are  particular  cases  of  a  general 
transformation,  obtainable  by  the  method  of  conjugate  functions 
and  equivalent  to  the  projection  of  the  original  figure  on  a  new 
plane.  Appropriate  transformations,  on  these  general  lines,  provide 
for  the  cases  of  a  coaxial  system  where  the  Cartesian  coordinates 
are  replaced  by  coaxial  circles,  or  a  confocal  system  in  which  they 
are  replaced  by  confocal  ellipses  and  hyperbolas. 


Fig.  504.  Outline  of  a  compound  leaf, 
like  a  horse-chestnut,  based  on  a 
composite  sine-curve,  of  the  form 
r  =  sin  012  .  sin  nd. 


Fig.  505. 


Yet  another  curious  and  important  transformation,  belonging  to 
the  same  class,  is  that  by  which  a  system  of  straight  lines  becomes 
transformed  into  a  conformal  system  of  logarithmic  spirals:  the 
straight  line  Y  —  AX  =  c  corresponding  to  the  logarithmic  spiral 
^  —  ^  log  r  =  c  (Fig.  505).  This  beautiful  and  simple  transforma- 
tion lets  us  at  once  convert,  for  instance,  the  straight  conical  shell 
of  the  Pteropod  or  the  Orthoceras  into  the  logarithmic  spiral  of  the 
Nautiloid;  it  involves  a  mathematical  symbolism  which  is  but  a 
slight  extension  of  that  which  we  have  employed  in  our  elementary 
treatment  of  the  logarithmic  spiral. 

These  various  systems  of  coordinates,  which  we  have  now  briefly 
considered,  are  sometimes  called  "isothermal  coordinates,"  from  the 
fact  that,  when  employed  in  this  particular  branch  of  physics,  they 
perfectly  represent  the  phenomena  of  the  conduction  of  heat,  the 


1048       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

contour  lines  of  equal  temperature  appearing,  under  appropriate 
conditions,  as  the  orthogonal  lines  of  the  coordinate  system.  And 
it  follows  that  the  "law  of  growth"  which  our  biological  analysis 
by  means  of  orthogonal  coordinate  systems  presupposes,  or  at 
least  foreshadows,  is  one  according  to  which  the  organism  grows  or 
develops  along  stream-lines,  which  may  be  defined  by  a  suitable 
mathematical  transformation. 

When  the  system  becomes  no  longer  orthogonal,  as  in  many 
of  the  following  illustrations — for  instance,  that  of  Orthagoriscus 
(Fig.  526) — then  the  transformation  is  no  longer  within  the  reach 
of  comparatively  simple  mathematical  analysis.  Such  departure 
from  the  typical  symmetry  of  a  "stream-line"  system  is,  in  the 
first  instance,  sufficiently  accounted  for  by  the  simple  fact  that 
the  developing  organism  is  very  far  from  being  homogeneous  and 
isotropic,  or,  in  other  words,  does  not  behave  like  a  perfect  fluid. 
But  though  under  such  circumstances  our  coordinate  systems  may 
be  no  longer  capable  of  strict  mathematical  analy'sis,  they  will  still 
indicate  graphically  the  relation  of  the  new  coordinate  system  to 
the  old,  and  conversely  will  furnish  us  with  some  guidance  as  to 
the  "law  of  growth,"  or  play  of  forces,  by  which  the  transformation 
has  been  effected. 

Before  we  pass  from  this  brief  discussion  of  transformations  in 
general,  let  us  glance  at  one  or  two  cases  in  which  the  forces  applied 
are  more  or  less  intelligible,  but  the  resulting  transformations  are, 
from  the  mathematical  point  of  view,  exceedingly  compHcated. 

The  "marbled  papers"  of  the  bookbinder  are  a  beautiful  illustra- 
tion of  visible  "stream-lines."  On  a  dishful  of  a  sort  of  semi-liquid 
gum  the  workman  dusts  a  few  simple  lines  or  patches  of  colouring 
matter;  and  then,  by  passing  a  comb  through  the  Hquid,.he  draws 
the  colour-bands  into  the  streaks,  waves,  and  spirals  which  con- 
stitute the  marbled  pattern,  and  which  he  then  transfers  to  sheets 
of  paper  laid  down  upon  the  gum.  By  some  such  system  of  shears, 
by  the  effect  of  unequal  traction  or  unequal  growth  in  various 
directions  and  superposed  on  an  originally  simple  pattern,  we  may 
account  for  the  not  dissimilar  marbled  patterns  which  we  recognise, 
for  instance,  on  a  large  serpent's  skin.  But  it  must  be  remarked,  in 
the  case  of  the  marbled  paper,  that  though  the  method  of  application 


XVII]     THE  COMPAKISON  OF  RELATED  FORMS     1049 

of  the  forces  is  simple,  yet  in  the  aggregate  the  system  of  forces 
set  up  by  the  many  teeth  of  the  comb  is  exceedingly  complex,  and 
its  complexity  is  revealed  in  the  complicated  "diagram  of  forces" 
which  constitutes  the  pattern.  ' 

To  take  another  and  still  more  instructive  illustration.  To  turn 
one  circle  (or  sphere)  into  two  circles  (or  spheres)  would  be,  from 
the  point  of  view  of  the  mathematician,  an  extraordinarily  difficult 
transformation;  but,  physically  speaking,  its  achievement  may  be 
extremely  simple.  The  little  round  gourd  grows  naturally,  by  its 
symmetrical  forces  of  expansive  growth,  into  a  big,  round,  or  some- 
what oval  pumpkin  or  melon*.  But  the  Moorish  husbandman  ties  a 
rag  round  its  middle,  and  the  same  forces  of  growth,  unaltered  save 
for  the  presence  of  this  trammel,  now  expand  the  globular  structure 
into  two  superposed  and  connected  globes.  And  again,  by  varying 
the  position  of  the  encircling  band,  or  by  applying  several  such 
ligatures  instead  of  one,  a  great  variety  of  artificial  forms  of  "gourd" 
may  be,  and  actually  are,  produced.  It  is  clear,  I  think,  that  we 
may  account  for  many  ordinary  biological  processes  of  development 
or  transformation  of  form  by  the  existence  of  trammels  or  lines 
of  constraint,  which  limit  and  determine  the  action  of  the  expansive 
forces  of  growth  that  would  otherwise  be  uniform  and  symmetrical. 
This  case  has  a  close  parallel  in  the  operations  of  the  glass-blower, 
to  which  we  have  already,  more  than  once,  referred  in  passing  f. 
The  glass-blower  starts  his  operations  with  a  tube,  which  he  first 
closes  at  one  end  so  as  to  form  a  hollow  vesicle,  within  which  his 
blast  of  air  exercises  a  uniform  pressure  on  all  sides ;  but  the  spherical 
conformation  which  this  uniform  expansive  force  would  naturally 
tend  to  produce  is  modified  into  all  kinds  of  forms  by  the  trammels 
or  resistances  set  up  as  the  workman  lets  one  part  or  another  of 
his  bubble  be  unequally  heated  or  cooled.     It  was  Oliver  Wendell 

*  Analogous  structural  differences,  especially  in  the  fibrovascular  bundles,  help 
to  explain  the  differences  between  (e.g.)  a  smooth  melon  and  a  cantelupe,  or  between 
various  elongate,  flattened  and  globular  varieties.  These  breed  true  to  type,  and 
obey,  when  crossed,  the  laws  of  Mendelian  inheritance.  Cf.  E.  W.  Sinnett,  Inherit- 
ance of  fruit-shape  in  Cucurbita,  Botan.  Gazette,  lxxiv,  pp.  95-103,  1922,  and  other 
papers. 

t  Where  gourds  are  common,  the  glass-blower  is  still  apt  to  take  them  for 
a  prototype,  as  the  prehistoric  potter  also  did.  For  instance,  a  tall,  annulated 
Florence  oil-flask  is  an  exact  but  no  longer  a  conscious  imitation  of  a  gourd  which 
has  been  converted  into  a  bottle  in  the  manner  described. 


1050       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

Holmes  who  first  shewed  this  curious  parallel  between  the  operations 
of  the  glass-blower  and  those  of  Nature,  when  she  starts,  as  she  so 
often  does,  with  a  simple  tube*.  The  alimentary  canal,  the  arterial 
system  including  the  heart,  the  central  nervous  system  of  the 
vertebrate,  including  the  brain  itself,  all  begin  as  simple  tubular 
structures.  And  with  them  Nature  does  just  what  the  glass-blower 
does,  and,  we  might  even  say,  no  more  than  he.  For  she  can  expand 
the  tube  here  and  narrow  it  there;  thicken  its  walls  or  thin  them; 
blow  off  a  lateral  offshoot  or  caecal  diverticulum;  bend  the  tube, 
or  twist  and  coil  it;  and  infold  or  crimp  its  walls  as,  so  to  speak, 
she  pleases.  Such  a  form  as  that  of  the  human  stomach  is  easily 
explained  when  it  is  regarded  from  this  point  of  view ;  it  is  simply 
an  ill-blown  bubble,  a  bubble  that  has  been  rendered  lopsided 
by  a  trammel  or  restraint  along  one  side,  such  as  to  prevent  its 
symmetrical  expansion — such  a  trammel  as  is  produced  if  the  glass- 
blower  lets  one  side  of  his  bubble  get  cold,  and  such  as  is  actually 
present  in  the  stomach  itself  in  the  form  of  a  muscular  band. 

The  Florence  flask,  or  any  other  handiwork  of  the  glass-blower, 
is  always  beautiful,  because  its  graded  contours  are,  as  in  its  living 
analogues,  a  picture  of  the  graded  forces  by  which  it  was  conformed. 
It  is  an  example  of  mathematical  beauty,  of  which  the  machine-made, 
moulded  bottle  has  no  trace  at  all.  An  alabaster  bottle  is  different 
again.  It  is  no  longer  an  unduloid  figure  of  equilibrium.  Turned 
on  a  lathe,  it  is  a  solid  of  revolution,  and  not  without  beauty ;  but 
it  is  not  near  so  beautiful  as  the  blown  flask  or  bubble. 

The  gravitational  field  is  part  of  the  complex  field  of  force  by 
which  the  form  of  the  organism  is  influenced  and  determined.  Its 
share  is  seldom  easy  to  define,  but  there  is  a  resultant  due  to  gravity 
in  hanging  breasts  and  tired  eyelids  and  all  the  sagging  wrinkles 
of  the  old.  Now  and  then  we  see  gravity  at  work  in  the  normal 
construction  of  the  body,  and  can  describe  its  effect  on  form  in 
a  general,  or  qualitative,  way.  Each  pair  of  ribs  in  man  forms 
a  hoop  which  droops  of  its  own  weight  in  front,  so  flattening  the 
chest,  and  at  the  same  time  twisting  the  rib  on  either  hand  near  its 
point  of  suspension  f.     But  in  the  dog  each  costal  hoop  is  dragged 

*  Cf.  Elsie  Venner,  chap.  ii. 

t  See  T.  P.  Anderson  Stuart,  How  the  form  of  the  thorax  is  partly  determined 
by  gravitation,  Proc.  R.S.  xlix,  p.l43,  1891. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1051 

straight  downwards,  into  a  vertical  instead  of  a  transverse  ellipse, 
and  is  even  narrowed  to  a  point  at  the  sternal  border. 

We  may  now  proceed  to  consider  and  illustrate  a  few  permutations 
or  transformations  of  organic  form,  out  of  the  vast  multitude  which 
are  equally  open  to  this  method  of  enquiry. 

We  have  already  compared  in  a  preliminary  fashion  the  metacarpal 
or  cannon-bone  of  the  ox,  the  sheep,  and  the  giraffe  (Fig,  493);  and 
we  have  seen  that  the  essential  difference  in  form  between  these 
three  bones  is  a  matter  of  relative  length  and  breadth,  such  that, 
if  we  reduce  the  figures  to  an  identical  standard  of  length  (or  identical 
values  of  y),  the  breadth  (or  value  of  x)  will  be  approximately 
two-thirds  that  of  the  ox  in  the  case  of  the  sheep  and  one-third 
that  of  the  ox  in  the  case  of  the  giraffe.  We  may  easily,  for  the 
sake  of  closer  comparison,  determine  these  ratios  more  accurately, 
for  instance,  if  it  be  our  purpose  to  compare  the  different  racial 
varieties  within  the  limits  of  a  single  species.  And  in  such  cases, 
by  the  way,  as  when  we  compare  with  one  another  various  breeds 
or  races  of  cattle  or  of  horses,  the  ratios  of  length  and  breadth  in 
this  particular  bone  are  extremely  significant*. 

If,  instead  of  limiting  ourselves  to  the  cannon-bone,  we  inscribe 
the  entire  foot  of  our  several  Ungulates  in  a  coordinate  system,  the 
same  ratios  of  x  that  served  us  for  the  cannon-bones  still  give  us 
a  first  approximation  to  the  required  comparison;  but  even  in  the 
case  of  such  closely  allied  forms  as  the  ox  and  the  sheep  there  is 
evidently  something  wanting  in  the  comparison.  The  reason  is  that 
the  relative  elongation  of  the  several  parts,  or  individual  bones,  has 
not  proceeded  equally  or  proportionately  in  all  cases ;  in  other  words, 
that  the  equations  for  x  will  not  suffice  without  some  simultaneous 
modification  of  the  values  of  y  (Fig.  506).  In  such  a  case  it  may  be 
found  possible  to  satisfy  the  varying  values  of  y  by  some  logarithmic 

*  This  significance  is  particularly  remarkable  in  connection  with  the  develop- 
ment of  speed,  for  the  metacarpal  region  is  the  seat  of  very  important  leverage 
in  the  propulsion  of  the  body.  In  a  certain  Scottish  Museum  there  stand  side  by 
side  the  skeleton  of  an  immense  carthorse  (celebrated  for  having  drawn  all  the 
stones  of  the  Bell  Rock  Lighthouse  to  the  shore),  and  a  beautiful  skeleton  of 
a  racehorse,  long  supposed  to  be  the  actual  skeleton  of  Eclipse.  When  I  was 
a  boy  my  grandfather  used  to  point  out  to  me  that  the  cannon-bone  of  the  little 
racer  is  not  only  relatively,  but  actually,  longer  than  that  of  the  great  Clydesdale. 


1052       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

or  other  formula ;  but,  even  if  that  be  possible,  it  will  probably  be 
somewhat  difficult  of  discovery  or  verification  in  such  a  case  as  the 
present,  owing  to  the  fact  that  we  have  too  few  well-marked  points 
of  correspondence  between  the  one  object  and  the  other,  and  that 
especially  along  the  shaft  of  such  long  bones  as  the  cannon-bone 
of  the  ox,  the  deer,  the  llama,  or  the  giraffe  there  is  a  complete  lack 
of  easily  recognisable  corresponding  points.  In  such  a  case  a  brief 
tabular  statement  of  apparently  corresponding  values  of  y,  or  of 
those  obviously  corresponding  values  which  coincide  with  the 
boundaries  of  the  several*  bones  of  the  foot,  will,  as  in  the  following 
example,  enable  us  to  dispense  with  a  fresh  equation. 


y  (Ox) 

y'  (Sheep) 
y"  (Giraffe) 


b 

27 
19 
10 


42 
36 
24 


100 
100 
100 


This  summary  of  values  of  y' ,  coupled  with  the  equations  for  the 
value  of  X,  will  enable  us,  from  any  drawing  of  the  ox's  foot,  to  con- 
struct a  figure  of  that  of  the  sheep  or  of  the  giraffe  with  remarkable 
accuracy. 


^n 


i 


100 


Sheep  Giraffe 


100 


Fig.  506. 


Fig.  507. 


That  underlying  the  varying  amounts  of  extension  to  which  the 
parts  or  segments  of  the  limb  have  been  subject  there  is  a  law, 
or  principle  of  continuity,  may  be  discerned  from  such  a  diagram 
as  the  above  (Fig.  507),  where  the  values  of  y  in  the  case  of  the  ox 
are  plotted  as  a  straight  line,  and  the  corresponding  values  for  the 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1053 

sheep  (extracted  from  the  above  table)  are  seen  to  form  a  more  or 
less  regular  and  even  curve.  This  simple  graphic  result  implies 
the  existence  of  a  comparatively  simple  equatiqn  between  y 
and  y'. 

An  elementary  application  of  the  principle  of  coordinates  to  the 
study  of  proportion,  as  we  have  here  used  it  to  illustrate  the  varying 
proportions  of  a  bone,  was  in  common  use  in  the  sixteenth  and 
seventeenth  centuries  by  artists  in  their  study  of  the  human  form. 
The  method  is  probably  much  more  ancient,  and  may  even  be 
classical*;  it  is  fully  described  and  put  in  practice  by  Albert  Diirer 
in  his  Geometry,  and  especially  in  his  Treatise  on  Proportio7f\ .  In 
this  latter  work,  the  manner  in  which  the  human  figure,  features, 
and  facial  expression  are  all  transformed  and  modified  by  slight 
variations  in  the  relative  magnitude  of  the  parts  is  admirably  and 
copiously  illustrated  (Fig.  508). 


:^Z3t 


Fig.  508.     (After  Albert  Durer.) 

In  a  tapir's  foot  there  is  a  striking  difference,  and  yet  at  the  same 
time  there  is  an  obvious  underlying  resemblance,  between  the  middle 
toe  and  either  of  its  unsymmetrical  lateral  neighbours.  Let  us 
take  the  median  terminal  phalanx  and  inscribe  its  outline  in  a  net 
of  rectangular  equidistant  coordinates  (Fig.  509,  a).  Let  us  then 
make  a  similar  network  about  axes  which  are  no  longer  at  right 
angles,  but  inclined  to  one  another  at  an  angle  of  about  50°  (h). 


*  Cf.  Vitruvius,  ni,  1. 

t  Les  quatres  livres  d' Albert  Diirer  de  la  proportion  des  parties  et  pourtraicts  des 
corps  humains,  Arnheim,  1613,  folio  (and  earlier  editions) .  Cf.  also  Lavater, 
Essays  on  Physiognomy,  iii,  p.  271,  1799;  also  H.  Meige,  La  geometrie  des  visages 
d'apres  Albert  Diirer,  La  Nature,  Dec.  1927.  On  Diirer  as  mathematician,  cf. 
Cantor,  ii,  p.  459;  S.  Giinther,  Die  geometrische  Ndherungsconstructione  Albrecht 
DUrers,  Ansbach,  1866;   H.  Staigmuller,  Diirer  als  Mathematiker,  Stuttgart,  1891. 


1054      THE  THEORY  OF  TRANSFORMATIONS        [ch. 

If  into  this  new  network  we  fill  in,  point  for  point,  an  outline 
precisely  corresponding  to  our  original  drawing  of  the  middle  toe, 
we  shall  find  that  we  have  already  represented  the  main  features 
of  the  adjacent  lateral  one.  We  shall,  however,  perceive  that  our 
new  diagram  looks  a  httle  too  bulky  on  one  side,  the  inner  side,  of 
the  lateral  toe.     If  now  we  substitute  for  our  equidistant  ordinates, 


' 

2    ( 

\   ^ 

, 

^ 

a 
b 
c 
d 

<' 

K 

•-^ 

N, 

^ 

/ 

\ 

/ 

(^ 

J 

^ 

^^ 

— ' 

Fig.  509. 

ordinates  which  get  gradually  closer  and  closer  together  as  we  pass 
towards  the  median  side  of  the  toe,  then  we  shall  obtain  a  diagram 
which  differs  in  no  essential  respect  from  an  actual  outline  copy 
of  the  lateral  toe  (c).  In  short,  the  difference  between  the  outline 
of  the  middle  toe  of  the  tapir  and  the  next  lateral  toe  may  be  almost 
completely  expressed  by  saying  that  if  the  one  be  represented  by 
rectangular  equidistant  coordinates,  the  other  will  be  represented 
by  oblique  coordinates,  whose  axes  make  an  angle  of  50°,  and  in 


i 


(After  Albert  Diirer.) 

which  the  abscissal  interspaces  decrease  in  a  certain  logarithmic 
ratio.  We  treated  our  original  complex  curve  or  projection  of  the 
tapir's  toe  as  a  function  of  the  form  F  (x,  y)  =  0.  The  figure  of 
the  tapir's  lateral  toe  is  a  precisely  identical  function  of  the  form 
f  (e*,  ^i)  =  0,  where  x^,  2/1  ^^®  oblique  coordinate  axes  inclined  to 
one  another  at  an  angle  of  50°. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1055 

Diirer  was  acquainted  with  these  oblique  coordinates  also,  and  I 
have  copied  two  illustrative  figures  from  his  book*. 

In  Fig.  511  I  have  sketched  the  common  Copepod  Oiihona  nana, 
and  have  inscribed  it  in  a  rectangular  net,  with  abscissae  three-fifths 
the  length  of  the  ordinates.  Side  by  side  (Fig.  512)  is  drawn  a  very 
different  Copepod,  of  the  genus  Sapphirina ;  and  about  it  is  drawn 
a  network  such  that  each  coordinate  passes  (as  nearly  as  possible) 
through  points  corresponding  to  those  of  the  former  figure.  It  will 
be  seen  that  two  differences  are  apparent.  (1)  The  values  of  y 
in  Fig.  512  are  large  in  the  upper  part  of  the  figure,  and  diminish 


Fig.  511.     Oithona  nana. 


Fig.  512.     Sapphirina. 


rapidly  towards  its  base.  (2)  The  values  of  x  are  very  large  in  the 
neighbourhood  of  the  origin,  but  diminish  rapidly  as  we  pass  towards 
either  side,  away  from  the  median  vertical  axis;  and  it  is  probable 
that  they  do  so  according  to  a  definite,  but  somewhat  complicated, 

*  It  was  these  very  drawings  of  Diirer's  that  gave  to  Peter  Camper  his  notion 
of  the  "facial  angle."  Camper's  method  of  comparison  was  the  very  same  as  ours, 
save  that  he  only  drew  the  axes,  without  filling  in  the  network,  of  his  coordinate 
system ;  he  saw  clearly  the  essential  fact,  that  the  skull  varies  as  a  whole,  and  that 
the  "facial  angle"  is  the  index  to  a  general  deformation.  "The  great  object  was  to 
shew  that  natural  differences  might  be  reduced  to  rules,  of  which  the  direction  of 
the  facial  line  forms  the  norma  or  canon ;  and  that  these  directions  and  inclinations 
are  always  accompanied  by  correspondent  form,  size  and  position  of  the  other 
parts  of  the  cranium,"  etc.;  from  Dr  T.  Cogan's  preface  to  Camper's  work  On  the, 
Connexion  between  the  Science  of  Anatomy  and  the  Arts  of  Drawing,  Painting  and 
Sculpture  (1768?),  quoted  in  Dr  R.  Hamilton's  Memoir  of  Camper,  in  lAves  of 
Eminent  Naturalists  (Nat.  Libr.),  Edinburgh,  1840.  See  also  P.  Camper,  Dissertation 
sur  les  differences  reelles  que  presentent  les  Traits  du  Visage  chez  les  hommes  de 
differents  pays  et  de  differents  ages,  Paris,  1791  {op.  posth.);  cf.  P.  Topinard,  ]6tudes 
8ur  Pierre  Camper,  et  sur  Tangle  facial  dit  de  Camper,  Bev.  d'Anthropol.  n.  1874. 


1056       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

ratio.  If,  instead  of  seeking  for  an  actual  equation,  we  simply 
tabulate  our  values  of  x  and  y  in  the  second  figure  as  compared  with 
the  first  (just  as  we  did  in  comparing  the  feet  of  the  Ungulates), 
we  get  the  dimensions  of  a  net  in  which,  by  simply  projecting  the 
figure  of  Oithona,  we  obtain  that  of  Sapphirina  without  further 
trouble,  e.g.: 


X  (Oithona) 

x'  {Sapphirina) 

0 
0 

3 

8 

6 
10 

9 
12 

12 
13. 

