There  is  no  lack  of  examples  among  geometers  of  inquisition  into
subjects  wholly  inaccessible  to  human  reason ;  nor  of  obstinacy  in
substituting  argument  for  observation. Sound  biological  specula-
tion, on  the  contrary,  perceives  philosophy  to  rest  on  an  historical
basis  ;  and  this  fulfilment  of  the  first  condition  of  positivism  suffices


THE   MATHEMATICAL  ELEMENT. 791

to  give  sociology  the  supremacy. Again,  the  sense  of  the  invari-
ableness  of  natural  laws  can  not  be  much  developed  in  mathemati-
cal researches,  though  it  originated  there ;  because  the  extreme
simplicity  of  geometrical  and  mechanical  phenomena  hardly  admits
of  a  full  and  practical  generalization  of  this  great  philosophical
idea,  notwithstanding  the  valuable  confirmation  arising  from  its  ex-
tension to  celestial  phenomena. Hence  it  is,  that  mathematicians
drop  the  supposition  of  natural  laws  as  soon  as  they  encounter
phenomena  of  any  considerable  degree  of  complexity,  and  espe-
cially when  human  action  is  in-  any  way  concerned ;  as  we  see  by
their  pretended  calculation  of  chances,  through  a  special  applica-
tion of  mathematical  analysis, — an  extravagance  which  is  wholly
incompatible  with  true  positivity,  but  from  which  the  vulgar  of  our
algebraists  still  expect,  after  a  century  of  wasted  labor,  the  per-
fecting of  some  of  the  most  difScult  of  human  studies. In  the
other  sciences  we  find  still  increasing  manifestations  of  the  invari-
ableness  of  natural  laws ;  but  in  sociology  alone  we  find  the  full
illustration  of  it,  because  there  it  is  extended  to  the  most  complex
of  all  events,  which  were  excluded  even  by  the  Cartesian  philos-
ophy. Whichever  way  we  look  at  the  positive  method;  we  shall
perceive  the  eminent  logical  superiority  of  the  sociological  over  the
mathematical  point  of  view. All  the  logical  resources  that  the  hu-
man mind  can  employ  are  exemplified  in  mathematical  practice ;
but,  through  the  extreme  simplicity  of  the  subjects  to  which  they
are  applied,  the  most  important  of  the  means  can  not  be  defined  ;
and  their  scope  can  be  duly  estimated  only  when  their  chief  destina-
tion is  found,  amidst  the  difficulties  of  an  increasing  complexity  of
phenomena,  in  the  series  of  departments  of  natural  philosophy. A
reaction  ensues,  which  can  not  but  be  highly  favorable  to  mathemati-
cal science,  while  it  exposes  the  precise  value  of  its  claims. 