Method  of  Urn.       Uudcr  this  method,  the  auxiliaries  introduced  are
it.^-  the  limits  of  the  ratios  of  the  simultaneous  increments

of  the  primitive  quantities ;  or,  in  other  words,  the  final  ratios  of
these  increments  ;  limits  or  final  ratios  which  we  can  easily  show
to  have  a  determinate  and  finite  value.  A  special  calculus,  which
is  the  equivalent  of  the  infinitesimal  calculus,  is  afterward  em-
ployed, to  rise  from  the  equations  betwi3en  these  limits  to  the
corresponding  equations  between  the  primitive  quantities  them-
selves.
The  power  of  easy  expression  of  the  mathematical  laws  of  phe-
nomena given  by  this  analysis  arises  from  the  calculus  applying,
not  to  the  increments  themselves  of  the  proposed  quantities,  but  to
the  limits  of  the  ratios  of  those  increments ;  and  from  our  being
therefore  able  always  to  substitute  for  each  increment  any  other
magnitude  more  easy  to  treat,  provided  their  final  ratio  is  the  ratio
of  equality ;  or,  in  other  words,  that  the  limit  of  their  ratio  is  unity.
It  is  clear,  in  fact,  that  the  calculus  of  limits  can  be  in  no  way
affected  by  this  substitution.  Starting  from  this  principle,  we  find
nearly  the  equivalent  of  the  facilities  offered  by  the  analysis  of
Leibnitz,  which  are  merely  considered  from  another  point  of  view.
Thus,  curves  will  be  regarded  as  the  limits  of  a  series  of  rectilinear
polygons,  and  variable  motions  as  the  limits  of  an  aggregate  of  uni-
form motions  of  continually  nearer  approximation,  etc.,  etc.  Such
is,  in  substance,  Newton's  conception ;  or  rather,  that  which  Mac-
laurin  and  D'Alembert  have  offered  as  the  most  rational  basis  of


LAGRANGE'S   METHOD.  75

the  transcendental  analysis,  in  the  endeavor  to  fix  and  arrange
Newton's  ideas  on  the  subject.
