On  the  Arithmetic  of  impoffible  Quantities.  319 
the  geometer  is  never  permitted  to  reafon  about  the  rela- 
tions of  things  which  do  not  exift,  or  cannot  be  exhi- 
bited. In  algebra  again  every  magnitude  being  denoted 
by  an  artificial  fymbol,  to  which  it  has  no  refemblance, 
is  liable,  on  fome  occafions,  to  be  negleCted,  while  the 
fymbol  may  become  the  foie  object  of  attention.  It  is 
not  perhaps  obferved  where  the  cenneCtion  between 
them  ceafes  to  exift,  and  the  analyft  continues  to  reafon 
about  the  characters  after  nothing  is  left  which  they  can 
poffibly  exprefs:  if  then,  in  the  end,  the  conclufions 
which  hold  only  of  the  characters  be  transferred  to  the 
quantities  themfelves,  obfcurity  and  paradox  muft  of  ne- 
ceffity  enfue.  The  truth  of  thefe  obfervations  will  be 
rendered  evident  by  confidering  the  nature  of  imaginary 
expreffions,  and  the  different  ufes  to  which  they  have 
been  applied. 
2.  Thofe  expreffions,  as  is  well  known,  owe  their  ori- 
gin to  a contradiction  taking  place  in  that  combination 
of  ideas  which  they  were  intended  to  denote.  Thus,  if 
it  be  required  to  divide  the  given  line  ab  (fig.  1.)  = a 
in  c,  fo  that  ac  x cb  may  be  equal  to  a given  fpace  b\ 
and  if  AC=ar,  then  x-\a±\l^a1-b7'\  which  value  of  x is 
imaginary  when  tr  is  greater  than  i cf ; now  to  fuppofe 
that  b1  is  greater  than  is  to  fuppofe  that  the  rectangle 
ac  x cb  is  greater  than  the  fquare  of  half  the  line  ab, 
which 