15 
14 

— 

y  {Oithona) 

y'  {Sapphirina) 

0 
0 

5- 

2 

10 

7 

15 
3 

20 
23 

25 
32 

30 

40 

In  this  manner,  with  a  single  model  or  type  to  copy  from,  we 
may  record  in  very  brief  space  the  data  requisite  for  the  production 
of  approximate  outlines  of  a  great  number  of  forms.  For  instance, 
the  difference,  at  first  sight  immense,  between  the  attenuated  body 
of  a  Caprella  and  the  thick-set  body  of  a  Cyamus  is  obviously  httle, 
and  is  probably  nothing  more  than  a  difference  of  relative  mag- 
nitudes, capable  of  tabulation  by  numbers  and  of  complete  expression 
by  means  of  rectihnear  coordinates. 

The  Crustacea  afford  innumerable  instances  of  more  complex 
deformations.  Thus  we  may  compare  various  higher  Crustacea 
with  one  another,  even  in  the  case  of  such  dissimilar  forms  as 
a  lobster  and  a  crab.  It  is  obvious  that  the  whole  body  of  the 
former  is  elongated  as  compared  with  the  latter,  and  that  the  crab 
is  relatively  broad  in  the  region  of  the  carapace,  while  it  tapers  off 
rapidly  towards  its  attenuated  and  abbreviated  tail.  In  a  general 
way,  the  elongated  rectangular  system  of  coordinates  in  which  we 
may  inscribe  the  outhne  of  the  lobster  becomes  a  shortened  triangle 
in  the  case  of  the  crab.  In  a  little  more  detail  we  may  compare 
the  outline  of  the  carapace  in  various  crabs  one  with  another :  and 
the  comparison  will  be  found  easy  and  significant,  even,  .in  many 
cases,  down  to  minute  details,  such  as  the  number  and  situation 
of  the  marginal  spines,  though  these  are  in  other  cases  subject  to 
independent  variabihty. 

If  we  choose,  to  begin  with,  such  a  crab  as  Geryon  (Fig.  513,  1) 
and  inscribe  it  in  our  equidistant  rectangular  coordinates,  we  shall 
see  that  we  pass  easily  to  forms  more  elongated  in  a  transverse 
direction,  such  as  Matuta  or  Lupa  (5),  and  conversely,  by  transverse 
compression,  to  such  a  form  as  Corystes  (2).     In  certain  other  cases 


I 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1057 

the  carapace  conforms  to  a  triangular  diagram,  more  or  less  curvi- 
linear, as  in  Fig.  513,  4,  which  represents  the  genus  Paralomis.  Here 
we  can  easily  see  that  the  posterior  border  is  transversely  elongated 
as  compared  with  that  of  Geryon,  while  at  the  same  time  the  anterior 


, ^ 


Fig.  513.     Carapaces  of  various  crabs.     I,  Geryon;  2,  Corystes;  3,  Scyramathia; 
4,  Paralomis;   5,  Lupa;   6,  Chorinus. 

part  is  longitudinally  extended  as  compared  with  the  posterior. 
A  system  of  slightly  curved  and  converging  ordinates,  with  ortho- 
gonal and  logarithmically  interspaced  abscissal  hues,  as  shewn  in 
the  figure,  appears  to  satisfy  the  conditions. 

In  an  interesting  series  of  cases,  such  as  the  genus  Chorinus,  or 
Scyramathia,  and  in  the  spider-crabs  generally,  we  appear  to  have 


1058       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

just  the  Qonverse  of  this.  While  the  carapace  of  these  crabs  presents 
a  somewhat  triangular  form,  which  seems  at  first  sight  more  or  less 
similar  to  those  just  described,  we  soon  see  that  the  actual  posterior 
border  is  now  narrow  instead  of  broad,  the  broadest  part  of  the 
carapace  corresponding  precisely,  not  to  that  which  is  broadest  in 
Paralomis,  but  to  that  which  was  broadest  in  Geryon;  while  the 
most  striking  difference  from  the  latter  Hes  in  an  antero-posterior 
lengthening  of  the  forepart  of  the  carapace,  culminating  in  a  great 
elongation  of  Ihe  frontal  region,  with  its  two  spines  or  "horns." 
The  curved  ordinates  here  converge  posteriorly  and  diverge  widely 
in  front  (Fig.  513,  3  and  6),  while  the  decremental  interspacing  of 
the  abscissae  is  very  marked  indeed. 

We  put  our  method  to  a  severer  test  when  we  attempt  to  sketch 
an  entire  and  compHcated  animal  than  when  we  simply  compare 
corresponding  parts  such  as  the  carapaces  of  various  Malacostraca, 
or  related  bones  as  in  the  case  of  the  tapir's  toes.  Nevertheless,  up 
to  a  certain  point,  the  method  stands  the  test  very  well.  In  other 
words,  one  particular  mode  and  direction  of  variation  is  often  (or 
even  usually)  so  prominent  and  so  paramount  throughout  the  entire 
organism,  that  one  comprehensive  system  of  coordinates  suffices  to 
give  a  fair  picture  of  the  actual  phenomenon.  To  take  another 
illustration  from  the  Crustacea,  I  have  drawn  roughly  in  Fig.  514,  1 
a  little  amphipod  of  the  family  Phoxocephalidae  (Harpinia  sp.). 
Deforming  the  coordinates  of  the  figure  into  the  curved  orthogonal  * 
system  in  Fig.  514,  2,  we  at  once  obtain  a  very  fair  representation  of 
an  alhed  genus,  belonging  to  a  different  family  of  amphipods,  namely 
Stegocephalus.  As  we  proceed  further  from  our  type  our  coordinates 
will  require  greater  deformation,  and  the  resultant  figure  will  usually 
be  somewhat  less  accurate.  In  Fig.  514,  3  I  shew  a  network,  to 
which,  if  we  transfer  our  diagram  of  Harpinia  or  of  Stegocephalus, 
we  shall  obtain  a  tolerable  representation  of  the  aberrant  genus 
Hyperia'f,  with  its  narrow  abdomen,  its  reduced  pleural  lappets,  its 
great  eyes,  and  its  inflated  head. 

*  Similar  coordinates  are  treated  of  by  Lame,  Lemons  sur  les  coordonnees  curvilignes, 
Paris,  1859. 

t  For  an  analogous,  but  more  detailed  comparison,  see  H.  Mogk,  Versuch  einer 
Formanalyse  bei  Hyperiden,  Int.  Rev.  d.  ges.  Hydrobiol.,  etc.,  xiv,  pp.  276-311, 
1923;  xvn,  pp.  1-98,  1926. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1059 

The  hydroid  zoophytes  constitute  a  "polymorphic"  group,  within 
which  a  vast  number  of  species  have  already  been  distinguished; 
and  the  labours  of  the  systematic  naturahst  are  constantly  adding 
to  the  number.  The  specific  distinctions  are  for  the  most  part 
based,  not  upon  characters  directly  presented  by  the  living  animal, 
but  upon  the  .form,  size  and  arrangement  of  the  little  cups,  or 
"calycles,"  secreted  and  inhabited  by  the  little  individual  polyps 


Fig.  514.     1,  Harpinia  plumosa  Kr.;   2,  Stegocephalus  infiatus  Kr.; 
3,  Hyperia  galha. 

which  compose  the  compound  organism.  The  variations,  which  are 
apparently  infinite,  of  these  conformations  are  easily  seen  to  be 
a  question  of  relative  magnitudes,  and  are  capable  of  complete 
expression,  sometimes  by  very  simple, 'sometimes  by  somewhat  more 
complex,  coordinate  networks. 

For  instance,  the  varying  shapes  of  the  simple  wineglass-shaped 
cups  of  the  Campanularidae  are  at  once  sufficiently  represented  and 
compared  by  means  of  simple  Cartesian  coordinates  (Fig.  515).  In 
the  two  allied  famihes  of  Plumulariidae  and  Aglaopheniidae  the 


1060       THE  THEORY  OF  TRANSFOEMATIONS        [ch. 

calycles  are  set  unilaterally  upon  a  jointed  stem,  and  small  cup-like 
structures  (holding  rudimentary  polyps)  are  associated  with  the 
large  calycles  in  definite  number  and  position.     These  small  calycuh 


IVWIl/W 


\ 


^ 


I 


yw\mj 

\-       -4 

\    1 

Ai- 

V\/VTl/V/V/| 

\ 

\ 

\ 

/ 

\ 

\ 

1 

/ 

\ 

L 

• 

ah  c 

Fig.  515.     a,  Campanularia  macroscyphus  Allm.;   b,  Gonothyraea  hyalina 
Hincks;  c,  Clytia  Johnstoni  Alder. 

are  variable  in  number,  but  in  the  great  majority  of  cases  they 
accompany  the  large  calycle  in  groups  of  three — two  standing  by 
its  upper  border,  and  one,  which  is  especially  variable  in  form  and 
magnitude,  lying  at  its  base.     The  stem  is  liable  to  flexure  and. 


Fig.  516.     a,  Cladocarpus  crenatus  F.;   &,  Aglaophenia  plurna  X,.; 
c,  A.  rhynchocarpa  A.;   d,  A.  cornuta  K.;   e,  A.  ramulosa  K. 

in  a  high  degree,  to  extension  or  compression;  and  these  variations 
extend,  often  on  an  exaggerated  scale,  to  the  related  calycles.  As 
a  result  we  find  that  we  can  draw  various  systems  of  curved  or 
sinuous  coordinates,  which  express,  all  but  completely,  the  con- 
figuration of  the  various  hydroids  which  we  inscribe  therein  (Fig.  516). 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1061 

The  comparative  smoothness  of  denticulation  of  the  margin  of  the 
calycle,  and  the  number  of  its  denticles,  constitutes  an  independent 
variation,  and  requires  separate  description;  we  have  already  seen 
(p.  391)  that  this  denticulation  is  in  all  probability  due  to  a  par- 
ticular physical  cause. 

Among  countless  other  invertebrate  animals  which  we  might 
illustrate,  did  space  and  time  permit,  we  should  find  the  bivalve 
molluscs  shewing  certain  things  extremely  well.  If  we  start  with 
a  more  or  less  oblong  shell,  such  as  Anodon  or  Mya  or  Psammobia, 
we  can  see  how  easily  it  may  be  transformed  into  a  more  circular 
or  orbicular,  but  still  closely  related  form;  while  on  the  other  hand 
a  simple  shear  is  well-nigh  all  that  is  needed  to  transform  the 
oblong  Anodon  into  the  triangular,  pointed  Mytilus,  Avicula  or 
Pinna.  Now  suppose  we  draw  the  shell  of  Anodon  in  the  usual 
rectangular  coordinates,  and  deform  this  network  into  the  corre- 
sponding oblique  coordinates  of  Mytilus,  we  may  then  proceed  to 
draw  within  the  same  two  nets  tEe  anatomy  of  the  same  two  molluscs. 
Then  of  the  two  adductor  muscles,  coequal  in  Anodon,  one  becomes 
small,  the  other  large,  when  transferred  to  the  oblique  network  of 
Mytilus ;  at  the  same  time  the  foot  becomes  stunted  and  the  siphonal 
aperture  enlarged.  In  short,  having  "transformed"  one  shell  into 
the  other  we  may  perform  an  identical  transformation  on  their 
contained  anatomy :  and  so  (provided  the  two  are  not  too  distantly 
related)  deduce  the  bodily  structure  of  the  one  from  our  knowledge 
of  the  other,  to  a  first  but  by  no  means  negligible  approximation. 

Among  the  fishes  we  discover  a  great  variety  of  deformations, 
some  of  them  of  a  very  simple  kind,  while  others  are  more  striking 
and  more  unexpected.  A  comparatively  simple  case,  involving  a 
simple  shear,  is  illustrated  by  Figs.  517  and  518.  The  one  represents; 
within  Cartesian  coordinates,  a  certain  little  oceanic  fish  known  as 
Argyropelecus  Olfersi.  The  other  represents  precisely  the  same  out- 
line, transferred  to  a  system  of  oblique  coordinates  whose  axes  are 
inclined  at  an  angle  of  70° ;  but  this  is  now  (as  far  as  can  be  seen  on 
the  scale  of  the  drawing)  a  very  good  figure  of  an  allied  fish,  assigned 
to  a  different  genus,  under  the  name  of  Sternoptyx  diaphana.  The 
deformation  illustrated  by  this  case  of  Argyropelecus  is  precisely 
analogous  to  the  simplest  and  commonest  kind  of  deformation  to 


1062       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

which  fossils  are  subject  (as  we  have  "seen  on  p.  811)  as  the  result 
of  shearing-stresses  in  the  solid  rock. 

Fig.  519  is  an  outUne  diagram  of  a  typical  Scaroid  fish.  Let  us 
deform  its  rectihnear  coordinates  into  a  system  of  (approximately) 
coaxial  circles,  as  in  Fig.  520,  and  then  filling  into  the  new  system. 


Fig.  517.     Argyropelecua  Olfersi. 


Fig.  518.     Stemoptyx  diaphana. 


space  by  space  and  point  by  point,  our  former  diagram  of  Scarus, 
we  obtain  a  very  good  outline  of  an  allied  fish,  belonging  to  a  neigh- 
bouring family,  of  the  genus  Pomacanihus.  This  case  is  all  the  more 
interesting,  because  upon  the  body  of  our  Pomacanihus  there  are 
striking  colour  bands,  which  correspond  in  direction  very  closely 


Fig.  519.     Scarus  sp. 


Fig.  520.     Pomacanihus. 


to  the  lines  of  our  new  curved  ordinates.  In  like  manner,  the  still 
more  bizarre  outUnes  of  other  fi.shes  of  the  same  family  of  Chaetodonts 
will  be  found  to  correspond  to  very  slight  modifications  of  similar 
coordinates;  in  other  words,  to  small  variations  in  the  values  of 
the  constants  of  the  coaxial  curves. 

In  Figs.  521-524  I  have  represented  another  series  of  Acantho- 
pterygian  fishes,  not  very  distantly  related  to  the  foregoing.     If  we 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1063 

start  this  series  with  the  figure  of  Polyprion,  in  Fig.  521,  we  see  that 
the  outlines  of  Pseudopriacanthus  (Fig.  522)  and  of  Sebastes  or 
Scorpaena  (Fig.  523)  are  easily  derived  by  substituting  a  system 


Fig.  521.     Polyprion. 


Fig.  522.     Pseudopriacanthus  alius. 


of  triangular,  or  radial,  coordinates  for  the  rectangular  ones  in  which 
we  had  inscribed  Polyprion.  The  very  curious  fish  Antigonia  capros, 
an  oceanic  relative  of  our  own  boar-fish,  <Jonforms. closely  to  the 
peculiar  deformation  represented  in  Fig.  524. 


Fig.  523.     Scorpaena  sp. 


Fig.  524.     Antigonia  capros. 


Fig.  525  is  a  common,  typical  Diodon  or  porcupine-fish,  and  in 
Fig.  526  I  have  deformed  its  vertical  coordinates  into  a  system  of 
concentric  circles,  and  its  horizontal  coordinates  into  a  system  of 
curves  which,  approximately  and  provisionally,  are  made  to  resemble 


1064      THE  THEORY  OF  TRANSFORMATIONS        [ch. 

a  system  of  hyperbolas  *.  The  old  outline,  transferred  in  its  integrity 
to  the  new  network,  appears  as  a  manifest  representation  of  the 
closely  alUed,  but  very  different  looking,  sunfish,  Orthagoriscus  mola. 
This  is  a  particularly  instructive  case  of  deformation  or  transforma- 
tion. It  is  true  that,  in  a  mathematical  sense,  it  is  not  a  perfectly 
satisfactory  or  perfectly  regular  deformation,  for  the  system  is  no 


Fig.  525.     Diodon. 


Fig,  526.     Orthagoriscus. 


longer  isogonal;  but  nevertheless,  it  is  symmetrical  to  the  eye,  and 
obviously  approaches  to  an  isogonal  system  under  certain  conditions 
of  friction  or  constraint.  And  as  such  it  accounts,  by  one  single 
integral  transformation,  for  all  the  apparently  separate  and  distinct 
external  differences  between  the  two  fishes.     It  leaves  the  pa-rts 

*  The  coordinate  system  of  Fig.  526  is  somewhat  different  from  that  which 
I  first  drew  and  published.  It  is  not  unlikely  that  further  investigation  will 
further  simplify  the  comparison,  and  shew  it  to  involve  a  still  more  symmetrical 
system. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1065 

near  to  the  origin  of  the  system,  the  whole  region  of  the  head,  the 
opercular  orifice  and  the  pectoral  fin,  practically  unchanged  in  form, 
size  and  position;  and  it  shews  a  greater  and  greater  apparent 
modification  of  size  and  form  as  we  pass  from  the  origin  towards 
the  periphery  of  the  system. 

In  a  word,  it  is  sufficient  to  account  for  the  new  and  striking 
contour  iri  all  its  essential  details,  of  rounded  body,  exaggerated 
dorsal  and  ventral  fins,  and  truncated  tail.  In  like  manner,  and 
using  precisely  the  same  coordinate  networks,  it  appears  to  me 
possible  to  shew  the  relations,  almost  bone  for  bone,  of  the  skeletons 
of  the  two  fishes;  in  other  words,  to  reconstruct  the  skeleton  of 
the  one  from  our  knowledge  of  the  skeleton  of  the  other,  under 
the  guidance  of  the  same  correspondence  as  is  indicated  in  their 
external  configuration. 

The  family  of  the  crocodiles  has  had  a  special  interest  for  the 
evolutionist  ever  since  Huxley  pointed  out  that,  in  a  degree  only 
second  to  the  horse  and  its  ancestors,  it  furnishes  us  with  a  close 
and  almost  unbroken  series  of  transitional  forms,  running  down 
in  continuous  succession  from  one  geological  formation  to  another. 
I  should  be  inclined  to  transpose  this  general  statement  into  other 
terms,  and  to  say  that  the  Crocodilia  constitute  a  case  in  which, 
with  unusually  little  complication  from  the  presence  of  independent 
variants,  the  trend  of  one  particular  mode  of  transformation  is 
visibly  manifested.  If  we  exclude  meanwhile  from  our  comparison 
a  few  of  the  oldest  of  the  crocodiles,  such  as  Belodon,  which  differ 
more  fundamentally  from  the  rest,  w^e  shall  find  a  long  series  of 
genera  in  which  we  can  refer  not  only  the  changing  contours  of  the 
skull,  but  even  the  shape  and  size  of  the  many  constituent  bones 
and  their  intervening  spaces  or  "vacuities,"  to  one  and  the  same 
simple  system  of  transformed  coordinates.  The  manner  in  which 
the  skulls  of  various  Crocodilians  differ  from  one  another  may  be 
sufficiently  illustrated  by  three  or  four  examples. 

Let  us  take  one  of  the  typical  modern  crocodiles  as  our  standard 
of  form,  e.g.  C.  porosus,  and  inscribe  it,  as  in  Fig.  527,  a,  in  the 
usual  Cartesian  coordinates.  By  deforming  the  rectangular  network 
into  a  triangular  system,  with  the  apex  of  the  triangle  a  little  way 
in  front  of  the  snout,  as  in  b,  we  pass  to  such  a  form  as  C.  americanus. 


1066       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

By  an  exaggeration  of  the  same  process  we  at  once  get  an  approxima- 
tion to  the  form  of  one  of  the  sharp-snouted,  or  longirostrine, 
crocodiles,  such  as  the  genus  Tomistoma;  and,  in  the  species  figured, 
the  obUque  position  of  the  orbits,  the  arched  contour  of  the  occipital 
border,  and  certain  other  characters  suggest  a  certain  amount  of 
curvature,  such  as  I  have  represented  in  the  diagram  (Fig.  527,  b), 
on  the  part  of  the  horizontal  coordinates.  In  the  still  more  elongated 
skull  of  such  a  form  as  the  Indian  Gavial,  the  whole  skull  has  under- 
gone a  great  longitudinal  extension,  or,  in  other  words,  the  ratio 
of  x/y  is  greatly  diminished ;  and  this  extension  is  not  uniform,  but 
is  at  a  maximum  in  the  region  of  the  nasal  and  maxillary  bones. 


/^ 

•\ 

1 

\\ 

;) 

N 

r 

K 

\ 

L  I 

m 

'Jr 

^ 

/- 

l^} 

a  b  c 

Fig.  527,     a,  Crocodilus  porosus;  b,  C.  americanus;  c,  Notosuchus  terrestris. 

This  especially  elongated  region  is  at  the  same  time  narrowed  in  an 
exceptional  degree,  and  its  excessive  narrowing  is  represented  by 
a  curvature,  convex  towards  the  median  axis,  on  the- part  of  the 
vertical  ordinates.  Let  us  take  as  a  last  illustration  one  of  the 
Mesozoic  crocodiles,  the  little  Notosuchus,  from  the  Cretaceous  for- 
mation. This  little  crocodile  is  very  different  from  our  type  in  the 
proportions  of  its  skull.  The  region  of  the  snout,  in  front  of  and 
including  the  frontal  bones,  is  greatly  shortened;  from  constituting 
fully  two-thirds  of  the  whole  length  of  the  skull  in  Crocodilus,  it 
now  constitutes  less  than  half,  or,  say,  three-sevenths  of  the  whole ; 
and  the  whole  skull,  and  especially  its  posterior  part,  is  curiously 
compact,  broad,  and  squat.     The  orbit  is  unusually  large.     If  in 


XVII]     THE  COMPARISON  OF  RELATED  FORMS     1067 

the  diagram  of  this  sktill  we  select  a  number  of  points  obviously 
corresponding  to  points  where  our  rectangular  coordinates  intersect 
particular  bones  or  other  recognisable  features  in  our  typical 
crocodile,  we  shall  easily  discover  that  the  Hues  joining  these  points 
in  Notosuchus  fall  into  such  a  coordinate  network  as  that  which 
is  represented  in  Fig.  527,  c.  To  all  intents  and  purposes,  then,  this 
not  very  complex  system,  representing  one  harmonious  "deforma- 
tion," accounts  for  all  the  differences  between  the  two  figures,  and  is 
sufficient  to  enable  one  at  any  time  to  reconstruct  a  detailed  drawing, 
bone  for  bone,  of  the  skull  of  Notosuchus  from  the  model  furnished 
by  the  common  crocodile. 


,^ 

^ 

1 

y 

^^ 

\lj 

V 

A 

rx^ 

^^ 

P^ 

L/ 

/. 

^ — \ — ■ 

C/ 

1 

Fig.  528.     Pelvis  of  (A)  Stegosaurus;   (B)  Camptosaurus. 

The  many  diverse  forms  of  Dinosaurian  reptiles,  all  of  which 
manifest  a  strong  family  likeness  underlying  much  superficial 
diversity,  furnish  us  with  plentiful  material  for  comparison  by  the 
method  of  transformations.  As  an  instance,  I  have  figured  the 
pelvic  bones  of  Stegosaurus  and  of  Camptosaurus  (Fig.  528,  a,  b)  to 
shew  that,  when  the  former  is  taken  as  our  Cartesian  type,  a  slight 
curvature  and  an  approximately  logarithmic  extension  of  the  x-axis, 
brings  us  easily  to  the  configuration  of  the  other.  In  the  original 
specimen  of  Camptosaurus  described  by  Marsh*,  the  anterior  portion 
of  the  iliac  bone  is  missing;  and  in  Marsh's  restoration  this  part 
of  the  bone  is  drawn  as  though  it  came  somewhat  abruptly  to 
a  sharp  point.     In  my  figure  I  have  completed  this  missing  part 

*  Dinosaurs  of  North  America,  pi.  Lxxxi,  etc.,  1896. 


