382  Mr.  milner’s  Obfervations  on  the 
§ 3.  It  will  not  be  difficult  to  fee  the  reafon  of  this,  if 
we  examine  the  demonftration,  which  is  ufually  given 
us  of  this  fecond  propofition. 
The  roots  of  the  biquadratic  equation 
cx  + t )=o  are  fuppofed  to  be  a,  b,  c,  d,  and  the  refults 
which  arife  by  fucceffively  fubffituting  them  for  x 
in  4ArJ- 3 AAf  + 2 b x-e  are  fuppofed  to  be  -R,  +s, 
-t,  + z.  From  which  maclaurin  concludes,  that 
when  abed  are  fubffituted  for  x in  the  quantity 
/+4  mxx^-l  +3  mx  aa;3+/  + 2?KB^-/+«cjf  + /D,  the 
quantities  that  refult  will  become  -mv.x,  +msx , -mrx, 
+mzx,  where,  fays  he,  the  figns  ^eing  alternately  nega- 
tive and  pofitive,  it  follows,  that  a,  b,  c,  d,  mull  be  limits 
of  the  equation  /+4«a,-/x  ^mAx3*  &c-  = °- 
Here  it  is  taken  for  granted,  that  the  quantities  -mRx, 
+ m sx,  -M  TX,  + m z x,  are  alternately  negative  and  po- 
fitive, which  Is  not  true,  unlefs  the  roots  a,  b,  c,  d,  be 
either  all  pofitive  or  all  negative. 
For  fuppofe  a , b,  c,  to  be  pofitive  (a>  quantities,  and  d a 
(a)  Philofophical  Tranfaftions,  vol.  XXXVI.  Mr.  MACLAURIN,  who  is 
here  very  diffufe  upon  this  fubje£l:,  never  mentions  any  exception  of  this  fort. 
In  his  Algebra,  art.  44.  part  2.  he  fays,  he  fhall  only  treat  of.fuch  equations 
as  have  their  roots  pofitive;  but  it  may  be  obferved,  that,  his  reafoning  from 
art.  45.  to  50.  holds  in  all  equations,  the  roots  of  which  are  real.  The  theorem 
in  p.  182.  of  that  treatife  is  not  general,  though  applied  in,  the  eleventh  chapter 
to  the  cfemonilration  of  newton’s  rule  for  finding  impohible  roots  in  all 
equations.  M . ) ' i -i  . j* , -•  r;  rr;  7 -•  V;ov 
negative 
