386  Mr.  milner’s  Obfervations  on-  the 
obvious,  that  the  fame  reafoning  holds  in  equations  of 
higher  dimenfions. 
Thefe  obfervations,  as  far  as  I know,  are  intirely  new. 
The  fundamental  propofition  (.§  4.)  was,  in  the  year 
1775,  communicated  to  Dr.  waring,  Lucafian  profeflor 
of  mathematics  in  this  univerfity,  and  by  him  inferted 
among  the  additions  to  his  Meditationes  Algebraic^ h‘ 
§ 11.  M.  euler,  at  the  conclufion  of  his  13th  chap. 
Calcul.  Different,  has  given  a demonftration  of  des 
cartes’s  rule  for  finding  the  number  of  affirmative  and 
of  negative  roots  in  any  equation,  the  roots  of  which  are 
real.  From  what  I have  already  faid,  his  reafonings  will 
appear  inconclufive,  though  I freely  own,  that  what  he 
has  done  fuggefted  the  following  different  method. 
Suppofe  (d)  l+«^+»/  + pa,!  . . . +xn=o,  and  the 
roots  of  the  equation  (e)  m+  mx +nxn~l-o  will 
be  limits  of  the  roots  of  the  equation  (d)  ; and  therefore 
there  muft  be  at  leaf!  as  many  pofitive  roots  in  the 
equation  (d)  as  there  are  in  the  equation  (e).  The  fame 
may  be  faid  of  the  negative  roots:  for  lince  every 
root  of  the  equation  (e)  lies  between  the  different 
roots  of  the  equation  (d),  it  is  -impoffihle  that  the  num- 
ber of  roots  fhould  be  lefs  in  either  cafe.  Suppofe 
L and  m x to  be  both  pofitive,  and  fince  the  laft  term  in 
i , . 
(c)  See  the  end  of  Proprietates  Cum 
any 