1068       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

of  the  bone  in  harmony  with  the  general  coordinate  network  which 
is  suggested  by  our  comparison  of  the  two  entire  pelves;  and  I 
venture  to  think  that  the  result  is  more  natural  in  appearance,  and 
more  hkely  to  be  correct  than  was  Marsh's  conjectural  restoration. 
It  would  seem,  in  fact,  that  there  is  an  obvious  field  for  the  employ- 
ment of  the  method  of  coordinates  in  this  task  of  reproducing  missing 

2        I        0 


Fig.  529.     Shoulder-girdle  of  Cryptocleidus.     a,  young;   6,  adult. 

portions  of  a  structure  to  the  proper  scale  and  in  harmony  with 
related  types.     To  this  subject  we  shall  presently  return. 

In  Fig.  529,  a,  b,  I  have  drawn  the  shoulder-girdle  of  Cryptocleidus, 
a  Plesiosaurian  reptile,  half-grown  in  the  one  case  and  full-grown 
in  the  other.  The  change  of  form  during  growth  in  this  region  of 
the  body  is  very  considerable,  and  its  nature  is  well  brought  out 

l^ --^,0 

2. 


Fig.  530.     Shoulder-girdle  of  Ichthyosaurus. 

by  the  two  coordinate  systems.  In  Fig.  530  I  have  drawn  the 
shoulder-girdle  of  an  Ichthyosaur,  referring  it  to  Cryptocleidus  as 
a  standard  of  comparison.  The  interclavicle,  which  is  present  in 
Ichthyosaurus,  is  minute  and  hidden  in  Cryptocleidus]  but  the 
numerous  other  diiferences  between  the  two  forms,  chief  among 
which  is  the  great  elongation  in  Ichthyosaurus  of  the  two  clavicles, 
are  all  seen  by  our  diagrams  to  be  part  and  parcel  of  one  general 
and  systematic  deformation. 


XVII]    THE  COMPARISON  DF  RELATED  FORMS     1069 

Before  we  leave  the  group  of  reptiles  we  may  glance  at  the  very 
strangely  modified  skull  of  Pteranodon,  one  of  the  extinct  flying 
reptiles,  or  Pterosauria.  In  this  very  curious  skull  the  region  of 
the  jaws,  or  beak,  is  greatly  elongated  and  pointed;  the  occipital 
bone  is  drawn  out  into  an  enormous  backwardly  directed  crest ;  the 
posterior  part  of  the  lower  jaw  is  similarly  produced  backwards; 
the  orbit  is  small;  and  the  quadrate  bone  is  strongly  inclined  down- 
wards and  forwards.  The  whole  skull  has  a  configuration  which 
stands,  apparently,  in  the  strongest  possible  contrast  to  that  of 
a  more  normal  Ornithosaurian  such  as  Dimorphodon.  But  i£  we 
inscribe  the  latter  in  Cartesian  coordinates  (Fig.  531,  a),  and  refer 


/ 

-:^ 

^H 

A   ( 

1 

r 

/ 

1 

r-^-i 

// 

y^ 

Qf^ 

\ 

Fig.  531.    a,  skull  of,  Dimorphodon;  b,  skull  of  Pteranodon. 

our  Pteranodon  to  a  system  of  oblique  coordinates  (6),  in  which  the 
two  coordinate  systems  of  parallel  lines  become  each  a  pencil  of 
diverging  rays,  we  make  manifest  a  correspondence  which  extends 
uniformly  throughout  all  parts  of  these  very  different-looking  skulls. 


We  have  dealt  so  far,  and  for  the  most  part  we  shall  continue 
to  deal,  with  our  coordinate  method  as  a  means  of  comparing  one 
known  structure  with  another.  But  it  is  obvious,  as  I  have  said, 
that  it  may  also  be  employed  for  drawing  hypothetical  structures, 
on  the  assumption  that  they  have  varied  from  a  known  form  in 
some  definite  way.  And  this  process  may  be  especially  useful,  and 
will  be  most  obviously  legitimate,  when  we  apply  it  to  the  particular 
case  of  representing  intermediate  stages  between  two  forms  which 


1070       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

are  actually  known  to  exist,  in  other  words,  of  reconstructing  the 
transitional  stages  through  which  the  course  of  evolution  must  have 
successively  travelled  if  it  has  brought  about  the  change  from  some 
ancestral  type  to  its  presumed  descendant.  Some  years  ago 
I  sent  my  friend,  Mr  Gerhard  Heilmann  of  Copenhagen,  a  few  of 


1        3       •; 

•>          6 

— . 

1 

1 

i  a— 

^1^ 

\^  \ 

J 

i 

i 

■     V 

Vi 

(" 

■ 

^AJ 

A 

^ 

/ 

y 

!  / 

y 

T 

1^ 

'/ 

33*S6         7e<f 


Fig.  5,32.     Pelvis  of  Archaeopteryx. 

my  own  rough  coordinate  diagrams,  including  some  in  which  the 
pelves  of  certain  ancient  and  primitive  birds  were  compared  one 
with  another.  Mr  Heilmann,  who  is  both  a  skilled  draughtsman 
and  an  able  morphologist,  returned  me  a  set  of  diagrams  which  are 


Fig.  533.     Pelvis  of  Apatornis. 

a  vast  improvement  on  my  own,  and  which  are  reproduced  in 
Figs.  532-537.  Here  we  have,  as  extreme  cases,  the  pelvis  of 
Archaeopteryx,  the  most  ancient  of  known  birds,  and  that  of  Apa- 
tornis, one  of  the  fossil  "toothed"  birds  from  the  North  American 
Cretaceous  formations — a  bird  shewing  some  resemblance  to  the 
modern  terns.  The  pelvis  of  Arckaeopteryx  is  taken  as  our  type, 
and  referred  accordingly  to  Cartesian  coordinates  (Fig.  532) ;   while 


XVII]     THE  COMPARISON  OF  ITELATED  FORMS     1071 

the  corresponding  coordinates  of  the  very  different  pelvis  of  Apatornis 
are  represented  in  Fig.  533.     In  Fig.  534  the  outUnes  of  these  two 


6         7         8         cj 

Fig.  534.     The  coordinate  systems  of  Figs.  532  and  533,  with  three 
intermediate  systems  interpolated. 

coordinate  systems  are  superposed  upon  one  another,  and  those 
of  three  intermediate  and  equidistant  coordinate  systems  are 
interpolated  between  them.     From  each  of  these  latter  systems, 


/      7     i    '-      s    6      r     9      9 
Fig.  535.     The  first  intermediate  coordinate  network,  with  its 
corresponding  inscribed  pelvis. 

SO  determined  by  diredt  interpolation,  a  complete  coordinate  diagram 
is  draw^n,  and  the  corresponding  outline  of  a  pelvis  is  found  from 
each  of  these  systems  of  coordinates,  as  in  Figs.  535,  536.     Finally, 


1072       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

in  Fig.  537  the  complete  series  is  represented,  beginning  witli  the 
known  pelvis  of  Archaeopteryx,  and  leading  up  by  our  three  inter- 
mediate hj^othetical  types  to  the  known  pelvis  of  Apatornis. 

Among  mammalian  skulls  I  will  take  two  illustrations  only,  one 
drawn  from  a  comparison  of  the  human  skull  with  that  of  the 
higher  apes,  and  another  from  the  group  of  Perissodactyle  Ungulates, 
the  group  which  includes  the  rhinoceros,  the  tapir,  and  the  horse. 


Fig.  536.     The  second  and  third  intermediate  coordinate  networks, 
with  their  corresponding  inscribed  pelves. 

Let  US  begin  by  choosing  as  our  type  the  skull  of  Hyrachyus 
agrarius  Cope,  from  the  Middle  Eocene  of  North  America,  as  figured 
by  Osborn  in  his  Monograph  of  the  Extinct  Rhinoceroses*  (Fig.  538). 

The  many  other  forms  of  primitive  rhinoceros  described  in 
the  monograph  differ  from  Hyrachyus  in  various  details — in  the 
characters  of  the  teeth,  sometimes  in  the  number  of  the  toes,  and 
so  forth;  and  they  also  differ  very  considerably  in  the  general 
appearance  of  the  skull.  But  these  differences  in  the  conformation 
*  Mem.  Amer.  Mies,  of  Nat.  Hist,  i,  in,  1898. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1073 

of  the  skull,  conspicuous  as  they  are  at  first  sight,  will  be  found 
easy  to  bring  under  the  conception  of  a  simple  and  homogeneous 
transformation,  such  as  would  result  from  the  application  of  some  not 
very  comphcated  stress.  For  instance,  the  corresponding  coordinates 
of  Aceratherium  tridactylum,  as  shewn  in  Fig.  539,  indicate  that  the 


Fig,  537,     The  pelvis  of  Archaeopteryx  and  of  Apatornis,  with  three 
transitional  types  interpolated  between  them. 

essential  difference  between  this  skull  and  the  former  one  may  be 
summed  up  by  saying  that  the  long  axis  of  the  skull  of  Aceratherium 
has  undergone  a  slight  double  curvature,  while  the  upper  parts  of 
the  skull  have  at  the  same  time  been  subject  to  a  vertical  expansion, 
or  to  growth  in  somewhat  greater  proportion  than  the  lower  parts. 
Precisely  the  same  changes,  on  a  somewhat  greater  scale,  give  us 
the  skull  of  an  existing  rhinoceros. 


1074       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

Among  the  species  of  Aceratherium,  the  posterior,  or  occipital, 
view  of  the  skull  presents  specific  differences  which  are  perhaps 
more  conspicuous  than  those  furnished  by  the  side  view;  and  these 


I 


Fig.  538,     Skull  of  Hyrachyus  agmrius.     After  Osborn. 

differences  are  very  strikingly  brought  out  by  the  series  of  conformal 
transformations  which  I  have  represented  in  Fig.  540.  In  this  case 
it  will  perhaps  be  noticed  that  the  correspondence  is  not  always 
quite  accurate  in  small  details.     It  could  easily  have  been  made 


Fig.  539.     Skull  of  Aceratherium  tridactylum.     After  Osborn. 

much  more  accurate  by  giving  a  slightly  sinuous  curvature  to  certain 
of  the  coordinates.  But  as  they  stand,  the  correspondence  indicated 
is  very  close,  and  the  simplicity  of  the  figures  illustrates  all  the 
better  the  general  character  of  the  transformation. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1075 

By  similar  and  not  more  violent  changes  we  pass  easily  to  suca 
allied  forms  as  the  Titanotheres  (Fig,  541);  and  the  well-known 
series  of  species  of  Titanotherium,  by  which  Professor  Osborn  has 


JFig.  540.     Occipital  view  of  the  skulls  of  various  extinct  rhinoceroses 
(Aceratherium  spp.).     After  Osborn. 

illustrated  the  evolution  of  this  genus,  constitutes  a  simple  and 
suitable  case  for  the  application  of  our  method. 

But  our  method  enables  us  to.  pass  over  greater  gaps  than  these, 
and  to  discern  the  general,  and  to  a  very  large  extent  even  the 


Fig.  541.     Titanotherium  rohiLstum.  Fig.  542.     Tapir's  skull. 

detailed,  resemblances  between  the  skull  of  the  rhinoceros  and  those 
of  the  tapir  or  the  horse.  From  the  Cartesian  coordinates  in  which 
we  have  begun  by  inscribing  the  sfcull  of  a  primitive  rhinoceros, 
we  pass  to  the  tapir's  skull  (Fig.  542),  firstly,  by  converting  the 


1076       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

rectangular  into  a  triangular  ;ietwork,  by  which  we  represent  the 
depression  of  the  anterior  and  the  progressively  increasing  elevation 
of  the  posterior  part  of  the  skull;  and  secondly,  by  giving  to 
the  vertical  ordinates  a  curvature  such  as  to  bring  about  a  certain 
longitudinal  compression,  or  condensation,  in  the  forepart  of  the 
skull,  especially  in  the  nasal  and  orbital  regions. 

The  conformation  of  the  horse's  skull  departs  from  that  of  our 
primitive  Perissodactyle  (that  is  to  say  our  early  type  of  rhinoceros, 
Hyrachyus)  in  a  direction  that  is  nearly  the  opposite  of  that  taken 
by  Titanotherium  and  by  the  recent  species  of  rhinoceros.  For  we 
perceive,  by  Fig.  543,  that  the  horizontal  coordinates,  which  in  these 
latter  cases  become  transformed  into  curves  with  the  concavity 


Fig.  543.     Horse's  skull. 

upwards,  are  curved,  in  the  case  of  the  horse,  in  the  opposite  direc- 
tion. And  the  vertical  ordinates,  which  are  also  curved,  somewhat 
in  the  same  fashion  as  in  the  tapir,  are  very  nearly  equidistant, 
instead  of  being,  as  in  that  animal,  crowded  together  anteriorly. 
Ordinates  and  abscissae  fol-m  an  oblique  system,  as  is  shewn  in  the 
figure.  In  this  case  I  have  attempted  to  produce  the  network 
beyond  the  region  which  is  actually  required  to  include  the  diagram 
of  the  horse's  skull,  in  order  to  shew  better  the  form  of  the  general 
transformation,  with  a  part  only  of  which  we  have  actually  to  deal. 
It  is  at  first  sight  not  a  little  surprising  to  find  that  we  can  pass,  by 
a  cognate  and  even  simpler  transformation,  from  our  Perissodactyle 
skulls  to  that  of  the  rabbit ;  but  the  fact  that  we  can  easily  do  so  is 
a  simple  illustration  of  the  undoubted  affinity  which  exists  between 
the  Rodentia,  especially  the  family  of  the  Leporidae,  and  the  more 
primitive  Ungulates.     For  my  part,  I  would  go  further;  for  I  think 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1077 

there  is  strong  reason  to  believe  that  the  Perissodactyles  are  more 
closely  related  to  the  Leporidae  than  the  former  are  to  the  other 
Ungulates,  or  than  the  Leporidae  are  to  the  rest  of  the  Rodentia. 
Be  that  as  it  may,  it  is  obvious  from  Fig.  544  that  the  rabbit's  skull 


Fig.  544.     Rabbit's  skuU. 

conforms  to  a  system  of  coordinates  corresponding  to  the  Cartesian 
coordinates  in  which  we  have  inscribed  the  skull  of  Hyrachyus,  with 
the  difference,  firstly,  that  the  horizontal  ordinates  of  the  latter  are 


Fig.  545.  A,  outline  diagram  of  the  Cartesian  coordinates  of  the  skull  of  Hyra- 
cotherium  or  Eohippus,  as  shewn  in  Fig.  546,  A.  H,  outline  of  the  corresponding 
projection  of  the  horse's  skull.     B-G,  intermediate,  or  interpolated,  outlines. 

transformed  into  equidistant  curved  lines,  approximately  arcs  of 
circles,  with  their  concavity  directed  downwards ;  and  secondly,  that 
the  vertical  ordinates  are  transformed  into  a  pencil  of  rays  approxi- 


1078       THE  THEORY  OF  TRANSFORMATIONS        [ch. 


9  10 


Fig.  546.     A,  skull  oi  Hymcotherium,  from  the  Eocene,  after  W.  B.  Scott;   H,  skull 
of  horse,  represented  as  a  coordinate  transformation  of  that  of  Hyracotherium, 


xvri]    THE  COMPARISON  OF  RELATED  FORMS     1079 


_2_.i r_.l_   »_   .7_8 


and  to  the  same  scale  of  magnitude;  B-G,  various  artificial  or  imaginary- 
types,  reconstructed  as  intermediate  stages  between  A  and  H;  M,  skull  of 
Mesohippus,  from  the  Oligocene,  after  Scott,  for  comparison  with  C;  P,  skull 
of  Protohippus,  from  the  Miocene,  after  Cope,  for  comparison  with  E;  Pp, 
lower  jaw  of  Protohippus  placidits  (after  Matthew  and  Gidley),  for  comparison 
with  F;  Mi,  Miohippus  (after  Osborn),  Pa,  Parahipptis  (after  Peterson), 
shewing  resemblance,  but  less  perfect  agreement,  with  C  and  D. 


1080      THE  THEORY  OF  TRANSFORMATIONS        [ch 

mately  orthogonal  to  the  circular  arcs.  In  short,  the  configuration 
of  the  rabbit's  skull  is  derived  from  that  of  our  primitive  rhinoceros 
by  the  unexpectedly  simple  process  of  submitting  the  latter  to  a 
strong  and  uniform  flexure  in  the  downward  direction  (cf.  Fig.  538, 
p.  1074).  In  the  case  of  the  rabbit  the  configuration  of  the  individual 
bones  does  not  conform  quite  so  well  to  the  general  transformation 
as  it  does  when  we  are  comparing  the  several  Perissodactyles  one 
with  another;  and  the  chief  departures  from  conformity  will  be 
found  in  the  size  of  the  orbit  and  in  the  outline  of  the  immediately 
surrounding  bones.  The  simple  fact  is  that  the  relatively  enormous 
eye  of  the  rabbit  constitutes  an  independent  variation,  which  cannot 
be  brought  into  the  general  and  fundamental  transformation,  but 
must  be  dealt  with  separately.  The  enlargement  of  the  eye,  like 
the  modification  in  form  and  number  of  the  teeth,  is  a  separate 
phenomenon,  which  supplements  but  in  no  way  contradicts  our 
general  comparison  of  the  skulls  taken  in  their  entirety. 

Before  we  leave  the  Perissodactyla  and  their  allies,  let  us  look 
a  little  more  closely  into  the  case  of  the  horse  and  its  immediate 
relations  or  ancestors,  doing  so  with  the  help  of  a  set  of  diagrams 
which  I  again  owe  to  Mr  Gerard  Heilmann*.  Here  we  start  afresh, 
with  the  skull  (Fig.  546,  A)  of  Hyracotherium  (or  Eohijypus),  inscribed 
in  a  simple  Cartesian  network.  At  the  other  end  of  the  series  (H) 
is  a  skull  of  Equus,  in  its  own  corresponding  network;  and  the 
intermediate  stages  (B-G)  are  all  drawn  by  direct  and  simple  inter- 
polation, as  in  Mr  Heilmann's  former  series  of  drawings  of  Archaeop- 
teryx  and  Apatornis.  In  this  present  case,  the  relative  magnitudes 
are  shewn,,  as  well  as  the  forms,  of  the  several  skulls.  Alongside 
of  these  reconstructed  diagrams  are  set  figures  of  certain  extinct 
"horses"  (Equidae  or  Palaeotheriidae),  and  in  two  cases,  viz.  Meso- 
hippus  and  Protohippus  (M,  P),  it  will  be  seen  that  the  actual 
fossil  skull  coincides  in  the  most  perfect  fashion  with  one  of  the 
hypothetical  forms  or  stages  which  our  method  shews  to  be  implicitly 
involved  in  the  transition  from  Hyracotherium  to  Equus  f.  In  a  third 
case,  that  of  Parahippus  (Pa),  the  correspondence  (as  Mr  Heilmann 

*  These  and  also  other  coordinate  diagrams  will  be  found  in  Mr  G.  Heilmann's 
beautiful  and  original  hook  Fuglenes  Afstamning,  398  pp.,  Copenhagen,  1916;  see 
especially  pp.  368-380. 

t  Cf.  Zittel,  Grundziige  d.  Palaeontologie,  1911,  p.  463. 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1081 

points  out)  is  by  no  means  exact.  The  outline  of  this  skull  comes 
nearest  to  that  of  the  hypothetical  transition  stage  D,  but  the  "fit" 
is  now  a  bad  one ;  for  the  skull  of  Parahippus  is  evidently  a  longer, 
straighter  and  narrower  skull,  and  differs  in  other  minor  characters 
besides.  In  short,  though  some  writers  have  placed  Parahippus  in 
the  direct  line  of  descent  between  Equus  and  Eohippus,  we  see  at 
once  that  there  is  no  place  for  it  there,  and  that  it  must,  accordingly, 
represent  a  somewhat  divergent  branch  or  offshoot  of  the  Equidae*. 
It  .may  be  noticed,  especially  in  the  case  of  Protohippus  (P),  that 
the  configuration  of  the  angle  of  the  jaw  does  not  tally  quite  so 
accurately  with  that  of  our  hypothetical  diagrams  as  do  other  parts 
of  the  skull.  As  a  matter  of  fact,  this  region  is  somewhat  variable, 
in  different  species  of  a  genus,  and  even  in  different  individuals  of 
the  same  species;  in  the  small  figure  (Pp)  of  Protohippus  placidus 
the  correspondence  is  more  exact. 

In  considering  this  series  of  figures  we  cannot  but  be  struck, 
not  only  with  the  regularity  of  the  succession  of  "transformations," 
but  also  with  the  slight  and  inconsiderable  differences  which  separate 
each  recorded  stage  from  the  next,  and  even  the  two  extremes  of 
the  whole  series  from  one  another.  These  differences  are  no  greater 
(save  in  regard  to  actual  magnitude)  than  those  between  one  human 
skull  and  another,  at  least  if  we  take  into  account  the  older  or 
remoter  races ;  and  they  are  again  no  greater,  but  if  anything  less, 
than  the  range  of  variation,  racial  and  individual,  in  certain  other 
human  bones,  for  instance  the  scapula  f. 

The  variability  of  this  latter  bone  is  great,  but  it  is  neither  sur- 
prising nor  peculiar;   for  it  is  linked  with  all  the  considerations  of 

*  Cf.  W.  B.  Scott  (Amer.  Journ.  of  Science,  XLVin,  pp.  335-374,  1894),  "We 
find  that  any  mammalian  series  at  all  cojnplete,  such  as  that  of  the  horses,  is 
remarkably  continuous,  and  that  the  progress  of  discovery  is  steadily  filling  up 
what  few  gaps  remain.  So  closely  do  successive  stages  follow  upon  one  another 
that  it  is  sometimes  extremely  difficult  to  arrange  them  all  in  order,  and  to 
distinguish  clearly  those  members  which  belong  in  the  main  line  of  descent,  and 
those  which  represent  incipient  branches."  Some  phylogenies  actually  suffer  from 
an  embarrassment  of  riches." 

t  Cf.  T.  Dwight,  The  range  of  variation  of  the  human  scapula,  Amer.  Nat. 
XXI,  pp.  627-638,  1887.  Cf.  also  Turner,  Challenger  Rep.  xlvii,  on  Human  Skele- 
tons, p.  86,  1886:  "I  gather  both  from  my  own  measurements,  and  those  of  other 
observers,  that  the  range  of  variation  in  the  relative  length  and  breadth  of  the 
scapula  is  very  considerable  in  the  same  race,  so  that  it  needs  a  large  number  of 
bones  to  enable  one  to  obtain  an  accurate  idea  of  the  mean  of  the  race." 


1082       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

mechanical  efficiency  and  functional  modification  which  we  dealt 
with  in  our  last  chapter.  The  scapula  occupies,  as  it  were,  a  focus 
in'a  very  important  field  of  force ;  and  the  lines  of  force  converging 
on  it  will  be  very  greatly  modified  by  the  varying  development  of 


Fig.  547.     Human  scapulae  (after  D wight).     A,  Caucasian;    B,  Negro; 
C,  North  American  Indian  (from  Kentucky  Mountains). 

the  muscles  over  a  large  area  of  the  body  and  of  the  uses  to  which 
they  are  habitually  put. 

