From  the  foregoing  it  is  plain  that  the  minuend  is  necessarily-
larger  than  the  subtrahent.  For  in  the  process  of  addition  the  minu-
end was  the  sum,  and  the  sum  grew  out  of  the  union  of  tivo  natural
number-pictures.*  Thus  5  minus  g,  or  1 1  minus  12,  or  8  minus  8,
are  combinations  of  numbers  lulioUy  destitute  of  inea)2ing ;  for  no
number,  that  is,  no  result  of  counting,  exists  that  added  to  9  gives
the  sum  5,  or  added  to  12  gives  the  sum  11,  or  added  to  8  gives  8.
What,  then,  is  to  be  done?  Shall  we  banish  entirely  from  arith-
metic such  meaningless  combinations  of  numbers  ;  or,  since  they
have  no  meaning,  shall  we  rather  invest  them  with  one?  If  we  do
the  first,  arithmetic  will  still  be  confined  in  the  strait-jacket  into
which  it  was  forced  by  the  original  definition  of  number  as  the  re-
sult of  counting.  If  we  adopt  the  latter  alternative  we  are  forced
to  extend  our  notion  of  number.  But  in  doing  this,  we  sow  the
first  seeds  of  the  science  of  pure  arithmetic,  an  organic  body  of
knowledge  which  fructifies  all  other  provinces  of  science.