Let  us  now  inscribe  in  our  Cartesian  coordinates  the  outline  of 
a  human  skull  (Fig.  548),  for  the  purpose  of  comparing  it  with  the 
skulls  of  some  of  the  higher  apes.  We  know  beforehand  that  the 
main  differences  between  the  human  and  the  simian  types  depend 


0  12         3  4         5 

Fig.  548.     Human  skull. 

upon  the  enlargement  or  expansion  of  the  brain  and  braincase  in 
man,  and  the  relative  diminution  or  enfeeblement  of  his  jaws. 
Together  with  these  changes,  the  "facial  angle"  increases  from  an 
oblique  angle  to  nearly  a  right  angle  in  man,  and  the  configuration 
of  every  constituent  bone  of  the  face  and  skull  undergoes  an  altera- 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1083 

tion.  We  do  not  know  to  begin  with,  and  we  are  not  shewn  by  the 
ordinary  methods  of  comparison,  how  far  these  various  changes 
form  part  of  one  harmonious  and  congruent  transformation,  or 
whether  we  are  to  look,  for  instance,  upon  the  changes  undergone 
by  the  frontal,  the  occipital,  the  maxillary,  and  the  mandibular 


Fig.  549.     Coordinates  of  chimpanzee's  skull,  as  a  projection  of 
the  Cartesian  coordinates  of  Fig.  548. 

regions  as  a  congeries  of  separate  modifications  or  independent 
variants.  But  as  soon  as  we  have  marked  out  a  number  of  points 
in  the  gorilla's  or  chimpanzee's  skull,  corresponding  with  those  which 
our  coordinate  network  intersected  in  the  human  skull,  we  find  that 
these  corresponding  points  may  be  at  once  linked  up  by  smoothly 
curved  lines  of  intersection,  which  form  a  new  system  of  coordinates 


Fig.  550.     Skull  of  chimpanzee. 


Fig.  551. 


of  baboon. 


and  constitute  a  simple  "projectipn"  of  our  human  skull.  The 
network  represented  in  Fig.  549  constitutes  such  a  projection  of 
the  human  skull  on  what  we  may  call,  figuratively  speaking,  the 
"plane"  of  the  chimpanzee;  and  the  full  diagram  in  Fig.  550 
demonstrates  the  correspondence.  In  Fig.  551  I  have  shewn  the 
similar  deformation  in  the  case  of  a  baboon,  and  it  is  obvious  that 
the  transformation  is  of  precisely  the  same  order,  and  differs  only 


1084       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

in  an  increased  intensity  or  degree  of  deformation* .  These  anthropoid 
skulls,  then,  which  we  can  transform  one  into  another  by  a  "con- 
tinuous transformation,"  are  admirable  examples  of  what  Listing 
called  "topological  similitude." 

In  both  dimensions,  as  we  pass  from  above  downwards  and  from 
behind  forwards,  the  corresponding  areas  of  the  network  are  seen 
to  increase  in  a  gradual  and  approximately  logarithmic  order  in  the 
lower  as  compared  with  the  higher  type  of  skull;  and,  in  short, 
it  becomes  at  once  manifest  that  the  modifications  of  jaws,  brain-case, 
and  the  regions  between  are  all  portions  of  one  continuous  and 
integral  process.  It  is  of  course  easy  to  draw  the  inverse  diagrams, 
by  which  the  Cartesian  coordinates  of  the  ape  are  transformed  into 
curvilinear  and  non-equidistant  coordinates  in  manf. 

From  this  comparison  of  the  gorilla's  or  chimpanzee's  with  the 
human  skull  we  realise  that  an  inherent  weakness  underlies  the 
anthropologist's  method  of  comparing  skulls  by  reference  to  a  small 
number  of  axes.  The  most  important  of  these  are  the  "facial '"and 
" basicranial "  axes,  which  include  between  them  the  "facial  angle." 
But  it  IS,  in  the  first  place,  evident  that  these  axes  are  merely  the 
principal  axes  of  a  system  of  coordinates,  and  that  their  restricted 
and  isolated  use  neglects  all  that  can  be  learned  from  the  filling  in 
of  the  rest  of  the  coordinate  network.  And,  in  the  second  place,  the 
"facial  axis,"  for  instance,  as  ordinarily  used  m  the  anthropological 
comparison  of  one  human  skull  with  another,  or  of  the  human  skull 
with  the  gorilla's,  is  in  all  cases  treated  as  a  straight  line;  but  our 
investigation  has  shewn  that  rectilinear  axes  only  meet  the  case  in 
the  simplest  and  most  closely  related  transformations ;  and  that,  for 
instance,  in  the  anthropoid  skull  no  rectilinear  axis  is  homologous 

*  The  empirical  coordinates  which  I  have  sketched  in  for  the  chimpanzee  as  a 
conformal  transformation  of  the  Cartesian  coordinates  of  the  human  skull  look  as 
if  they  might  find  their  place  in  an  equipotential  elliptic  field.  They  are  indeed 
closely  analogous  to  some  already  figured  by  MM.  Y.  Ikada  and  M.  Kuwaori, 
Some  conformal  representations  by  means  of  the  elliptic  integrals,  Sci.  Papers 
Inst.  Phys.  Research,  Tokyo,  xxvi,  pp.  208-215,  1936:   e.g.  pi.  xxxi&. 

t  Speaking  of  "diagrams  in  pairs,"  and  doubtless  thinking  of  his  own  "reciprocal 
diagrams,"  Clerk  Maxwell  says  (in  his  article  Diagrams  in  the  Encyclopaedia  Britan- 
nica):  "The  method  in  which  we  simultaneously  contemplate  two  figures,  and 
recognise  a  correspondence  between  certain  points  in  the  one  figure  and  certain 
points  in  the  other,  is  one  of  the  most  powerful  and  fertile  methods  hitherto  known 
in  science. . .  .It  is  sometimes  spoken  of  as  the  method  or  principle  of  duality." 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1085 

with  a  rectilinear  axis  in  a  man's  skull,  but  what  is  a  straight  line 
in  the  one  has  become  a  certain  definite  curve  in  the  other. 

Mr  Heilmann  tells  me  that  he  has  tried,  but  without  success, 
to  obtain  a  transitional  series  between  the  human  skull  and  some 
prehuman,  anthropoid  type,  which  series  (as  in  the  case  of  the 
Equidae)  should  be  found  to  contain  other  known  types  in  direct 
linear  sequence.  It  appears  impossible,  however,  to  obtain  such  a 
series,  or  to  pass  by  successive  and  continuous  gradations  through 
such  forms  as  Mesopithecus,  Pithecanthropus,  Homo  neanderthalensis, 
and  the  lower  or  higher  races  of  modern  man.  The  failure  is  not 
the  fault  of  our  method.  It  merely  indicates  that  no  one  straight 
line  of  descent,  or  of  consecutive  transformation,  exists;  but  on 
the  contrary,  that  among  human  and  anthropoid  tjrpes',  recent  and 
extinct,  we  have  to  do  with  a  complex  problem  of  divergent,  rather 
than  of  continuous,  variation.  And  in  like  manner,  easy  as  it  is  to 
correlate  the  baboon's  and  chimpanzee's  skulls  severally  with  that 
of  man,  and  easy  as  it  is  to  see  that  the  chimpanzee's  skull  is  much 
nearer  to  the  human  type  than  is  the  baboon's,  it  is  also  not  difficult 
to  perceive  that  the  series  is  not,  strictly  speaking,  continuous,  and 
that  neither  of  our  two  apes  lies  precisely  on  the  same  direct  line 
or  sequence  of  deformation  by  which  we  may  hypothetically  connect 
the  other  with  man. 

After  easily  transforming  our  coordinate  diagram  of  the  human 
skull  into  a  corresponding  diagram  of  ape  or  of  baboon,  we  may 
effect  a  further  transformation  of  man  or  monkey  into  dog  no  less 
easily;  and  we  are  thereby  encouraged  to  believe  that  any  two 
mammalian  skulls  may  be  compared  with,  or  transformed  into,  one 
another  by  this  method.  There  is  something,  an  essential  and 
indispensable  something,  which  is  common  to  them  all,  something 
which  is  the  subject  of  all  our  transformations,  and  remains  invariant 
(as  the  mathematicians  say)  under  them  all.  In  these  transforma- 
tions of  ours  every  point  may  change  its  place,  every  line  its 
curvature,  every  area  its  magnitude;  but  on  the  other  hand  every 
point  and  every  hne  continues  to  exist,  and  keeps  its  relative  order 
and  position  throughout  all  distortions  and  transformations.  A  series 
of  points,  a,  6,  c,  along  a  certain  line  persist  as  corresponding  points 
a',  h\  c\  however  the  line  connecting  them  may  lengthen  or  bend; 
and  as  with  points,  so  with  lines,  and  so  also  with  areas.     Ear, 


1086       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

eye  and  nostril,  and  all  the  other  great  landmarks  of  cranial  anatomy, 
not  only  continue  to  exist  but  retain  their  relative  order  and  position 
throughout  all  our  transformations. 

We  can  discover  a  certain  invariance,  somewhat  more  restricted 
than  before,  between  the  mammahan  skull  and  that  of  fowl,  frog 
or  even  herring.  We  have  still  something  common  to  them  all; 
and  using  another  mathematical  term  (somewhat  loosely  perhaps) 
we  may  speak  of  the  discriminant  characters  which  persist  unchanged, 
and  continue  to  form  the  subject  of  our  transformation.  But  the 
method,  far  as  it  goes,  has  its  limitations.  We  cannot  fit  both 
beetle  and  cuttlefish  into  the  same  framework,  however  we  distort 
it;  nor  by  any  coordinate  transformation  can  we  turn  either  of 
them  into  one  another  or  into  the  vertebrate  type.     They  are 


Fig.  552.     Skull  of  dog,  compared  with  the  human  .skull  of  Fig.  548. 

essentially  different;  there  is  nothing  about  them  which  can  be 
legitimately  compared.  Eyes  they  all  have,  and  mouth  and  jaws; 
but  what  we  call  by  these  names  are  no  longer  in  the  same  order 
or  relative  position;  they  are  no  longer  the  same  thing,  there  is  no 
invariant  basis  for  transformation.  The  cuttlefish  eye  seems  as 
perfect,  optically,  as  our  own ;  but  the  lack  of  an  invariant  relation 
of  position  between  them,  or  lack  of  true  homology  between  them 
(as  we  naturalists  say),  is  enough  to  shew  that  they  are  unrelated 
things,  and  have  come  into  existence  independently  of  one  another. 
As  a  final  illustration  I  have  drawn  the  outline  of  a  dog's  skull 
(Fig.  552),  and  inscribed  it  in  a  network  comparable  with  the  Car- 
tesian network  of  the  human  skull  in  Fig.  548.  Here  we  attempt  to 
bridge  over  a  wider  gulf  than  we  have  crossed  in  any  of  our  former 
comparisons.  But,  nevertheless,  it  is  obvious  that  our  method  still 
holds  good,  in  spite  of  the  fact  that  there  are  various  specific 
differences,  s.uch  as  the  open  or  closed  orbit,  etc.,  which  have  to  be 


XVII]    THE  COMPARISON  OF  RELATED   FORMS    1087 

separately  described  and  accounted  for.  We  see  that  the  chief 
essential  differences  in  plan  between  the  dog's  skull  and  the  man's 
lie  in  the  fact  that,  relatively  speaking,  the  former  tapers  away  in 
front,  a  triangular  taking  the  place  of  a  rectangular  conformation; 
secondly,  that,  coincident  with  the  tapering  off,  there  is  a  progressive 
elongation,  or  pulling  out,  of  the  -whole  forepart  of  the  skull;  and 
lastly,  as  a  minor  difference,  that  the  straight  vertical  ordinates  of 
the  human  skull  become  curved,  with  their  convexity  directed  for- 
wards, in  the  dog.  While  the  net  result  is  that  in  the  dog,  just  as 
in  the  chimpanzee,  the  brain-pan  is  smaller  and  the  jaw^s  are  larger 
than  in  man,  it  is  now  conspicuously  evident  that  the  coordinate 
network  of  the  ape  is  by  no  means  intermediate  between  those  which 
fit  the  other  two.  The  mode  of  deformation  is  on  different  Hnes; 
and,  while  it  may  be  correct  to  say  that  the  chimpanzee  and  the 
baboon  are  more  brute-like,  it  would  be  by  no  means  accurate  to 
assert  that  they  are  more  dog-like,  than  man. 

In  this  brief  account  of  coordinate  transformations  and  of  their 
morphological  utility  I  have  dealt  with  plane  coordinates  only,  and 
have  made  no  mention  of  the  less  elementary  subject  of  coordinates 
in  three-dimensional  space.  In  theory  there  is  no  difficulty  what- 
soever in  such  an  extension  of  our  method;  it  is  just  as  easy  to  refer 
the  form  of  our  fish  or  of  our  skull  to  the  rectangular  coordinates 
X,  y,  z,  or  to  the  polar  coordinates  ^,  rj,  ^,  as  it  is  to  refer  their  plane 
projections  to  the  two  axes  to  which  our  investigation  has  been 
confined.  And  that  it  w^ould  be  advantageous  to  do  so  goes  w^ithout 
saying,  for  it  is  the  shape  of  the  sohd  object,  not  that  of  the  mere 
drawing  of  the  object,  that  we  want  to  understand;  and  already 
we  have  found  some  of  our  easy  problems  in  solid  geometry  leading 
us  (as  in  the  case  of  the  form  of  the  bivalve  and  even  of  the  univalve 
shell)  quickly  in  the  direction  of  coordinate  analysis  and  the  theory 
of  conformal  transformations.  But  this  extended  theme  I  have  not 
attempted  to  pursue,  and  it  must  be  left  to  other  times,  and  to  other 
hands.  Nevertheless,  let  us  glance  for  a  moment  at  the  sort  of  simple 
cases,  the  simplest  possible  cases,  with  which  such  an  investigation 
might  begin;  and  we  have  found  our  plane  coordinate  systems  so 
easily  and  effectively  applicable  to  certain  fishes  that  we  may  seek 
among  them  for  our  first  and  tentative  introduction  to  the  three- 
dimensional  field. 


1088       THE  THEORY  OF  TRANSFORMATIONS        [ch. 

It  is  obvious  enough  that  the  same  method  of  description  and 
analysis  which  we  have  appUed  to  one  plane,  we  may  apply  to 
another:  drawing  by  observation,  and  by  a  process  of  trial  and 
error,  our  various  cross-sections  arid  the  coordinate  systems  which 
seem  best  to  correspond.  But  the  new  and  important  problem 
which  now  emerges  is  to  correlate  the  deformation  or  transformation 
which  we  discover  in  one  plane  with  that  which  we  have  observed  in 
another:  and  at  length,  perhaps,  after  grasping  the  general  prin- 
ciples of  such  correlation,  to  forecast  approximately  what  is  likely 
to  take  place  in  the  third  dimension  when  we  are  acquainted  with 
two,  that  is  to  say,  to  determine  the  values  along  one  axis  in  terms 
of  the  other  two. 

Let  us  imagine  a  common  "round"  fish,  and  a  common  "flat" 
fish,  such  as  a  haddock  and  a  plaice.  These  two  fishes  are  not  as 
nicely  adapted  for  comparison  by  means  of  plane  coordinates  as 
some  which  we  have  studied,  owing  to  the  presence  of  essentially 
unimportant,  but  yet  conspicuous  differences  in  the  position  of  the 
eyes,  or  in  the  number  of  the  fins — that  is  to  say  in  the  manner  in 
which  the  continuous  dorsal  fin  of  the  plaice  appears  in  the  haddock 
to  be  cut  or  scolloped  into  a  number  of  separate  fins.  But  speaking 
broadly,  and  apart  from  such  minor  differences  as  these,  it  is  manifest 
that  the  chief  factor  in  the  case  (so  far  as  we  at  present  see)  is  simply 
the  broadening  out  of  the  plaice's  body,  as  compared  with  the 
haddock's,  in  the  dorso-ventral  direction,  that  is  to  say,  along  the 
y  axis;  in  other  words,  the  ratio  x/y  is  much  less  (and  indeed  little 
more  than  half  as  great)  in  the  haddock  than  in  the  plaice.  But 
we  also  recognise  at  once  that  while  the  plaice  (as  compared  with 
the  haddock)  is  expanded  in  one  direction,  it  is  also  flattened,  or 
thinned  out,  in  the  other:  y  increases,  but  z  diminishes,  relatively 
to  X.  And  furthermore,  we  soon  see  that  this  is  a  common  or  even 
a  general  phenomenon.  The  high,  expanded  body  in  our  Antigonia 
or  in  our  sun-fish  or  in  a  John  Dory  is  at  the  same  time  flattened 
or  compressed  from  side  to  side,  in  comparison  with  the  related 
fishes  which  we  have  chosen  as  standards  of  reference  or  comparison ; 
and  conversely,  such  a  fish  as  the  skate,  while  it  is  expanded  from 
side  to  side  in  comparison  with  a  shark  or  dogfish,  is  at  the  same 
time  flattened  or  depressed  in  its  vertical  section.  We  hasten  to 
enquire  whether  there  be  any  simple  relation  of  magnitude  dis- 


XVII]    THE  COMPARISON  OF  RELATED  FORMS     1089 

cernible  between  these  twin  factors  of  expansion  and  compression; 
and  the  very  fact  that  the  two  dimensions  of  breadth  and  depth 
tend  to  vary  inversely  assures  us  that,  in  the  general  process 
of  deformation,  the  volume  and  the  area  of  cross-section  are  less 
affected  than  are  those  two  linear  dimensions.  Some  years  ago, 
when  I  was  studying  the  weight-length  coefficient  in  fishes  (of  which 
we  have  already  spoken  in  chapter  iii),  that  is  to  say  the  coefficient 
k  in  the  formula  W  =  kL^,  I  was  not  a  Httle  surprised  to  find  that  k 
(let  us  call  it  in  this  case  ki)  was  all  but  identical  in  two  such  different 
looking  fishes  as  the  haddock  and  the  plaice:  thus  indicating  that 
these  two  fishes  have  approximately  the  same  voluyne  when  they 
are  equal  in  length ;  or,  in  other  words,  that  the  extent  to  which  the 
plaice  has  broadened  is  just  about  compensated  for  by  the  extent 
to  which  it  has  also  got  flattened  or  thinned.  In  short,  if  we  might 
conceive  of  a  haddock  being  transformed  directly  into  a  plaice, 
a  very  large  part  of  the  change  would  be  accounted  for  by  supposing 
the  round  fish  to  be  "rolled  out"  into  the  flat  one,  as  a  baker  rolls 
a  piece  of  dough.  This  is,  as  it  were,  an  extreme  case  of  the  halance- 
ment  des  organes,  or  "compensation  of  parts." 

We  must  not  forget,  while  we  consider  the  "deformation"  of 
a  fish,  that  the  fish,  like  the  bird,  is  subject  to  certain  strict  limita- 
tions of  form.  What  we  happen  to  have  found  in  a  particular 
case  was  observed  fifty  yearg  ago,  and  brought  under  a  general  rule, 
by  a  naval  engineer  studying  fishes  from  the  shipbuilder's  point  of 
view.  Mr  Parsons  *  compared  the  contours  and  the  sectional  areas 
of  a  number  of  fishes  and  of  several  whales ;  and  he  found  the  sec- 
tional areas  to  be  always  very  much  the  same  at  the  same  proportional 
distances  from  the  front  end  of  the  body|.  Increase  in  depth  was 
balanced  (as  we  also  have  found)  by  diminution  of  breadth;  and 
the  magnitude  of  the  "entering  angle"  presented  to  the  water  by 
the  advancing  fish  was  fairly  constant.  Moreover,  according  to 
Parsons,  the  position  of  the  greatest  cross-section  is  fixed  for  all 
species,  being  situated  at  36  per  cent,  of  the  length  behind  the 

*  H.  de  B.  Parsons,  Displacements  and  area-curves  of  fish,  Trans.  Amer.  Soc. 
of  Mechan.  Engineers,  ix,  pp.  679-695,  1888. 

t  That  is  to  say,  if  the  areas  of  cross-section  be  plotted  against  their  distances 
from  the  front  end  of  the  body,  the  results  are  very  much  alike  for  all  the  species 
examined.  See  also  Selig  Hecht,  Form  and  growth  in  fishes,  Journ.  Morph. 
xxvn,  pp.  379-400,  1916. 


1090       THE  THEORY  OF  TRANSFORMATIONS       [ch. 

snout.  We  need  not  stop  to  consider  such  extreme  cases  as  the  eel 
or  the  globefish  (Diodon),  whose  ways  of  propulsion  and  locomotion 
are  materially  modified.  But  it  is  certainly  curious  that  no  sooner 
do  we  try  to  correlate  deformation  in  one  direction  with  deformation 
in  another,  than  we  are  led  towards  a  broad  generalisation,  touching 
on  hydrodynamical  conditions  and  the  limitations  of  form  and 
structure  which  are  imposed  thereby. 

Our  simple,  or  simplified,  illustrations  carry  us  but  a  little  way, 
and  only  half  prepare  us  for  much  harder  things.  But  interesting 
as  the  whole  subject  is  we  must  meanwhile  leave  it  alone ;  recognising, 
however,  that  if  the  difiiculties  'of  description  and  representation 
could  be  overcome,  it  is  by  means  of  such  coordinates  in  space  that 
we  should  at  last  obtain  an  adequate  and  satisfying  picture  of  the 
processes  of  deformation  and  the  directions  of  growth. 

A  Note  on  Pattern 

We  have  had  so  much  to  do  with  the  study  of  Form  that  pattern 
has  been  wellnigh  left  out  of  the  account,  although  it  is  part  of 
the  same  story.  Like  any  other  aspect  of  form,  pattern  is  correlated 
with  growth,  and  even  determined  by  it.  A  feather,  for  example, 
which  is  equally  and  equidistantly  striped  to  begin  with,  may  have 
this  simple  striping  transformed  into  a  more  complex  pattern  by 
the  unequal  hut. graded  elongation  of  the  feather.  We  need  not  go 
farther  than  the  zebra  for  a  characteristic  pattern  of  stripes,  nor 
need  we  seek  a  better  illustration  of  how  a  common  pattern  may 
vary  in  related  species. 

A  zebra's  stripes  may  be  broad  or  narrow,  uniform  or  alternately 
dark  and  pale — these  are  minor  or  secondary  diversities;  but  the 
pattern  of  the  stripes  shews  more  conspicuous  differences  than  these, 
though  the  differences  remain  of  a  simple  kind.  A  zebra's  stripes 
fall  into  several  series.  One  set  covers  the  neck,  including  the  mane, 
and  extends  backwards  over  the  body  and  forwards  on  to  the  face; 
and  these  "body-stripes"  are  all  that  the  extinct  Quagga  possessed. 
On  the  head  they  are  interrupted  by  the  ears  and  eyes,  and  end  at  a 
definite  vertex  on  the  forehead:  from  which,  however,  they  run 
down  the  face  in  pairs,  of  which  the  first  pair  of  all  may  jr  may  not 
coalesce  into  a  single  median  stripe  (Fig.  553).  A  second  series 
runs  up  the  foreleg,  and  where  it  meets  the  body  we  have  the 


xvii]    THE  COMPARISON  OF  RELATED  FORMS     1091 

problem  of  how  best  to  fit  the  horizontal  leg-stripes  and  the  vertical 
body-stripes  together.  There  is  only  one  way.  A  pair  of  body- 
stripes  diverge  apart  and  the  upper  leg-stripes  fit  in  between-, 
becoming  at  the  same  time  chevron-shaped  so  as  to  adapt  them- 
selves to  the  space  they  have  come  to  occupy.  The  stripes  of  the 
forelegs,  and  their  manner  of  fitting  on  to  the  body-stripes,  vary 
very  little  in  the  several  species  or  varieties. 

A  third  series  of  stripes  ascends  the  hindlegs,  in  a  fashion  iden- 
tical to  begin  with  for  all,  but  open  to  modification  where  these 
leg-stripes  spread  over  the  hauncljes;   for  here  there  hiay  be  great 


Fig.  553.     Zebra's  head,  to  shew  how  the  body-stripes 
extend  to  the  face.     From  A.  Rzasnicki. 

differences  in  the  extent  to  which  the  leg-stripes  compete  with  and 
interfere  with,  or  (so  to  speak)  encroach  upon,  the  stripes  of  the 
body.  Tne  typical  Equus  zebra  is  easily  recognised  by  the  so-called 
"gridiron"  on  its  rump;  this  is  a  dorsal  continuation  of  the  body- 
stripes,  extending  to  the  tail,  but  sharply  cut  off  on  either  side  by 
the  stripes  ascending  from  the  leg  (Fig.  554,  C).  In  Burchell's  zebra 
the  hindleg-stripes  encroach  still  farther  on  the  body,  and  even 
reach  up  to  the  rump,  so  that  the "■  gridiron  is  entirely  cut  away*. 

*  Ward's  zebra  and  Grant's  zebra  are  varieties  of  Equus  zebra,  the  former  with 
a  very  strong  "gridiron,"  the  latter  with  a  mere  vestige  of  the  same:  which  is  as 
much  as  to  say  that  the  leg-stripes  encroach  little  in  the  one,  and  much  in  the 
other,  on  the  hindmost  body-stripes.  Chapman's  zebra  is  a  form  of  E.  Burchelli, 
with  well-striped  legs  and  faint  intermediate  striping.  Cf.  W.  Ridgeway,  on  The 
differentiation  of  the  three  species  of  Zebra,  P.Z.S.  1909,  pp.  547-563;  also  {int.  al) 
Adolf  Rzasnicki,  Zebry,  Warsaw,  1931. 


1092       THE  THEORY  OF  TRANSFORMATIONS        [qh. 

In  the  Abyssinian  Equus  Grevyi,  all  the  stripes  are  very  numerous, 
narrow  and  close-set.  The  body-stripes  refuse,  as  it  were,  to  be 
encroached  on  or  obliterated  by  those  of  the  hindlegs ;  which  latter 
are  merely  intercalated  between  them,  chevron  fashion,  wedging 


Fig.  554.     Zebra  patterns.     A,  B,  Equus  Burchelli;  C,  E.  zebra;  D,  E.  Grevyi. 

in  between  the  body-stripes  as  the  foreleg-stripes  are  wont  to  do. 
It  follows  that  in  the  middle  of  the  haunch,  over  the  region  of  the 
hip- joint,  there  is  in  this  species  a  characteristic  "focus,"  where  the 
leg-stripes  fit  in  between  the  lumbar  and  the  caudal  sections  of  the 
body-stripes.  We  may  now  add,  as  a  fourth  and  last  series,  common 
to  all  kinds,  the  few  stripes  which  surround  the  lips  on  either  side, 
and  wedge  in  between  the  stripes  upon  the  face. 

Conclusion 

There  is  one  last  lesson  which  coordinate  geometry  helps  us  to 
learn;  it  is  simple  and  easy,  but  very  impprtant  indeed.  In  the 
study  of  evolution,  and  in  all  attempts  to  trace  the  descent  of  the 
animal  kingdom,  fourscore  years'  study  of  the  Origin  of  Species 
has  had  an  unlooked-for  and  disappointing  result.     It  was  hoped 


XVII]  THE  COMPARISON  OF  RELATED  FORMS  1093 

to  begin  with,  and  within  my  own  recollection  it  was  confidently 
believed,  that  the  broad  lines  of  descent,  the  relation  of  the  main 
branches  to  one  another  and  to  the  trunk  of  the  tree,  would  soon 
be  settled,  and  the  lesser  ramifications  would  bp  unravelled  bit 
by  bit  and  later  on.  But  things  have  turned  out  otherwise.  We 
have  long  known,  in  more  or  less  satisfactory  detail,  the  pedigree  of 
horses,  elephants,  turtles,  crocodiles  and  some  few  more;  and  our 
conclusions  tally  as  to  these,  again  more  or  less  to  our  satisfaction, 
with  the  direct  evidence  of  palaeontological  succession.  But  the 
larger  and  at  first  sight  simpler  questions  remain  unanswered;  for 
eighty  years'  study  of  Darwinian  evolution  has  not  taught  us  how 
birds  descend  from  reptiles,  mammals  from  earlier  quadrupeds, 
quadrupeds  from  fishes,  nor  vertebrates  from  the  invertebrate  stock. 
The  invertebrates  themselves  involve  the  selfsame  difficulties,  so 
that  we  do  not  know  the  origin  of  the  echinoderms,  of  the  molluscs, 
of  the  coelenterates,  nor  of  one  group  of  protozoa  from  another. 
The  difficulty  is  not  always  quite  the  same.  We  may  fail  to  find 
the  actual  links  between  the  vertebrate  groups,  but  yet  their  re- 
semblance and  their  relationship,  real  though  indefinable,  are  plain 
to  see;  there  are  gaps  between  the  groups,  but  we  can  see,  so  to 
speak,  across  the  gap.  On  the  other  hand,  the  breach  between 
vertebrate  and  invertebrate,  worm  and  coelenterate,  coelenterate 
and  protozoon,  is  in  each  case  of  another  order,  and  is  so  wide  that 
we  cannot  see  across  the  intervening  gap  at  all. 

This  failure  to  solve  the  cardinal  problem  of  evolutionary  biology 
is  a  very  curious  thing;  and  we  may  well  wonder  why  the  long 
pedigree  is  subject  to  such  breaches  of  continuity.  We  used  to  be 
told,  and  were  content  to  believe,  that  the  old  record  was  of  necessity 
imperfect — we  could  not  expect  it  to  be  otherwise;  the  story  was 
hard  to  read  because  every  here  and  there  a  page  had  been  lost  or 
torn  away,  like  some  hiatus  valde  deflendus  in  an  ancient  manuscript. 
But  there  is  a  deeper  reason.  When  we  begin  to  draw  comparisons 
between  our  algebraic  curves  and  attempt  to  transform  one  into 
another,  we  find  ourselves  limited  by  the  very  nature  of  the  case 
to  curves  having  some  tangible  degree  of  relation  to  oner  another; 
and  these  "degrees  of  relationship"  imply  a  classification  of  mathe- 
matical forms,  analogous  to  the  classification  of  plants  or  animals 
in  another  part  of  the  Systema  Naturae. 


1094     THE  THEORY  OF  TRANSFORMATIONS      [ch. 

Ah  algebraic  curve  has  its  fundamental  formula,  which  defines 
the  family  to  which  it  belongs;  and  its  parameters,  whose  quantita- 
tive variation  admits  of  infinite  variety  within  the  limits  which 
the  formula  prescribes.  With  some  extension  of  the  meaning  of 
parameters,  we  may  say  the  same  of  the  families,  or  genera,  or  other 
classificatory  groups  of  plants  and  animals.  We  cross  a  boundary 
every  time  we  pass  from  family  to  family,  or  group  to  group.  The 
passage  is  easy  at  first,  and  we  are  led,  along  definite  lines,  to  more 
and  more  subtle  and  elegant  comparisons.  But  we  come  in  time 
to  forms  which,  though  both  may  still  be  simple,  yet  stand  so  far 
apart  that  direct  comparison  is  no  longer  legitimate.  We  never 
think  of  "transforming"  a  helicoid  into  an  ellipsoid,  or  a  circle  into 
a  frequency-curve.  So  it  is  with  the  forms  of  animals.  We  cannot 
transform  an  invertebrate  into  a  vertebrate,  nor  a  coelenterate  into 
a  worm,  by  any  simple  and  legitimate  deformation,  nor  by  anything 
short  of  reduction  to  elementary  principles. 

A  "principle  of  discontinuity,"  then,  is  inherent  in  all  our  classifi- 
cations, whether  mathematical,  physical  or  biological;  and  the 
infinitude  of  possible  forms,  always  limited,  may  be  further  reduced 
and  discontinuity  further  revealed  by  imposing  conditions — as,  for 
example,  that  our  parameters  must  be  whole  numbers,  or  proceed 
by  quanta,  as  the  physicists  say.  The  lines  of  the  spectrum,  the  six 
families  of  crystals,  Dalton's  atomic  law,  the  chemical  elements 
themselves,  all  illustrate  this  principle  of  discontinuity.  In  short, 
nature  proceeds  from  one  type  to  another  among  organic  as  well  as 
inorganic  forms;  and  these  types  vary  according  to  their  own 
parameters,  and  are  defined  by  physico-mathematical  conditions  of 
possibility.  In  natural  history  Cuvier's  "types"  may  not  be  per- 
fectly chosen  nor  numerous  enough,  but  types  they  are;  and  to  seek 
for  stepping-stones  across  the  gaps  between  is  to  seek  in  vain,  for 
ever. 

This  is  no  argument  against  the  theory  of  evolutionary  descent. 
It  merely  states  that  formal  resemblance,  which  we  depend  on  as 
our  trusty  guide  to  the  affinities  of  animals  within  certain  bounds  or 
grades  of  kinship  and  propinquity,  ceases  in  certain  other  cases  to 
serve  us,  because  under  certain  circumstances  it  ceases  to  exist.  Our 
geometrical  analogies  weigh  heavily  against  Darwin's  conception  of 
endless  small  continuous  variations;    they  help  to  show  that  dis- 


XVII]  THE  COMPARISON  OF  RELATED  FORMS  1095 

continuous  variations  are  a  natural  thing,  that  "mutations" — or 
sudden  changes,  greater  or  less — are  bound  to  have  taken  place,  and 
new  "types "  to  have  arisen,  now  and  then.  Our  argument  indicates, 
if  it  does  not  prove,  that  such  mutations,  occurring  on  a  com- 
paratively few  definite  lines,  or  plain  alternatives,  of  physico- 
mathematical  possibility,  are  likely  to  repeat  themselves:  that  the 
"higher"  protozoa,  for  instance,  may  have  sprung  not  from  or 
through  one  another,  but  severally  from  the  simpler  forms ;  or  that 
the  worm-type,  to  take  another  example,  may  have  come  into  being 
again  and  again. 


EPILOGUE 

I N  the  beginning  of  this  book  I  said  that  its  scope  and  treatment 
were  of  so  prefatory  a  kind  that  of  other  preface  it  had  no  need; 
and  now,  for  the  same  reason,  with  no  formal  and  elaborate  con- 
clusion do  I  bring  it  to  a  close.  The  fact  that  I  set  little  store  by- 
certain  postulates  (often  deemed  to  be  fundamental)  of  our  present- 
day  biology  the  reader  will  have  discovered  and  I  have  not 
endeavoured  to  conceal.  But  it  is  not  for  the  sake  of  polemical 
argument  that  I»have  written,  and  the  doctrines  which  I  do  not 
subscribe  to  I  have  only  spoken  of  by  the  way.  My  task  is  finished 
if  I  have  been  able  to  shew  that  a  certain  mathematical  aspect  of 
morphology,  to  which  as  yet  the  morphologist  gives  little  heed,  is 
interwoven  with  his  problems,  complementary  to  his  descriptive 
task,  and  helpful,  nay  essential,  to  his  proper  study  and  com- 
prehension of  Growth  and  Form.     Hie  artem  remumque  repono. 

And  while  I  have  sought  to  shew  the  naturalist  how  a  few  mathe- 
matical concepts  and  dynamical  principles  may  help  and  guide  him, 
I  have  tried  to  shew  the  mathematician  a  field  for  his  labour — a  field 
which  few  have  entered  and  no  man  has  explored.  Here  may  be 
found  homely  problems,  such  as  often  tax  the  highest  skill  of  the 
mathematician,  and  reward  his  ingenuity  all  the  more  for  their 
trivial  associations  and  outward  semblance  of  simplicity.  Haec  utinam 
excolant,  utinam  exhauriant,  utinam  aperiant  nobis  Viri  mathematice 
docti*. 

That  I  am  no  skilled  mathematician  I  have  had  little  need  to 
confess.  I  am  "advanced  in  these  enquiries  no  farther  than  the 
threshold";  but  something  of  the  use  and  beauty  of  mathematics 
I  think  I  am  able  to  understand.  I  know  that  in  the  study  of 
material  things,  number,  order  and  position  are  the  threefold  clue 
to  exact  knowledge ;  that  these  three,  in  the  mathematician's  hands, 
furnish  the  "first  outlines  for  a  sketch  of  the  Universe";  that  by 
square  and  circle  we  are  helped,  like  Emile  Verhaeren's  carpenter, 
to  conceive  "Les  lois  indubitables  et  fecondes  Qui  sont  la  regie  et 
la  clarte  du  monde." 

For  the  harmony  of  the  world  is  made  manifest  in  Form  and 

*  So  Boerhaave,  in  his  Oratio  de  Usu  Ratiocinii  Mechanici  in  Medicina  (1703). 


EPILOGUE  1097 

Number,  and  the  heart  and  soul  and  all  the  poetry  of  Natural 
Philosophy  are  embodied  in  the  concept  of  mathematical  beauty. 
A  greater  than  Verhaeren  had  this  in  mind  when  he  told  of  "the 
golden  compasses  prepared  In  God's  eternal  store."  A  greater  than 
Milton  had  magnified  the  theme  and  glorified  Him  "that  sitteth 
upon  the  circle  of  the  earth,"  saying:  He  hath  measured  the  waters 
in  the  hollow  of  his  hand,  and  meted  out  heaven  with  the  span, 
and  comprehended  the  dust  of  the  earth  in  a  measure. 
,  Moreover,  the  perfection  of  mathematical  beauty  is  such  (as  Colin 
Maclaurin  learned  of  the  bee),  that  whatsoever  i&  most  beautiful 
and  regular  is  also  found  to  be  most  useful  and  excellent. 

Not  only  the  movements  of  the  heavenly  host  must  be 
determined  by  observation  and  elucidated  by  mathematics,  but 
whatsoever  else  can  be  expressed  by  number  and  defined  by 
natural  law.  This  is  the  teaching  of  Plato  and  Pythagoras,  and 
the  message  of  Greek  wisdom  to  mankind.  So  the  living  and 
the  dead,  things  animate  and  inanimate,  we  dwellers  in  the  world 
and  this  world  wherein  we  dwell — Travra  ya  fxav  ra  yiyvcoGKo^eva — 
are  bound  alike  by  physical  and  mathematical  law.  "  Conterminous 
with  space  and  coeval  with  time  is  the  kingdom  of  Mathematics; 
within  this  range  her  dominion  is  supreme ;  otherwise  than  according 
to  her  order  nothing  can  exist,  and  nothing  takes  place  in  contradiction 
to  her  laws."  So  said,  some  sixty  years  ago,  a  certain  mathe- 
matician* ;   and  Philolaus  the  Pythagorean  had  said  much  the  same. 

But  with  no  less  love  and  insight  has  the  science  of  Form  and 
Number  been  appraised  in  our  own  day  and  generation  by  a  very 
great  Naturalist  indeedf — by  that  old  man  eloquent,  that  wise 
student  and  pupil  of  the  ant  and  the  bee,  who  died  while  this  book 
was  being  written ;  who  in  his  all  but  saecular  Hfe  had  tasted  of  the 
firstfruits  of  immortality;  who  curiously  conjoined  the  wisdom 
of  antiquity  with  the  learning  of  today;  whose  Proven9al  verse 
seems  set  to  Dorian  music;  in  whose  plainest  words  is  a  sound 
as  of  bees'  industrious  murnmr;  and  who,  being  of  the  same 
blood  and  marrow  with  Plato  and  Pythagoras,  saw  in  Number  le 
comment  et  le  pourquoi  des  choses,  and  found  in  it  la  clef  de  voute  de 
V  Univers. 

*  William  Spottiswoode,  in  his  presidential  address  to  the' British  Association  at 
Dublin  in  1878.  t  Henri  Fabre. 


INDEX 


Abbe,  Ernst  304 

Abonyi,  A.   251,  254 

Acanthocystis   675,  700 

Acanthometridae   701,  730 

Aceratherium   1074 

Achlya  404 

Acineta  461 

Acromegaly  264 

Actinomma   708 

Actinophrys  342,  426,  470 

Actinosphaerium  470,  707 

Adam,  N.  K.   72,  359,  450 

Adams,  T.  W.   95 

Adamson,  Martin   607 

Adaptation  961 

Addison,  Joseph   5,  959 

Adiantum,  embryo  of  642 

Adsorption   355,  444,  etc. 

Aethalium   361 

Aethusa   267 

Agassiz,  Louis  .176 

Agates   661 

Age -composition    159 

Aglaophenia    1060 

Airy,  G.  B.    123;  H.   916 

Albertus  Magnus   938 

Albinus  532 

AlexeieflF,  A.   287,  295 

Alison,  W.  P.  955 

Allen,  Bennet  M.   265 

Allman,  William   923 

Allocapsa  614 

Alpheus,  claws  of  279 

Amans,  P.   45 

Amelung,  Erich   60 

Ammonites  796,  846,  1038 

Amoeba   16,  357,  363,  425,  458,  704,  746 

Amphioxus  490 

Ampullaria   777 

Anabaena  477 

Anaxagoras   809 

Anderson,  A.  P.   260 

Andrews,    C.    W.     1024;    G.    F.     294; 

Mrs  336 
Anikin,  W.  P.   254 
Ankylostoma  488 
Annandale,  Nelson   244 
Anodon    1061 
Anomia   827 

Anthogorgia,  spicules  of  647 
Antigonia   1063,  1088 


Antlers  892 

Antony,  R.  56,  117 

Ants,  trail- running   220 

Apatornis   1070 

Apocynum,  pollen  of  631 

Approximation   1029 

Apteryx   1011 

Aptychus  670,  838 

Arachnophyllum   513 

Aragonite   670 

Arcella  510 

Arcestes  797 

Archaeopteryx   50,  1070 

Archibald,  R.  T.   755,  812,  924 

Archimedean  bodies  552,  735 

Archimedes  22,  68,  734,  762,  767 

Argali  850 

Argonauta  800,  823 

Argus  pheasant  664 

Argyropelecus   1062 

Aristotle  3,  4,  6,  12,  14,  20,  81,  176,  180, 

248,  286,  357,  759,  830,  900,  935. 

1019,  1034 
Arnold,  A.   184 

Arrhenius,  Svant  221,  232,  302,  450 
Artemia  251,  338 
Ascaris  638 
Aschemonella  414 
Ascidia  483 
Ashby,  Eric   129 
Ashworth,  J.  H.   244 
Asparagus  553 
Assheton,  R.  561 
Astbury,  W.  T.  403,  670,  743 
Asterias  559 
Asterolepis  517 
Asters  299 
Asthenosoma   947 
Astrolampra   618 
Astrorhiza   703 
Astrosclera  672 
Atkins,  D.   71 
Atrypa   832 
Auerbach,  F.    14 
Aulacantha   699 
Aulastrura  711 
Aulonia  708,  737 
Aurelia  205 
Ausonius  527 
Autocatalysis   256 
Auxin  262,  281 


INDEX 


1099 


Avicula    1061 
Awerinzew,  S. 
Axolotl  265 
Azolla  67 


85: 


Babak,  E.   56,  265 

Baboon,  skull  of  1083 

Bacillus  64 

Backhouse,  T.  W.   353,  946 

Backman,  Gustav   98,  108,  115,  157 

Bacon,  Francis    1,  58,  79,  81,  255,  341, 

376,  384,  849,  939;  Roger   1,  8 
Baer,  K.  E.  von  3,  11,  83,  86,  286 
Baillanger,  J.    189 
Balfour,  F.  M.   85,  568 
Ball,  0.  M.  974,  986 
Balls,  W.  L.   746 
Baltzer,  Fr.   514,  753 
Bambeke,  C.  van   17 
Bamboo,  growth  of  29,  159,  217 
Bantam  fowl   117 
Barclay,  John   544 
Barcroft,  Joseph   249 
Barfurth,  E.    168 
Barlow,  W.   348 
Barr,  A.   26 
Barratt,  J.  O.  W.   495 
Barrow,  S.  487 
Barry,  Martin  341 
Bartholinus,  E.   527,  541,  695 
Basalt  520 
Basil,  St  527 

Bast-fibres,  strength  of  973 
Baster,  Job   271 

Bateson,  WQliam  208,  212,  340,  907 
Bather,  F.  A.   941 
Batsch,  A.  J.  G.  C.   868 
Baudrimont,  A.   243 
Baurmann,  M.   304 
Baylis,  H.  A.   897 
Bayliss,  W.  M.   264,  328 
Beam,  loaded  967 
Bean,  growth  of  191 
Beaumont,  Elie  de   785 
Becher,  A.   210;  S.   687 
Bechhold,  H.  665 
Beebe,  W.   424 
Bee's  cell  525,  etc. 

Begonia,  leaf  of   1042;  raphides  in   646 
Beijer,  J.  J.  480 
Beilby  layer  493 
Belar,  K.   307 
Belcher,  Sir  E.  411 
Belehradek,  J.   221,  232 
Belemnites  858 

Bell,  Sir  Charles  51,  956,  976,  987 
Bellerophon  809 
Belodon    1065 
Benard,  H.    418,  500,  853 
Benedict,  Fr.  G.    205,  241 


Bennett,  A.    115,161;   G.  T.    121,385, 

536,  610,  763,  783 
Benninghoff,  A.    979 
Bentlev,  W.  A.,  and  W.  J.  Humphreys 

4n,  696 
Benton,  J.  R.    387 
Berezowsky,  A.    62 
Berg,  W.  F.    304,  333,  661 
Berger,  C.  A.    243 
Bergmann,  H.  von    34,  35 
Bergson,  Henri   9,  193,  412,  873,  1029; 

Joseph    131 
Berkeley,  George    24 
Bermuda  life-plant   275 
Bernal,  J.  D.    303,  488 
Bernard,  Claude    2,  4,  5,  249,  256 
Bernhardt,  H.  979 
Bernoulli,  Daniel   923;  James    89,  118, 

755,  767,  961;  John    37,  82 
Bernstein,  Julius,  361 
Berthelot,  M.  464 
Berthqld,   G.     12,  475,  482,  567,   571, 

578,  593,  758 
Bertholf,  L.  M.    54 
Berzelius,  J.  J.    255 
Bethe,  A.    441 
Beudant,  F.  S.    661 
Bezold,  W.    393 
Bialaszewicz,  K.,  226,  229,  245 
Bichat,  Xavier    11,  97,  249,  956,  1029 
Bidder,  George    507 
Biedermann,  W.    664 
Bilharzia,  egg  of   943 
Biloculina,    858 
Biometrics    119 
Birds,  flight  of   41;  skeleton  of    1011; 

toothed    1070 
Bishop,  John    40 
Bison,    990 
Bivalve  shells    822 
Bjerknes,  V.    46,  59,  325 
Black,  James    956 
Blackman,  F.  F.    217,  224,  289;  V.  H. 

153,  154,  220 
Blackwall,  J.    387 
Blair,  G.  W.  S.    986;  Patrick    900 
Blake,  J.  F.    789,  793,  845,  1038 
Bliss,  G.  A.    382 
Blochmann,  Fr.    653 
Blood-corpuscles,  form  of   435;  size  of 

60,  69 
Blood-vessels    948 
Blowfly,  speed  of    162 
Blum,  E.    951 
Boas,  Fr.    125,  130,  416 
Bodo    383,  432 

Boerhaave,  Hermann    2,  604,  1096 
Boggild,  O.  B.    656 
Bogorrow,  B.  G.    219 
Bohn,  C.    245 


1100 


INDEX 


Bohr,  Nils    54 

Boltzmann,  Ludwig    19,  356 

Bolk,  L.    901 

Bonanni,  F.    499,  544 

Bonaventure,  St    19 

Bone,  structure  of    975  seq. 

Bonneson,  T.    382,  566 

Bonnet,  Charles  271,  275,  382,  543,  915 

Borelli,  J.  A.    11,  12,  36,  48,  82,  356, 

970,  988 
Born,  Max    1031 
Boscovich,  Father  R.  J.    13,  93,  286, 

346,  531 
Bose,  J.  C.    171 
Bothriolepis    517 
Bottazzi,  F.    243 
Bottomley,  W.  B.    262 
Boundary  layer    449 
Bourgery,  J.  M.    979 
Bourrelet,  Plateau's    475,  710 
Boveri,  Th.    63,  275,  300 
Bowditch,  H.  P.   63,  125,  275,  299,  306 
Bower,  F.  0.    640 
Bowman,  J.  H.    662 
Boycott,  A.  E.    64,  819,  822 
Boyer,  Carl  B.    346 
Boyle,  Robert  7,  370 
Brachiopods,    831 
Bradford,  S.  C.    302 
Brady,  H.  B.    700,  868 
Bragg,  Sir  W.  E.    348 
Brahe,  Tycho    11 

Brain,  growth  of  186-189;  weight  of  180 
Branchipus    559 
Brandt,  A.    188 
Brauer,  A.    314 
Braun,  A.    915 

Bravais,  L.  and  A.    237,  767,  915 
Brazier,  L.  G.    970 
Bredig,  G.    311 
Breeder,  G.  M.    50 
Breton,  E.  le    60,  162 
Brewster,  Sir  David   358,  549,  569,  656 
Bridge,  T.  W.    959 
Bridge- construction    989 
Bridgeman,  P.  W.    26 
Briggs,  C.  E.    158 
Brinckmann,  R.    457 
Brindley,  H.  H.    208,  212 
Brine-shrimps    253 
Brobdignag    24 
Brody,  8.    141 
Broglie,  Louis  de    74 
Brooke,  Sir  Vincent    877,  892,  895 
Brougham,  Lord    537,  544 
Brown,  A.  W.    113;  H.  T.   627;  Robert 

73,  75,  341 ;  Samuel    255 
Browne,  Sir  Thomas   495,  725,  932,  935 
Brownian  movement    75 
Brownlee,  John    76,  144 


Brucke,  C.    290,  344 

Bruckner,  Max  598 

Brunswick,  459 

Bryan,  G.  H.    44 

Bryophyllum,  regeneration  of   275 

Bubbles    350  seq.,  468 

Buch,  Leopold  von    785,  845 

Buddenbroek,  W.    756 

BufiFon    100,  536,  543,  970,  1020 

Bulimus    808 

Bull,  A.  J.    565 

BuUen,  A.  H.  R.    72 

Bullfrog,  growth  of    264 

Bunting,  Martha     559 

Burnet,  John    759 

Burr,  Malcolm    216 

Burton,  E.  F.    460 

Butler,  Samuel    6 

Biitschli,  C.    300,  330,  359,  361.  665, 

667,  853 
Biittel-Reepen,  H.  von    540 
Buxton,  P.  A.    961 
Byk,  A.    652 

Cajori,  Florian    135,  1031 

Calandrini,  G.  L.    915 

Calanus    219 

Calcospherites    655 

Calkins,  L.  A.    108,  184 

Callimitra    712 

Callitharanion,  spores  of   631 

Caiman,  W.  T.    278 

Calvert,  P.  P.    165,  214 

Calymene    707 

Calyptraea    802,  815 

Cambium    480,  503 

Camel    1002 

Camerer,  W.    108 

Campanularia    391,  422,  634,  1060 

Campbell,  D.  H.    593 

Camphor    361 

Camptosaurus    1067 

Campylodiscus    434 

Cannon,  H,  Graham    293,  316,  325 

Cannon-bone    1039 

Cantilever    972,  etc. 

Cantor,  Moritz    1053 

Capillarity    465,  etc. 

Caprella    1056 

Caprinella    829 

Carcinus    463 

Carey,  E.  J.    744 

Cariacus    893 

Ckrlier,  E.  W.    360 

Carlson,  T.    153 

Carnot,  Sadi    9,  464,  507,  752 

Carnoy,  J.  B.    707 

Carp,  growth  of    180 

Carpenter,  W.  B.    73,  704;  E.    654 

Carradini,  G.    361 


INDEX 


1101 


Cariuccio,  E.    544 

Caryocinesis  /  299,  304 

Cassini,  Dominic    528,  695 

Castagna    579 

Castillon    530 

Catalan,  E.  C.    735 

Catalytic  action  255 

Catenoid  369 

Cauliflower    914 

Cay  ley,  Arthur    318,  609;   Sir  George 

49,  612,  961 
Celestite  699 
Cell-size,  fixed    60 
Cell-theory    287,  341 
Cells,  collared    429;  stellate    547 
Cenosphaera    710 
Centres  of  force    2^6 
Centrosome    299,  306 
Cephalopods,  eggs  of  602 
Ceratites    845 

Ceratophyllura,  growth  of    193 
Cerebratulus,  egg  of   323 
Cerithium    787 
Cesaro,  G.    526,  544 
Chabry,  L.    37,  483,  648 
Chaetopeltis    593 
Chaetopterus,  egg  of   338 
Chambers,  Robert    302,  306,  326,  335, 

363,  388,  434 
Chancourtois,  A.  E.  B.    565 
Chandra,  Krishna    504 
Chank  sheU    805 
Chapman,  Abel    45,  960 
Chara    479 

Characters,  biological    1019,  1036 
Chelyosoma    517 
Chermak,  P.    393 
Child,  C.  M.    282 
Chilomonas    225 
Chimpanzee,  skull  of    1083 
Chladni,  E.  F.  F.    472,  719 
Chlamydomyxa    700 
Chodat,  R.    115,  256 
Chondriosomes    455 
Chorinus    1057 
Chossat,  Ch.    189 
Christison,  Sir  R.    235 
Chromatin    296,  305 
Chromosomes    338 
Chromulina    422 
Chrysaora    248 
Church,  A.   H.    915,  919 
Chydorus    214 
Chytridia    462 
Cicada  613 
Cicero    92 
('icinnus    767 
Cilia    406 

Ciliate  infusoria    425 
Circoporus    72U 


Cladocarpus    1060 

Cladocera    165,  506 

Cladonema    397 

Claparede,  E.  R.    656 

Clathrulina    710 

Claus,  F.    506 

Clausilia    758,  808 

Clausius,  R.    11,  358 

Cleland,  John    5 

Cleodora    833 

Clusters,  globular  star    68 

Clytia    1060 

Coan,  C.  A.    765,  931 

Coassus    893 

Codonella    408 

Codosiga    429 

Coe,  W:  B.    324 

Cogan,  Dr  T.    1055 

Cohen,  Ernst    266 

Cohesion  figures    723,  847 

Cole,  William    744 

Coleridge,  S.  T.    1034 

Collar-cells    429 

CoUosphaera    697 

Colman,  S.    765 

Colpodium    257 

Comoseris    514 

Compositae    930 

Conchospiral    788 

Conidiophore    479 

Conklin,  E.  G.  5,  60,  293,  483,  487,  557, 

601 
Contour  lines  904 
Contractile  vacuole    426 
Conus    813,  922 

Cooke,  Sir  T.  A.    749,  912,  919,  931 
Coordinates    1032,  etc. 
Cope,  E.  D.    517,  899,  906 
Copernicus    1 1 
Coplans,  Myer    151 
Corals    512-514,  621-624 
Cordylophora    397 
Cornea  of  insect    511 
Cornevin,  Charles    201 
Cornish,  Vaughan    965 
Cornuspira    857 
Corpora  Arantii    472 
Correns,  C.    743 
Corse,  John    903 
Corystes    1057 
CV)tes,  Roger    755 
Cotton,  A.    651 
Coulomb,  C.  A.    983 
Cowell,  J.  W.    100 
Oeodonta    1024 
Crepidula    60,  487,  557 
Creseis    833 
Crew,  F.  E.  A.    108 
Cristellaria    766,  792,  863 
Crocodile    1065 


1102 


INDEX 


Crocus,  growth  of    171 

Crookes,  Sir  Wm.    51 

Crozier,  W.  J.    198,  221 

Crum  Brown,  Alexander    1 

Cryptocleidus    1068 

Ctenophora    624 

Cube,  partition  of  a    567 

Cuboctahedron    734 

('ucurbita    1049 

(Julmann,  Prof.  C.    976,  997 

Curry,  Manfred,  961 

Curves,  of  distribution  135;  of  error 
131;  of  frequency  119;  of  growth 
95;  logistic    147;  multimodal    135 

Cushman,  J.  A.    510 

Cuvier,  G.    1036,  1094 

Cuvierina    416,  833 

Cyamus    1056 

Cyathophyllum    512 

Cycas    516 

Cyclammina    859 

Cyclol    737 

Cyclommatus    209 

Cyclostoma    813 

Cylinder    369 

Cymose  inflorescence    767 

Cynthia    601 

Cypraea    802,  805 

Cystolithmus    647 

Cytoplasm    347 

Daday  de  Dees,  E.  von    253 

Daffner,  Fr.    233 

Dakin,  W.  J.    336 

Dalcq,  A.    303 

D'Alembert,  Jean  le  R.    89 

DalyeU,  Sir  John  G.    274 

D'Ancona,  Umberto    159,  196,  274 

Dandolo,  Count  Vincentzio    163 

Danilewsky,  B.    262 

Dannevig,  Alf.    180 

Daphnia    253,  506 

Darboux,  G.    382 

Darling,  C.  R.    370,  416 

DarUngton,  C.  D.    305,  310,  347,  947 

D'Arsonval,  A.    449 

Darwin,  Charles    5,  86,  537-541,  664, 

704,  887-889,  959,  1019,  1094 
Dastre,  A.    268 

Davenport,  C.  V.    83,  243,  245,  360 
Da  vies,  G.  R.    156;  R.  A.    35 
Dawes,  Ben    253 
Dawson,  J.  A.    434 
De  Bary,  A.    345 
De  Candolle,  A.   546;  A.  P.   29;  Casimir 

916 
Deep-sea  fishes    423 
De  Heen,  P.    502 
Dehler,  Adolf   438 
Dehrael.  G.    83 


Delage,   Yves    270 

Delaunay,  C.  E.    368 

Delisle,  M.    40 

DeUinger,  O.  P.    361 

Delpino,  F.    916 

Democritus    6 

De  Morgan,  Augustus    9,  24,  733 

Dendy,  Arthur   269,  658,  671,  690,  693 

Dentalium    792,  805 

Dentine    657 

Deropyxis    422 

Descartes,  Rene  4,  7,  321,  733,  754,  809 

Desch,  C.  H.    554 

Desmids    579 

Des  Murs,  0.    935 

Devaux,  H.    69,  72,  464 

Deviation,  standard    119 

De  Vries,  H.    217 

Dexippus    164 

Diagrams    136;  reciprocal    997 

Diakonow,  D.  M.    212 

Diatoms    510 

Dickson,  A.   916;  H.   241;L.  E.  924 

Dicquemare,  J.  F.    271 

Dictyocha    717 

Dictyota    479 

Dietz,  J.  F.  G.    98 

Difflugia    422,  702,  705 

Diffraction  plates    510 

Dimensions    23 

Dimorphism    212 

Dimorphodon    1069 

Dinenympha    433 

Dinobryon    408 

Dinosaurs    1002,  1067 

Diodon    1064,  1090 

Dionaea    1043 

Diplodocus    1001,  1005,  1009 

Disc,  segmentation  of  a    594 

Discontinuity,  principle  of   1094 

Discorbina    865 

Distephanus    718 

Ditrupa    848 

Dixon,   A.    F.     979;    H.    H.     65,    238; 

K.  R.    438 
Dobell,  CUfford    339,  341,  433,  456 
Dodecahedron    526,  545 
Doflein,  F.  J.    75,  410,  429,  868,  899 
Dog,  skuU  of  211,  1087;  weight  of  187 
Dolium    780,  787 
Dolphin,  skeleton  of    1007 
Donaldson,  H.  H.    188 
Doncaster,  L.    316,  344 
Donnan  equilibrium    438,  463 
Dowding,  E.  S.    462 
Dorataspis    727,  738 
D'Orbigny,  Alcide    786,  854,  1038 
Douglas,  Jesse    382 
Douglass,  A.  E.    235,  239 
Dragonfly    476,  611 


IND'EX 


1103 


Draper,  J.  W.    422;  R.  L.    115 

Dreyer,  Fr.    671,  682,  707,  868,  870 

Driesch,  Hans  5,  60,  287,  483,  490,  602 

Dromia    441 

Drops    59,  467,  850,  946 

Drosophila    155,  231 

Drzwina,  Anna    245 

Duarte,  A.  J.    164 

Du  Bois  Reymond,  Emil    1,  189,  352, 

444 
Duckweed    263 
Duclaux,  J.    254 
Dudich,  E.    209 
Duerden,  J.  E.    654,  749 
Dujardin,  F.    17,  18,  415,  854 
Duke-Elder,  8ir  W.  S.    304 
Du  Monceau,  Duhamel    325 
Dumortier,  B.    312 
Dunan,  Charles    9 
Duncan,  P.  Martin    621 
Duncker,  G.    199,  819 
Dupres,  Athanase    446 
Durbin,  Marion  L.    271 
Diirer,  Albrecht   83,  89,  190,  372,  1053 
Duthie,  E.  S.    546 
Dutrochet,  R.  J.  H.    361,  888 
Dwight,  T.    979,  1081 
Dyar,  H.  J.    164 
Dye,  W.  D.    510 

Earwig,  dimorphism  in    212 

Ebner,  V.  von    679,  979 

Echinoderm  larvae    626 

Echinus    229,  602,  946 

Ectoplasm    359 

Edgerton,  H.  E.    390 

Edgeworth,  F.  H.    76 

Eel,  growth  of    169,  170 

Efficiency,  index  of    153 

Eggs  of  birds    934 

Eidmann,  W.    264 

Eiffel  tower    29,  961,  997 

Eimer,  Th.    868 

Einstein,  Albert    74,  76 

P^lasmotherium    905 

Elastic  curve    374 

Elderton,  Sir  W.  Palin    90,  98 

Elephant    203,  900,  905,  1004 

Eleutheria    619 

Elk,  antlers  of   892 

EllipsoHthes    1038 

Ellis,  M.   M.    276;    R.    Leslie    5,    123, 

532,  927 
Elodea    509 
Elsasser,  Th.    55 
Emarginula    815 
Emmel,  V.  E.    275 
Empedocles    12 
Enchelys    257 
Enestrom,  G.    566 


Engelmann,  T.  VV.    368 

Enriques,  Paolo    4,  61,  93,  970,  987 

Entelechy    5,  1020 

Entosolenia    682 

Entrophy    1 1 

Enzyme    264 

Eohippus    1077 

Epeira    386 

Ephyra    205 

Epicurus    6,  76 

Epidermis    493 

Epilobium,  pollen  of    631 

Epithelium    507 

Equiangular  spiral    748,  etc. 

Equisetum    460,  742 

Erica    631 

Eriocheir    250 

Errera,  Leo   53,  64,  222,  363,  482,  568 

Erythrotrichia    580,  623 

Estheria    828 

Ethmosphaera    708,  710 

Euastrum    364 

Eucharis    624 

EucUd    759 

Euglypha    316 

Eulaha    664 

Euler,  Leonard    4,  27,  250,  356,  529, 

609,  732 
Eunicea,  spicules  of   657 
Euphorbia    554 
Euplectella    698 
Euryale    61 
Euryhaline    423 
Eurypterus    517 
Eustomias    423 
Evelyn,  John    935 
Ewart,  A.  J.    29,  449 
Excess  and  defect    1035 
Exner,  Sigraund    74 

Fabre,  J.  H.    93,  1097 

Facial  angle    1055 

Fairy.flies    47 

Faraday,  Michael    286,  292,  297,  510, 

6'61,  719 
Farr,  Wm.    155 
Farrea    689 
Farrer,  Enoch    965 
Faure-Fremiet,  E.    108,  232 
Favaro,  A.    774 
Favosites    512 
Fechner,  G.  T.    936,  1027 
Fedorow,  E.  S.  von   348,  552,  662,  7.34 
Fehlmg,  H.    246 
Ferguson,  A.    946 
Fermat    Pierre  de    4,  356 
Fessard  and  Laufer   90,  98 
Festuca    507 
Fezzan  worms  251 
Fibonacci    923 


1104 


INDEX 


Fick,  L.    979 

Fickert,  C.    868 

Fiddler  crabs    206,  210 

Fidler,  T.  Claxton    967,  990 

Filter-passers    65 

Final  cause    4 

Fischel,  Alfred    184 

Fischer,  P^mU    651;  Hugo    631;  M.  H. 

327;  Otto    970,  999 
Fishes,  age  of   176;  eggs  of   135;  form 

of    1061 
Fissurella    815 

FitzGerald,  G.  F.   288,  449,  510,  675,  719 
Fixed  cell-size    103 
FlageUum    407,  506 
Flamingo    23,  960 
Fleming,  R.  M.    107,   125 
Flemming,  W.    296,  300,  304,  306,  314 
Flight    41 

FUnt,  Prof.  R.    966 
Florideae    639 
Flower,  Stanley  F.    172 
Flugel,  G.    72 
Fluid  crystals    437 
Fokker,  A.  P.    669 
Fol,  Hermann    299,  303,  335,  337 
FoUiculina    409 

Fontenelle,  B.  le  B.  de    530,  989 
Foot-and-mouth  disease    65 
Foraminifera    850 
Ford,  E.    191 
Fordham,  M.  G.  C.    336 
Forficula    212 
Forth  Bridge    975,  992 
Foster,  Michael    321 
Fourier,  J.  B.  G.    8 
Fox,  Munro    242 
Fraas,  E.    1024 
Fraenkel,  G.    35 
Frankenberger,  W.    256 
Eraser,  J.  H.    167 
Frazee,  O.  E.    282 
Fredericq,  Leon    255 
Frequency,  curve  of    119 
Frey-Wissling,  A.    302,  347 
Friant,  M.    903 
Frog,  egg  of   584,  603,  606;  growth  of 

226 
Froth  or  foam    432,  454,  482 
Froude,  W.    31,  33 
Fucus    576 
Fujiyama    121 
Fulton,  Angus  R.    974 
Fundulus    245,  323 
Furchgott,  R.  F.    438 
FusuUna    856 
Fusus    818 

Galathea    439 
Galen    4,  704,  939 


Galiani,  Abbe    19 

Galileo   9,  11,  12,  20,  27,  36,  152,  346, 

650,  825,  970,  997,  1026 
Gallardo,  A.    293 
Galloway,  Principal  G.    960 
GaUs    265 

Galton,  Fr.    119,  120,  135,  158 
Gamble,  F.  A.    695 
Ganglion-cells,  size  of    61 
Gans,  R.    75 
Garreau,  Dr   929 
Gaertner,  R.    201 
Gastrula    560 
Gaudry,  Albert    900 
Gauss,  K.  S.    121,  132,  354,  609,  1031 
Gaver,  F.  von    901 
Gavial    1065 
Gebhart,  W.    664,  979 
Generation-time    153 
Geodesies    741,  etc. 
Geoffroy  8t  Hilaire,  E.  de    1019 
Gerdy,  P.  N.    491,  745 
Geryon,  299,  1057 
Gestaltungskraft    731 
Ghetto,  A.    539 
Giard,  Alfred    286 
Giardina,  Andrea  330 
Gibbs,  Willard  356,  362,  471,  714,  719, 

733 
Gila  monster    959 
Gilmore,  C.  W.    1008 
Gilson,  E.    19 
Giraffe    1004,  1051 
Girders     991,    etc.;    continuous     1014; 

parabolic    1009 
Glaisher,  James  411;J.  W.  L.   531,536 
Glaser,  Otto    246 
Glassblower  744,  1049 
Gley,  E.    264,  268 
Globigerina    365,   387,   675,   700,   750, 

853 
Glock,  W.  S.    235 
Gnomon    759-766 
Goblet  cells    546 
Goebel,  K.  von    58,  546,  643 
Goethe    2,  29,  62,  344,  1019,  1026 
Goetsch,  W.    155 
Golden  mean    923,  932 
Goldfish    198 

Goldschmidt,  R.    187,  466 
Goldstein,  W.  C.    516 
Golgi  bodies    727 
Golightly,  W.  H.    219 
Gompertz,  Benjamin    156 
Goniatites    809 
Goniodoma    738 
Gonium    614 
Gonothyraea    1060 
Goodsir,  John   286,  341,  456,  697,  751, 

767 


INDEX 


1105 


Gordon,  Isabella    625 

Gorilla    186 

Goringe,  C.    202 

Gortner,  R.  A.    248 

Gott,  J.  P.    391 

Gottlieb,  H.    999 

Gough,  A.    436 

Goupilliere,  Haton  de  la   750,  778,  1030 

Gourd,  form  of  1049 

Gow,  H.  Z.    295 

Grabau,  A.  H.    788,  796 

Gradients,   193 

Graham,   A.     505;    Michael     179,    183, 

346,  666;  Thomas    292 
Grandi,  Guido    812,  1045 
Grant,  Kerr    418 
Graphic  statics    976 
Grassi,  G.  B.    170 
Graunt,  John    98 
Gravis,  A.    930 
Gray,  Asa    223;  James    156,  221,  228, 

243,  246,  306,  323,  335,  344,  407, 

450,  462,  587,  909 
Greenhill,  Sir  A.  G.    26,  28.  44 
Greenwood,  Major  132 
Gregory,  J.  W.    219;  R.  P.    63;  W.  K. 

900,  966,  969 
Greville,  R.  K.    617 
Grew,  Nehemiah    341,  482,  545,  912 
Gromia,  387,  415,  700 
Gryphaea^  802,  830 
Gudernatsch,  J.  S.    264 
Guillemot,  egg  of   936 
Guineapig    115 
Gulliver,  G.    61 
Gulliver's  Travels    22 
Giinther,  F.  C.    937;  S.    916-,  931,  1053 
Gurney,  R.    165 
Gurwitsch,  A.    328,  455 

Haberlandt,  0.    262 

Habermehl,  K.    483 

Hacker,  V.    695,  698 

Haddock    1008,  1088 

Haeckel,  Ernst    690 

Haeusler,  Rudolf   419 

Haldane,  J.  B.  S.    158;  J.  S.    956 

Hale  White,  Sir  WiUiam    949 

Hales,  Stephen,  88,  191,  955 

Halibut    180 

Haliotis    765,  802,  804,  813,  815,  822 

Hall,  C.  E.    236 

Haller,  A.  von    2,  82,  88,  93,  97,  184, 

•271,  279,  532,  956 
HaUey,  E.    755 
Halobates    67 
Halosphaera.  406 
Hamai,  Ikuso    167,   177 
Hambidge,  Jay    755,  924,  931 
Hamburger,  H.  J.    437 


Hamilton,  R.    1055;  Sir  W.  Rowan  356 

Hammett,  D.  W.    206 

Hankin,  E.  H.    46,  50 

Hanstein,   Johannes    494,  593 

Hardesty,  Irving    61 

Hardy,  Sir  W.   B.    73,  303,  304,  322 

Hargenvilliers,  M.     100 

Harmozome    264 

Harpa    780 

Harper,  R.  A.    615 

Harpinia    1059 

Harris,  J.    203 

Harrison,  Ross    986 

Harting,   P.    653 

Hartog,  M.    293,  514 

Hartridge,  H.    436-- 

Harvey,  E.  N.  and  H.  W.   322;  W.    12, 

955 
Hasikura    411 
Hastigerina    675,  851 
Hatai,  S.    256,  262 
Hatano,  T.    241 
Hatchett,  Chai'les    653   . 
Hatschek,  Emil    395,  437,  562,  681 
Haughton,  Samuel    544,  949 
Haushofer,  M.    147 
Hauy,  R.  J.    526,   1028 
Hawkins,  H.  L.    505 
Hay,  O.  P.    1008 
Haycraft,  J.  B.    360,  988 
Headley,  F.  W.    44 
Heart,  growth  of    186;  muscles  of   743 
Heath,  Sir  T.  L.    762 
Heaviside,..0.    24 
Hecht,  Selig    199,  206,  1089 
Hedges,  E.  S.    661 
Heel  bone    982 
Hegel,  G.  W.  S.    5 
Hegler,   R.    974,  985 
Heidenhain,  M.    85,  361 
Heilbronn,  L.  V.    232,  328,  350 
Heilmann,  Gerard    1080 
Hele-Shaw,  H.  S.    774 
Helianthus    913 
Helicina    777 
Helicometer   786 
Heliolites    513 
Heliozoa    465 
Heilmann,  G."    411 

Helmholtz,  H.  von    2,   10,  12,  42,  393 
Hemicardium    829 
Hemmingsen,  Axel  M.    133 
Henderson,  L.  H.   254;  L.  J.   27;  W.  P. 

510 
Henle,  F.  G.  J.    73 
Hennessy,  H.    532 
Henslow,  G.    916,  930 
Heredity    284,  etc. 
Hero  of  Alexandria    527,  759,  761 
Heron- AUen,  E.    649,  704,  786 


1106 


INDEX 


Herpetomonas    431 
Herrera,  A.  L.    649 
Herring,  growth  of    181,  183 
Herschel,    Sir   John    2,    24,    lo7,    266, 

976 
Hertwig,  C.  306;  G.  and  P.  221 ;  0.  85, 

226,  344,  605;  R.    60,  229,  456 
Hertzog,  R.  0.    858 
Hesse,  W.    952 
Heterogony    205,  209,  279 
Heterophy ilia    621 
Hexactinellids    662,  688 
Heymons,  R.    254 
Heyn,  A.  M.  J.    243 
Hexagonal  symmetry  496,  499,  507,  etc. 
Hickson,  S.  J.    657 
Hilb0rt,  David    382 
HiU,  A.  V.    248,  438,  449;  F.  J.    72; 

John    545 
Hime,  H.  W.  L.    811 
Hindekoper,  R.  S.    876 
Hippopus    219,  824 
Hirata,  M.    525 
His,  W.    83,  84,  85,  111 
Hobbes,  Thomas    290 
Hober,  R.    250,  304 
Hofbauer,  C.    180 
Hoffmann,  C.    892;  H.    21,  217 
Hofmeister,  F.    171,  344,  358,  387,  481, 

915,  919 
Hogben,  Lancelot    98,  147 
Hogue,  Mary  J.    363 
Holism    1020 

Holland,  R.  H.    260;  W.  J.    1006 
Holmes,  Eric    265;  0.  W.    969,  1050 
Holothurians    677,  718 
Hooke,  Robert   341,  348,  351,  358,  482, 

512,  545,  805 
Hooker,  R.  H.    217 
Hookham's  crystals    659 
Hop,    growth   of  238,    spirals    of    820, 

891 
Hormones    263 
Horns    874 

Horse,  skull  of    1076;  teeth  of    904-6 
Horse-chestnut,  leaf  of    1047 
Houssay,  F.    774 
Howes,  G.  B.    1022 
Hubbs,  L.    170 
Huia  bird    897 
Hukusina,  H.    523 
Hull,  C.  H.    98 
Hultmann,  F.  W.    544 
Humboldt,  A.  von    73,  251 
Hume,  David    49,  123,  143,  960 
Hunter,  John    266,  544,  897,  949,  955, 

1020 
Huntingdon,  E.    240 
Huxley,  Julian  S.   206;  T.  H.    17,  1020, 

1031,  1065 


Huygens,  Chr.  609,  1029 

Hyacinth  628 

Hyalaea  833 

Hyaloneraa    677 

Hyatt,  Alfred    806 

Hyde,  Ida  H.    243,  293,  323,  332 

Hydra    282,  294,  382 

Hydractinia    559 

Hyperia    1069 

Hypodermis    506 

Hyrachyus    1072,   1074 

Hyracotherinm    1080 

Hyrax    903 

Ichthyosaurus    1068 

Icosahedron    724,  735 

Iguanodon    1005 

Ikada,  Y.    1084 

Imbert,  A.    449 

Inachus,  sperm -cells  of   440 

Index,  ponderal    195 

Indol-acetic  acid    263 

Infusoria,  ciliate    296,  743 

Insect-eye    510,  512 

Insulin    264 

Intussusception    347 

Inulin    665 

Ionic  regulation    463 

Irvine,  Robert    648,  669 

Isely,  L.    768 

Ishikawa,  C.    339 

Isocardia    839 

Isoperimetrical  problems    566^ 

Jackson,  C.  H.    110,  184 

Jacob,  Captain    544 

Jaeger,  F.  M.    357 

Jamin,  J.  C.    651 

Janet,  Paul    6 

Janisch,  E.    221 

Japp,  F.  R.    651 

Jeffreys,  Harold    505 

Jenkin,  C.  F.    679,  997 

Jenkinson,  J.  W.    191,  746 

Jennings,  H.  F.   361,  746^  Vaughan  656 

Jensen,  Boyson    263,  360 

Johansen,  A.  C.    229 

Johnson,  E.  N.   649;  Dr  Samiiel   2,  90, 

92,  184 
Johnstone,  J.    189 
Joly,  John    14,  92,  238 
Jonquieres,  Alfred  de    733 
Jorgensen,  E.    617 
Jost,  L.    222,  483 
Juncus,  pith  of   546 
Jungermannia    638 
Juhn,  Mary    267 
JuU,  M.  A.    943 
Jurine,  Louis   11 
Just,  E.    335 


INDEX 


1107 


Kagi,  Nobumasa    165 

Kahlenberg,  Louis  457 

Kangaroo    1005 

Kant,  Immanuel    1,  4,  21,  1020 

Kappers,  J.  Ariens    186,  264 

Kapteyn,  J.  C,    133 

Keats,  John    949 

Keill,  James    11 

Keller,  H.    986 

KeUieott,  W.  E.    187 

Kellner,  G.  W.    21 

Kelvin  2,  14,  19,  26,  35,  137,  348,  548, 

551,  690 
Kent,  W.  SavUle    406,  408,  433 
Kepler,  J.    9,  284,  526,  528,  695,  724, 

731,  923,  931 
Kerr,  Sir  J.  Graham    806 
Keynes,  J.  M.    89,  121,  784,  927 
Kidney-beans    524 
Kielmeyer,  C.  F.  von   79 
Kienitz-GerlofiF,  F.    481,  595,  639 
Kieser,  D.  C.    532,  545 
Kirby  and  Spence    36 
Kircher,  Athanasius    45,  977 
Kirch  hofif,  G.    393 
Kirkman,  T.  P.    597 
Kirkpatrick,  R.    671 
Kitching,  J.  A.    295 
Klatt,  B.    189 
Klebs,  G.    482 
Klein,  Felix    2 
Klem,  Alfred    153 
Klugel,  G.  S.    531,  538,  544 
Knoll,  Philipp  949 
Kooh,  G.  von    218 
Koehler,  R.    687 
Koenig,  Samuel    529 
Kofoid,  C.  A.    431 
Kogl  and  Kostermans    263 
Koller,  Wilding    59 
KoUi  {or  CoUey),  R.  A.   64 
Kolliker,  A.  von    647 
Kollmann,  M.    300 
Koltzoff,  N.  K.    81,  428,  701 
Konungsberger,  W.  J.    170 
Koppen,  Vladimir    217,  222 
Korotneff,  A.    602 
Kostitzin,  V.  H.    146,  159 
Kraus,  W.    72 
Kreusler,  G.  A.  von    154 
Krizensky,  J.    265,  273 
Krogh,  A.    221,  229,  246,  956 
Kuenen,  D.  J.    251 
Kuhl,  Willi    216 
Kuhn,  A.    54 
Kiihne,  W.    346,  388 
Kukenthal,  Willy    900 
Krister,  E.    663 
Kuwada,  Y.    305 
Kuwana,  Z.    163 


Kuwaori,  M.    1084 

Lafay,  A.    502 

Lafresnaye,  F.  de    985 

Lagena    414,  564 

Lagrange,  J.  L.    27,  356,  931,  950 

Lalanne,  L.    544 

Lamarck,  J.  B.  de    808,  1023 

Lamarle,  E.    486,  497,  559,  595,  714 

Lamb,  A.  B.    293,  324 

Lame,  Gabriel    923,  1058 

Lamellaria    813 

Lammel,  R.    196 

Lamy's  theorem    468 

Lanchester,  F.  W.    26,  31,  41,  44,  45 

Langmuir,  Irving    72,  449,  464 

Lankester,  Sir  E.  Ray    5,  412,  704 

Lapierre    935 

Laplace,  P.  S.  de    1,  70,  89,  121,  354, 

368 
Larmor,  Sir  Joseph    14,  347,  380,  41^^ 
Lartigues,  Alfred  de    751,  1045 
Latrunculia    690 
Laue  effect    670 
LavPter,  J.  C.    1053 
Lavoisier,  A.  L.    11,  348 
Law,  Borelli's    36,  37;  Brandt's    727; 

Bergmann's      35;     Brooks's      165; 

Errera's  482;Euler's  733;Froude's 

31,  33;  Gibbs-Thomson  451;  Hert- 

wig's    306,   328;   Kielmeyer's    79; 

Lamarle's      487;     Liebig's       153; 

Maupertuis's     356;    Miiller's     725; 

Stokes's     71;    Van't    HoflF's     221; 

Verhulst- Pearl    141;  Weber's    133; 

Wolff's    3,  79,  285 
Lea,  Einar    182;  Rosa  M.    183 
Leaves,  forms  of    1041 
Leblond,  E.    328 
Le  Breton,  E.    34,  60,  162 
Le  Chatelier,  Louis    445 
Leche,  W.    906 

Ledingham,  J.  C.  G.    248,  360 
Leduc,   Stephane    283,   297,   324,   501, 

649 
Leeuwenhoek,  A.  van   60,  180,  358,  512 
Leger,  L.    653 
Le  Hello,  P.    37 

Lehmann,  O.    437,  563,  675,  788,  852 
Leibniz,   G.    W.    von    4,    6,    290,   356, 

609 
Leidenfrost,  J.  G.    446 
Leidy,  J.    433,  707 
Leineraann,  K.    512 
Leitch,  I.    223 
Leitgeb,  H.    481 
Lemna    67,  263 
Lendenfeld,  R.    46 
Lengerich,  Hans    619 
Length-weight  coefficient    194,  1089 


1108 


INDEX 


Leonardo  da  Vinci    1,  912,  932,  954 

Leonardo  of  Pisa    923 

Lepidesthes    505 

Lepidopus    1038 

Leptocephalus    169 

Lesage,  G.  L.    26 

Lesbia    267 

Leslie,  Sir  John    294,  751,  770,  815 

Lesqueureusia    706 

Lestiboudois,  T.    916 

Levers,  orders  of   988 

Levi,  G.    2,  60,  62 

Lewis,  C.  M.    447;  F.  T.    58,  357,  517, 

551,  774 
L'Heritier,  P.  L.    162 
Lhuiller,  S.  A.  J.    530,  536 
Liebig,  G.  W.    2,  153 
Liesegang's  rings    659 
Light,  pressure  of   337 
Lilienthal    Otto    41,  43,  961,  964 
Lillie,   F.    R.     5,    268,   275,   337,   558; 

R.  S.    288,  314,  322,  331,  649 
Lilliput    22 
Lima    827 
Limnaea    312 
Limpet    177 

Lines  of  force    294;  of  growth    825 
Lingelsheim,  A.    667 
Lingula    412 

Linnaeus    144,  735,  804,  964,  1028 
Lmsser,  C.    217 
Lippmann,  Gabriel    121,  464 
Lister,  J.  J.    671 
Listing,  J.  G.    609,  820 
Lith    488 

Lithocardium    824 
Lithocubus    715 
Lithostrotion    512,  513 
Lituites    802,  809 
Livi,  R.    194 
Lloyd,  LI.    219 
Lobatschevsky,  N.    11 
Lobster    162,  278 
Locust    164 
Loddigesia    267 
Lodge,  Oliver    288,  1027 
Loeb,  J.    230,  245,  250,  256,  266,  276, 

287,  330,  335,  626 
Loewy,  A.    450 
Logarithmic  spiral    1047,  etc. 
Logistic  curve    145 
Lo  Monaco,  Domenico    163 
Lonnberg,  E.    876,  878,  893,  895 
Loria,  Gino    368,  755 
Lotka,    A.    J.      11,    19,    76,    143,    158, 

2a5 
Lotze,  R.  H.    83 
Lucanus    208 

Lucas,  Eduard    609,  923;  F.  A.    271 
Luciani,  Luigi    163 


Lucretius  6,  76,  163,  183,  270,  290,  740 
Ludwig,   Carl   G.     341;     F.   916;     W. 

747 
Lumbri  cuius    275 
Lunar  influence    242 
Lund,  E.  J.    280 
Lupa    1057 

Lupine,  growth  of    159,'  223 
Luyten,  W.  J.    121 
Lychnocanna    721 
Lyon,  C.  J.    240 
Lyons,  C.  G.    72 

Macalister,  Alexander   816;   Sir  Donald 

967    979 
MacaUum,  A.  B.    447,  458,  629;    J.  B. 

745 
McClement,  W.  D.    65 
McCoy,  Frederick    621 
Macdonald,  J.    29 
McDougal,  D.  T.    235,  237,  241 
MacDowall,  E.  C.    162 
Mach,  Ernst    4,  9,  357 
Machairodus,  teeth  of    898 
Mcintosh,  W.  C.    ^97 
McKendrick,  J.  G.   .151,  341 
Mackinnon,  D.  L.    431,  853 
Mackintosh,  N.    176 
Macky,  W.  A.    435 
Maclagan,  D.  Stewart    155 
Maclaurin,  Colin    25,  530,  1096 
McMurrich,  J.  Playfair    239,  954 
Macrocheira    52 
Macroscaphites    809 
Mactra    125,   177 
Magnan,  A.    45 
Maier,  H.  N.    434 
Maize,  growth  of    224,  247,  471 
Malaria  equations    146 
Mall,  F.  P.    745 
Mallock,  A.    235,  238,  934     . 
Malpighi,  Marcello    341 
Maltaux,  Mile    225 
Malthus,  T.  R.    143 
Maraldi,  J.  P.    528,  695;    angle  of  498, 

682,  712,  720 
xMarbled  papers    1048 
Marriott,  R.  H.    350 
Marsh,  0.  C.    1067 
Marshall,  E.  K.    248 
Marsigli,  Comte  L.  F.  de    934,  971 
Martin,  C.  H.    431;  L.    525 
Mason,  C.  W.    55 
Massart,  J.    225 
Mast,  S.  0.    359 
Mastodon    899 
Matthew,  A.    1006 
Matthews,  A.  P.    280,  444,  455 
Maupertuis,  P.  L.  M.  de    4,  6,  7,  356, 

1031 


INDEX 


1109 


Maxwell,  J.  Clerk  12,  14,  26,  64,  71,  75, 
136,  291,  354,  368,  374,  378,  465, 
467,  609,  794,  820,  964,  989,  997, 
1084 

Mayer,  A.    356 

Mean  sea  level    137 

Medawar,  P.  B.    156 

Medusochloris    397 

Medusoid    120,  396,  619 

Medvedewa,  Mme    251 

Meek,  A.    169,  198 

Megusar,  F.    165 

Meige,  H.    1053 

Melanehthon,  Philip    5 

Meldola,  R.    958 

Melipona    525,  539 

MeUanby,  Kenneth    220 

Melo    779 

Melobesia    646 

Melssens,  L.  H.  F.    451,  482 

Mensbrugghe,  G.  van  der   361,  475,  579 

Meredith,  H.  V.    113 

Merke,  E.    678 

Merrill,  Margaret    101 

Mersenne,  Marin    754,  809 

Mesooarpus    460 

Mesohippus    1079 

Metzger,  E.  B.    553 

Meves,  F.    293,  438,  455 

Meyer,  Arthur  665;  G.  H.  12,  988; 
Hermann    976,  979 

MiceUae    347,  747 

Miehaelis,  L.    449 

Mieseher,  J.    339 

Miliolina    859 

Mill,  J.  S.    9 

Millikan,  R.  A.    74,  970,  997 

MiUer,  W.  J.    32 

Milner,  R.  S.    447 

Milton,  John     1097 

Mimicry    958 

Minchin,  E.  A.    430,  679 

Miner,  J.  R.    158 

Minimum,  Law  of    153 

Minkowski,  Hermann    566 

Minnow    117,  133 

Minot,  C.  S.    61,  110,  141,  1030 

Miohippus    1079 

Mitchell,  Sir  P.  Chalmers    1002 

Mitochondria    296,  455 

Mitosis    300 

Mitra    143 

Mobius,  K.    684;    M.    341 

Mogk,  H.    1058 

Mohl,  H.  von    888 

Moina    231 

Mole,  forelimb  of   963 

Mole-cricket,  chromosomes  of  315 

Moller,  V.  von    856 

Monad    435;    monadology    7 


Monnier,  A-    115,  256;   Denis    649 

Monolayers    352,  464 

Montbeliard,  P.  G.  de    100 

Moonfish    1021 

Moore,  A.  R.    347 

Morey,  S.    422 

Morgan,  T.  H.    262,  271,  275,  279,  340 

Morita    999 

Morosow,  A.  W.    666 

Morse,  Max    265 

Mortality,  curve  of    178 

Moseley,   H.     12,   661,   770,   784,   811, 

876;    H.  G.  J.    784;    H.  N.  784 
Moss,  gemma  of   595,  638 
MouiUard,  L.  P.    46,  205 
Mouse,  growth  of    161 
Moving  average    239 
Mrazek,  A.    327 
Mucor,  sporangium  of   479 
Mud,  cracks  in    516 
Mullen hof,  K.  von    42,  540 
Miiller,  Fritz   3;   Johannes  73,  697,  725 
MuUet    179 
Mummery,  J.  H.    657 
Munro,  H.    512 
Murie,  James    897 
Murex    180,  781 

Murray,  C.  D.    950,  957;    H.  A.    243 
Musk  ox,  horns  of    877 
Mutations     1095 
Myers,  J.  E.    661 
Mymaridae    47 
Myonemes    701 
Mytilus    1061 
i  Mj'xomycetes    346 

Naber,  H.  A.    762 

Nageh,  C.   242,  289,  347,  359,  387,  403, 

631,  747 
Nakaya,  U.    411 
Napier,  James  R.    32;    of  Merchiston 

68 
Narwhal    907 
Nassellaria    712 
Natica    613 

Naumann,  C.  F.    788,  796,  916 
Nautilus    749,  765,  813,  840,  843,  855, 

1047 
Necturus    265 

Needham,  Joseph  79,  98,  108,  141,  268 
Negligibility,  principle  of    1029 
Neottia,  pollen  of   488,  631 
Nereis,  egg  of   558,  633 
Neumayr,  M.    870 
Newell,  H.  F.    395 
Newton,  Sir  Isaac    1,  6,  9,  11,  20,  55, 

284,  288,  755,  989 
Nicholson,  H.  A.    621;    J.  W.  690 
Niclas,  P.    755 
Nielson,  N.    263 


1110 


INDEX 


Nino,  el    220 
Noboru  Abe    167 
Nodoid    369,  425 
Nodosaria    421 
Noll,  Fritz    154 
Norman,  A.  M.    704 
Norris,  Richard    437 
Northrop,  J.  H.    230 
Nostoc    477,  493 
Notosuchus'  1065 
Nuclear  spindle    299 
Nummulite    811,  855 
Niissbaum,  M.    343 

Oak,  an  ancient    240 

Odontoceti    1024 

Odquist,  G.    328 

Oedogonium    487 

Ogilvie-Gordon,  M.  M.    654 

Oithona    1055 

Okada,  Y.  K.    61 

Oken,  L.    5,  546,  751    888,  912,  1023 

Okuso  Hamai    167 

Olivier,  Theodore    812 

Oljhovikov,  M.  A.    241 

Onslow,  H.    55 

Onthophagus    214 

Operculina    857 

Operculum  of  gastropods    774 

Ophiurid  larvae    626 

Oppel,  A.    184 

Orbitolites    360,  852 

Orbulina    88,  376,  861 

Ord,  W.  M.    653 

Ornithosaurians    1069 

Orthagoriscus    1048,  1064 

Orthoceras    765,  809,  841,   1047 

Orthogenesis    807 

Orthotoluidene    370 

Orton,  J.  H.    165,   177 

Osborne,   H.  F.     147,   900,  ^05,    1019, 

1036,  1072 
Oscillatoria    477 
Osmosis    18,  243,  etc. 
Osmunda    63 1,  640 
Ostracoda,  165 
Ostraea    165,  219 
Ostrich    47,  264,  1011 
Ostwald,   Wilhelm    74,   171,   256,  658; 

Wolfgang    162,  253,  327,  450 
Otoliths    666 
Ottestad,  Per    153,   154 
Overbeck,  A.    393,  423 
Ovis  Ammon,  Poli  etc,    882 
Owen,  Sir  R.  29,  290,  806,  838,  900,  906, 

949,  955 
Ox,  growth  of   201 
Oxyotes    662 

Pacioli,  Luca    89 


Packard,  A.  S.    937 

Pai,  Kesava    151 

Palaeechinus    946 

Palm    888 

Paludina    777 

Pander,  C.  H.    83 

Pangenesis    788 

Pantin,  C.  J.    359;    F.  A.    254 

Paphia    177 

Papilio    664 

Papillon,  Femand    14 

Parabolic  girder    995 

Paralomis    1057 

Paramoecium    425 

Parker,  A.  S.    260;    G.  H.    172;    S.  L. 

155 
Parr,  Old    132 
Parsons,  H.  de  B.    1089 
Partitions,  theory  of    609 
Pascal,  Blaise    2,  11 
Pascher,  A.    398,  632 
Passiflora,  pollen  of   631 
Pasteur,  L.    650 
Patella    167,  805 
Pattern,  1090 
Pauli,  W.    360,  656,  669 
Paulian,  Rene    215 
Peari,   Raymond    118,    149,    155,    158, 

186,  193,  937,  959 
Pearsall,  W.  H.    223 
Pearse,  H.  L.    263 
Pearson,  G.  A.   236;  Kari  60,  119,  125, 

166,  190,  884 
Peas,  growth  of   223 
Pecten    830 

Peddie,  William    318,  561 
Pellew,  Ann    502 
Pellia,  spores  of   478 
Pelseneer,  Paul    833 
Pendulum    39 
Peneroplis    868 
Penguin    1015 
Penicillin m    244 
Penrose,  M.    449 
Periblem    494 
Peridinium    487,  615,  738 
Periploca,  pollen  of    631 
Permeability,  magnetic    311 
Perrin,  J.    9,  74 
Peter,  Kari    221,  229 
Peters,  R.  A.    257 
Petersen,  Chr.    758,  796,  808;   C.  J.  G. 

177 
Petsch,  T.    29 
Pettersson,  Otto    240 
Pettigrew,  J.  B.    744,  995 
Petty,  Sir  William    90,  98 
Pfeffer,  W.   159,  222,  335,  336,  440,  627, 

985 
Pfeiffer,  F.  X.    931;    N.    241 


INDEX 


1111 


Pfluger,  E.    974 

Phagocytosis    360 

Phascura    642 

Phase-beauty    194 

Phasianella    772,  816 

Phatnaspis    728 

Philip,  H.  F.    48 

Philipsastraea    514 

Philolaus,   1097 

Phormosoma    947 

Phractaspis    730 

Phvllophorus    718 

Phyllotaxis    912 

Phylogeny    340,  413,  etc. 

Picken,  L.  E.  R.    404 

Pike,  F.  H.    950 

Pileopsis    814 

Pinaeoceras    817 

Pine-cone    515,  916 

Pinna    1061 

Pirondin?,  G.    813 

Pisum    223 

Pith  of  rush    547 

Pithecanthropus    1085 

Pituitary  body    264 

Placuna    166  ' 

Plaice    195,  213,  229,  1088 

Planaria    282 

Planck,  Max    21 

Planorbis,  803,  813,  816 

Plateau,  Felix  38,  385;  Joseph  71, 
332,  351,  368,  373,  398,  465,  475, 
496,  549,  563,  596,  712,  710 

Plato    2,  152,  695,  724,  1026,  1097 

Platonic  bodies    724,  734 

Plerome    464 

Pleurocarpus    460 

Pleurotomaria    777 

Pliny    527,  749 

Plutarch    1026 

Pluteus  larva    648 

Podocoryne    559 

Podocyrtis    718,  721 

Poincare,  Henri    21,   108,  284,   1028 

PoiseuiUe,  J.  L.  M.    956 

Poisson,  S.  D.    89,  118 

Polar  furrow,  487 

Polarity,  morphological    280 

Polistes    488,  526 

Political  arithmetic    90 

Pollen    67,  630 

Polyhalite    667 

Polyhedra    550,  732,  etc. 

Poly  prion    1063 

Polyspecmy    335 

Polytrichum    576 

Poraacanthus    1062 

Ponder,  Eric    436,  438 

Ponderal  index    194 

Pony,  Shetland    268 


Popoff,  M.    60,  456 

Poppy  seed  564 

Population    142 

Porcelain,  crackles  in    523 

Porchet,  M.  S.    256 

Portier,  Paul    248 

Potamides    813 

Potassium    459 

Potter's  wheel    392 

Pottery,  Roman    415 

Potts,  R.    246 

Pouchet,  G.    648 

Poulton,  Sir  E.  B.    958;    E.  P.  95 

Powell,  B.    544 

Prenant,  A.    293,  299,  323,  456,  459; 

Marcel    670 
Preston,  F.  W.    521;    R.  D.    403,  743 
Price,  A.  T.    967 
Price-Jones,  C.    819 
Priestley,  J.  M.    223,  471,  480,  495 
Pringsheim,  N.    601 
Prismatium    718 

Probability,  theory  of   20,  118,  136 
Procetus    1024 
Productus    829 
Protococcus    80,  406,  477 
Protohippus    1080 
Przibram,    Hans     165,    194,    216,    275, 

278,  360,  858;    Karl    75,  76 
Psammobia    1061 
Pseudopriacanthus    1063 
Pteranodon    1069 
Pteridophora    267 
Pteris,  antheridia  of    643 
Pteropods    832 
Pucraria    159 
Puget,  M.    512 

Pulvinulina    750,  766,  858,  863 
Pupa    808 

Purkinje,  J.  E.    290 
Putter,  A.    56,  222,  360,  747 
Pyrosoma,  egg  of   602 
Pythagoras    2,  759,  1026,  1097 

Quadrant,  bisection  of   580 

Quadrula    702 

Quagga    1090 

Quastel,  J.  H.    449 

Quekett,  J.  T.    654 

QuetelH>r  A.    89,  95,   136 

Quincke,  H.  H.   322, 'S31,  359,  418,  447, 

502,  579,  659 
Quirang,  D.  P.    203 

Rabbit,  growth  of    117,   187;    skull  of 

1077 
Rabelais    5 
Rabl,  K.    60,  487 
Radau,  M.    38 
Radiolaria    694 


1112 


INDEX 


Rado,  Tibor    382 

Rainey,  George    10,  653 

Ram,  horns  of    878 

Raman,  8ir  C.  V.    656 

Rameau  and  Sarrus    33 

Rammer,  W.    165 

Ramsden,  W.    451 

Ramulina    413 

Ranella    781 

Rankine,    W.   J.   Macquorn    355,    755, 

997,  1017 
Ransom's  waves    294 
Ranzio,  Silvio    242 
Raphides    646 
Rashevsky,  N.    70,  333 
Rasumowsky,  V.  I.    979 
Rauber,  A.    490,  604,  633,  970,  979 
Ray,  John    4 
Rayleigh,    Lord     26,    55,    69,    72,    304, 

380,  398,  418,  449,  469,  502,  510, 

659 
Read,  L.  J.    149 
Reaumur,  R.  A.  de    12,  162,  271,  528, 

684,  781,  1031 
Redwood,  growth  of    237 
Reed,  H.  S.    260 
Regan,  C.  Tate    423 
Regeneration    271 
Reichenbach,  Hans    123 
Reichert,  C.  B.    290 
Reid,  E.  Way  mouth    437 
Reighart,  J,  E.    959 
Reindeer  beetle    209 
Reinecke,  J.  C.  M.    785 
Reinke,  J.    481,  576 
Relay  crystals    663 
Remak,  Robert    341,  456 
Reusch,  B.    55 
Revenstorf   979 
Reymond,  Du  Bois    73 
Rhabdammina    870 
Rheophax    422 
Rhinoceros    807,  874,  905 
Rhode,  von    194 
Rhumbler,  L.    293,  301,  419,  506,  704, 

807,  852,  863,  871,  892 
Riccia    593,  677 
Rice,  J.    400 
Richards,  0.  W.    153 
Richardson,  G.  M.    650 
Richet,  Charles    34 
Richtmeyer,  F.  K.    54 
Rideal,  E.  K.    72,  437 
Ridge  way.  Sir  W.    1091 
Riemann,  B.    609 
Ripples    510 
Rissoa    565 
Rivularia    477 
Robb,  R.  C.    187 
Robert,  A.    483,  556,  568,  601 


Robertson,  Muriel  43;    R.    162;  T.   B. 

113,  130,  257,  331 
Robin,  Ch.    1020 
Robinson,  Wilfred    746 
Roederer,  J.  B.    98 
Rohrig,  A.    437,  892     , 
Rollefson,  Gunner    179 
Rolleston,  G.    206 
Roosevelt,  Th.    961 
Rorqual,  Sibbald's    69,   173 
Rose,  Gustav    654 
Ross,  Ronald    146,   158 
Rotalia    365,  792,  865 
Rouleaux  of  blood-corpuscles    439 
Roulettes    368 
Roux,  W.    12,  84,  287,  337,  584,  607, 

952 
Rudolf,  G.  de  M.    980 
Runge,  F.  F.    659 
Runnstrom,  J.    438 
Rusconi,  M.    487 
Ruskin,  John    661,  932 
Russell,  E.  S.    167 
Russow,  110 
Rutherford,  Lord    286 
Rutimeyer,  L.    906 
Rvder,  J.  A.    330,  360,  899,  937 
Rzasnicki,  Adolf    1091 

Sachs,  J.  60,  63,  115,  222,  344,  933,  974 

Sachs's  rule    450,  475,  567,  601 

Sagrina    422 

St  Ange,  G.  J.  Martin    243 

St  Loup,  R.    162 

St  Venant,  Barre  de    884,  891,  967 

SaUsbury,  E.  J.    217 

Salpingoeca    408 

Salvia,  hairs  of   488 

Salvinia    601 

Sambon,  L.  W.    944 

Samec,  M.    656,  669 

Samter,  M.  and  Heymons    254 

Sandberger,  G.    796 

Saponin    446 

Sapphirina    1055 

Sarcode    18 

Sasaki,  Kichiro    198,  301 

Saunders,  A.  M.  Carr    148 

Savart,  F.    380,  771 

Saver,  A.  B.    68 

Scklaria    780,  804,  813 

Scale,  effect  of    23,  673 

Scammon,  R.  E.  100,  108 

Scaphites    809 

Scapholeberis    506 

Scapula,  human    1082 

Scarus    1062 

Schacko,  G.    866 

SchaefFer,  G.    60,  162 

Schafer,  E.  A.  Sharpey    264 


INDEX 


1113 


Schaper,  A.    168 

Schaudinn,  F,    75,  456 

Scheel,  K.    363 

Schellbach,  K.  H.    325,  544 

Schewiakoff,  W.    316,  701 

Schimper,  C.  F.    915 

Schistosoma    943 

Schleiden,  Matthias    341,  546 

Schmaltz,  A.    969 

Schmankewitz,  W.    251,  254 

Schmidt,  Johannes    169,  191,  238,  656; 

R.    979;    W.  J.    312,  647,  835 
Schofield,  R.  K.    986 
Schoneboom,  C.  G.    467 
Schonfliess,  A.    348 

Schultze,  F.  EQhard  692;   Max   17,510 
Schiitte,  D.  J.    234 
Schutz,  Fr.    445 
Schwalbe,  G.    951 
Schwan,  Albrecht    688  • 
Schwann,   Theodor     12,    17,    341,   508, 

603,  854 
Schwendener,  S.    359,  481,  973 
Scoresby,  William    411 
Scorpaena    1063 
Scorpioid  cyme    767 
Scot,  Michael    923 
Scott,  W.  B.    1081 
Scyromathia    1057  " 

Sea-anemones    184,  244 
Searle,  H.    745 
Seasonal  growth    232 
Sea-urchins  625,  944;  eggs  of  353,  602; 

growth  of    229 
Sectio  aurea,  s.  divina    923 
Sedgwick,  Adam    79,  342,  344 
Sedillot,  C.  E.    985 
Sedum    628 
Segner,  J,  A.  von    351 
Selaginella    638 

Selection,  natural    770,  840,  936 
Selenka,  L.    653,  689 
Seligmann,  G.    500 
Selous,  F.  C.    961 
Semi-permeable  membranes    437 
Semon,  R.    653,  687 
Seneca    2 
Senility    132 
Sepia    838 

Septa    840;    of  corals    621 
Sequoia    237,  240 
Serbetis,  C.  D.    179 
Serpula    865 
Serret,  Paul    813 
Shad,  growth  of    196 
Shapley,  Harlow    68,  221 
Sharpe,  D.    728;    Sam    544 
Sharpey,  William    955 
Shearing  stress    981,   1017,   1040 
Sheep    876 


Shinryo,  M.    265 

Ship,  speed  of    31 

Shrinkage    563 

Shuleikin,  W.    909 

Siegwart,  J.  E.    544 

Sigaretus    814 

Silicoflagellata    717 

Silkworm,  growth  of    163 

Similitude,  principle  of    25,  27 

Simmonds,  Katherine    98,  107 

Sinclair,  Mary  E.    363 

Sinnett,  E.  W.    1049 

Siphonogorgia    647 

Sitter,  W.  de    384 

Slack,  H.  J.    510 

Slator,  A.    153 

Slipper  limpet    166 

Sloan,  Tod    993 

Sloth    899,  1007 

Smeaton,  John    29 

Smellie,  W.    543 

Smilodon    271 

Smith,  Adam    143;   D.  B.   967;   Homer 

248;    Kenneth    65 
Smoke    504 

Smoluchowski,  M.  von    74,  439,  450 
Smuts,  J.  C.    1020 
Snow-crystals    411,  599,  695 
Soap-bubbles    600 
Socrates    695 
Soding,  H.    281 
Sohncke,  L.  A.    348 
Solanum  Dulcamara    889 
Solarium    804,  813 
Solecurtus    827 
Solen    828 
Solger,  B.    979 
Sollas,  W.  J.    675,  692 
Sorby,  H.  C.    646,   1037 
Sosman,  R.  B.    521 
Southwell,  R.  V.    502,  885,  967 
Spallanzani,  L.    271 
Span  of  arms    189 
Spandel,  Erich    419 
Spangenberg,  Fr.    559 
Spath,  L.  F.    806 
Spatial  complexes    610^ 
Spek,  J.    295,  330 
Spencer,  Herbert    26,   143,   194,  255 
Spermatozoon,  path  of    434 
Sperm -cells    441 
Sphacelaria    571 
Sphaerechinus    229,  -75 
Sphagnum    634,  641 
Spherocrystals    665 
Spherulites    655 
Sphodromantis    165 
Spicules    547;    of  sponges    679;    holo- 

thurians    687;    radiolaria    697 
Spider's  web    386 


1114 


INDEX 


Spindle,  nuclear    308 

Spira  mirabilis    758 

Spiral,  of  Archimedes  752 ;  logarithmic 

748 
Spireme    312 
Spirifer    829,  831 
Spirillum    433 
Spirochaetes    383,  410,  432 
Spirographis    848 
Spirogyra    16,  171,  371,  378,  401,  578, 

677 
Spirorbis    848,  865 
Spirula    805,  815 
Splashes    389,  393 
Sponge-spicules    675 
Spottiswoode,  W.    1097 
Squilla    164 
Srinavasam,  P.  S.    656 
Stability,  elastic    1015 
Stag-beetle    208 
Staigmiiller,  H.    1053 
Standard  deviation    124 
Starch    665 
Stark,  J.    263 
Starting,  E.  H.    264 
Starvation    189,  265 
Staudinger,  H.    303 
Stay,  Benedict    531 
Stefanowska,  Mile    115,   116 
Stegocephalus    1059 
Stegosaurus    1006,  1009,  1067 
Steinbach,  C.    973 
Steiner,  Jacob    357,  376,  734,  936 
Steinmann,  G.    665 
Stellate  cells    547 
Stenohaline    250 
Stentor    275,  424 

Stephenson,  John    341 ;    Robert    1012 
Stereochemistry    651 
Sternoptyx    1062 
Sternum  of  bird    1013 
Stevenson,  R.  L.    20 
Stillman,  J.  D.    993 
Stokes,  Sir  G.  G.    71 
Stomach,  muscles  of    743 
Stomata    627 
Stomatella    813 
Strangeways,  T.  S.    305 
Strasburger,  E.    642 
Straub,  J.    440 
Straus-Diirckheim,  H.  E.    38 
Stream-lines    941,  965,   1048 
Strehlke,  F.    385 
Strength  of  materials    973 
Strontium    697 

Stuart,  Norman    667;    T.  P.  A.    1050 
Studer,  T.    647,  657 
Succinea    815 
SuflFert,  F.    55 
Sun-animalcule    427 


Sunflower    258,  913 

Surface  energy    354 

Surirella    434 

Sutures  of  cephalopods    840 

Suzuki,  Kuro    241 

Svedberg,  T.    65 

Swammerdam,  Jan    12,  164,  512,  528, 

531,  538,  604,  785 
Swamy,  S.  R.    656 
Swezy,  Olive    431 
Swinnerton,  H.  H.    831 
Sydney,  G.  J.    77 
Sylvester,  J.  J.    2,  1031 
Symmetry,  meaning  of   357 
Synapta,  egg  of   689:    spicules  of   688 
Synge,  J.  L.    911 
Synhelia    514 
Szava-Kovatz,  J.    217 
Szent-Gyorgyi,  A.  von    457 
Szielasko,  A.    936 

Tabaka,  T.    231 

Tadpole,  growth  of    168,  271,  282 

Tait,  P.  G.    2,  59,  354,  459,  609,  924 

Takahasi,  K.    500 

Taonia    476 

Tapetum    641 

Tate,  John    10 

Tate  Regan,  C.    423 

Taylor,  G.  I.    398,  416;   Jeremy    526 

Teeth    846;    of  dolphin  .  899 ;    elephant 

901,    horse    908 
Teissier,  George    248 
Teixiera,  F.  G.  de    318,  755 
Temperature  coefficient    16,  221 
Tenebrio    272 
Teodoresco,  E.  C.    241 
Terao,  A.    231 
Terebra    772,  787,  911 
Terni,  L.    60 
Terquem,  Olry    378,  544 
Tesch,  J.  J.    199 
Tetracitula    487 
Tetractinellida    586 
Tetrahedron    497 
Tetrakaidekahedron    734 
Tetraspores    632 
Textularia    867 
Thamnastraea    514 
Thatcheria    805,  813 
Thayer,  J.  E.    960 
Thecosmiha    512 
Theel,  H.    678 
Theocyrtis    716 
Thevenot,  Jean    538 
Thienemann,  F.  A.  L.    936 
Thigh-bone    977 
Thigmotaxis    360 
Thimann,  K.  V.    263 
Thoma,  R.    951 


INDEX 


1115 


Thompson,  W.  R.    146 

Thomson,  Allen    956;    James    26,  355, 

418,    502,    521;     J.    Arthur     704; 

J.J.  395,447;  S.  H.   156;  Warren 

Sv   147;   C.  Wyville   705;    WiUiam, 

see  Kelvin 
Thornton,  H.  G.    151 
Threads    380 
Thy one    717 
Thyroid  gland    186,  265 
Tiegs,  0.  VV.    359 
Tilesius,  B.    939 
Time-element    79 
Tintenpilze    393 
Tintinnus    408 
Tischler,  G.    339 
Tissot,  A.    813,  1033 
Titanotherium    1003,  1075 
Todd,  T.  Wingate    98 
Tomistoma    1065 
Tomkins,  R.  G.    217,  244 
Tomlinson,  C.    418,  579,  661,  723 
Tomotika,  S.    398 
Topinard,  P.    1055 
Topology    609 
Tornier,  G.    1006 
Torpedo    449 
Torque  of  inertia    910 
Torre,  Father  G.  M.  Delia    358 
Torsion    887 
Tortoise    172,  517 
Townsend,  C.  H.    172 
Trachelophyllum    410 
Tradeseantia    388 
Transformations,  theory  of '  1032 
Traube,  M.    457,  649 
Trees,  growth  of    235;    height  of    29 
Trembley,  Abraham    27,  274 
Treutlein,  P.    760 
Trianaea,  hairs  of   387 
Triangular  numbers    512,  760 
Triceratium    511 
Trichodina    381 
Trichomonas    430 
Tridacna    219,  824 
Triepel,  H.    979 
Triolampas    716 
Tripocyrtis    721 
Triton    813 
Trituberculy    900 

Troehus  556,  601,  820;  egg  of  601,  602 
Trophon    780 

Trout,  growth  of  176,  191,  229,  288,  294 
Truman,  A.  E.    831 
Trypanosomes    430 
Tubular  structures    971 
Tubularia    245,  275 
Tur,  Jan    83 
Turbinella    805 
Turbo    771,  775 


Turgor    243,  245 

Turner,  Sir  William    1081 

Turritella    742,  771,  814 

Turtle    519 

Tutton,  A.  E.  H.    38,  696,  731 

Tylor,  E.    380 

Tyndall,  J.  504,  661;   Tyndali's  blue  55 

Types,  Cuvierian    1094 

Typha    631 

Uca    206 

Uhlenhuth,  E.    264 
Ultra-violet  light    241 
Unduloid    369 
Unio    558 
Univalves    812 
Urosalpinx    167 
Uvarow,  B.  P.    219 

V'acuole,  contractile    295 

Vaginicola    409 

Valin,  H.    649 

Vallisneri,  Antonio    271 

Vallisneria    161 

Valonia    403,  743 

Van  Bambeke,  Charles    17,  345 

Van  Beneden,  Eduard    300 

Van  BischofE,  Th.     189 

Van  der  Horst,  J.    54,  264 

V^andermonde,  N.    609 

Van  Heurck,  Henri    434 

Van  Iterson,  G.    553,  743,  85.7 

Van  Rees    550,  595 

Van't  Hoff,  J.  H.    1,  220,  255 

Variability    118 

Variot,  G.    100,  110 

Varnish,  cracks  in    517 

Varves    667 

Vaucheria    376 

Vejdovsky,  F.    327 

Verhaeren,  Emile    1097 

Verhulst,  P.  F.    145 

Vermetus    777 

Verworn,  Max    360,  706,  867 

Vesque,  Julien    646 

Vibrations    509 

Vicq  d'Azyrj,  Felix    9 

Vierordt,  Hermann    108 

Vincent,  J.  H.    510 

Vine-spirals    820 

Violet,  leaf  of    1045 

Virchow,  R.    341,  380,  456 

Virgil    527 

Vis  reyulsionis    279 

Viscose    264 

Vies,  Fred    70 

Vogt,  Carl    649;    H.    538 

Voice    53 

Volkamer,  A.  W.    251 

Voltaire    5,  274 


1116 


INDEX 


Volterra,  Vito    143,  159 

Volvox    48,  406,  613 

Vom  Rath,  0.    315 

Von  Baer,  K.  E.  3,  11,  83,  86,  286 

Von  Buch,  Leopold    785,  845 

Von  Mohl,  H.    290,  341 

Vorsteher,  H.    187 

Vortex  motion    393;    rings    437 

Vorticella    462 

Voss,  H.    63 

Vries,  H.  de    217 

Wager,  H.  W.  T.    418 

Warner,  C.  E.    457;    Gilbert    49,  505; 

G.  T.    418 
Wall,  W.  S.    30 

Wallace,  A.  Russell    544;    Robert    143 
Wallis,  J.    785 
Walton,  W.    27,  769 
Walton  and  Hammond    269 
Warburg,  O.    291 
Warnecke,  P.    189 
Warren,  Ernst    506 
Warren's  truss    981 
Wasp's  nest    540 
Wasteels,  C.  E.    924 
Watase,  S.    602 
Waterflea    214,  231 
Waterhouse,  G.  H.    541 
Watkin,  Emrys    183 
Watson,   C.   M.     146;     D.   S.   M.    960; 

E.  R.    510;    G.  I.    36 
Webb,  D.  A.    250,  463 
Weber,  E.  H.    418;    Max    187 
Weidenreich,  Fr    434,  981 
Weismann,  A.    288 
Wells,  G.  P.    248 
Wenham,  F.  H.    49 
Went,  F.  W.    263 
Werner,  A.  G.    27,  506;    Fritz    214 
Westergaard,  H.    98 
Westwood,  J.  0.    163 
Wettstein,  R,  von    1038 
Weymouth,  F.  W.    125,  156,  177 
Weyrauch,  W.  K.    214 
Whelpton,  P.  K.    147 
Wherry,  D.    881 
Whewell,  William    137 
White,  Gilbert    23;  M.  J.  D.    338 
Whitman,  C.  O.    287,  295,  336,  344 
Whitworth,  W.  A.    757 
Whyte,  R.  G.    241 
Wiener,  Chr.    74,  757 
Wiesner,  J.    216 
Wigglesworth,  V.  B.    245,  756 
Wilcox,  F,  W.    263 
Wildemann,  E.  de    483 
Wilhelmi,  H.    687 
Willcox,  W.  E.  F.    149 


Willem,  Victor    541 

Wiiley,  A.    613,  658,  813,  840 

"Williamson,  W.  C.    654,  871 

Willsia    620 

Wilson,  E.  B.    279,  293,  304,  339,  343, 

490,  558,  633,  689 
Winckworth,  R.    176,  777,  806 
Wingfield,  C.  A.    176 
Wings  of  insects    476,  611 
Winsor,  C.  P.    156 
Wislicenus,  H.    667 
Wissler,  Clark    125 
Withrow,  J.  R.    241 
Wittemore,  J.  K,    368 
Wodehouse,  R.  P.    67,  631 
Woken,  Gertrud    325 
Wolfermann,  H.    979 
Wolff,  C.  Freiherr  von  341;  Julius  976, 

979;    J.  C.  F.    3,  79,  285 
Wolffia    67 

Wood,  R.  W.    690,  853 
Woodcock,  ear  of    1015 
Woodhead,  Sims    648,  669 
Woodland,  W.    687 
Woods,  R.  H.    949 
Woodward,  H.    840 
Worthington,  A.  M.    389;    C.  B.    214 
Wren,  Christopher    785 
Wright,   Almroth    467;     Chauncy     93, 

544;   Sewell    156;   T.  Strethill   359 
Wrinch,  Dorothy    737 
Wyman,  Jeffries    531,  538,  976  ;  Morrill 

976 

Xylotrupa    212 

Year-classes    183 

Yeast    152,  363 

Young,  Thomas    11,  60,  61,  354,  954, 

989;    W.  H.    1032 
Yule,  Sir  G.  Udny    145 

Zalophus    264 
Zamia    515 
Zangger,  H.    451 
Zaphrentis    623 
Zebra    1090 
Zeising,  A.    915,  931 
Zeleny,  C.    277,  391 
Zeuglodon    1024 
Zeuthen,  H.  G.    762 
Ziehen,  Ch.    189 
Zimmermann,  P.  W.    263,  483 
Zinanni,  G.    634 
Zittel,  K.  A.  von    806 
Zoogloea    451 
Zschokke,  F.    979 
Zsigmondi,  Richard-  68 
Zweigelenkige  Muskejn    963