Arithmetic  of  impoffible  Quantities.  343 
_ 4 ^AkfAzLr ; and  the  fame  may  be  deduced  from  art.  4.. 
Conlidered  as  a quadrature  of  the  circle,  this  imaginary 
theorem  is  wholly  infignificant,  and  would  defervedly 
pafs  for  an  abufe  of  calculation;  at  the  lame  time  we 
learn  from  it,  that  if  in  any  equation  the  quantity 
~'V_ 7~  fhould  occur,  it  may  be  made  to  difappear,  by 
the  fubftitution  of  a circular  arch,  and  a property,  com- 
mon to  both  the  circle  and  hyperbola,  may  be  obtained. 
The  fame  is  to  be  obferved  of  the  rules  which  have  been 
invented  for  the  transformation  and  reduction  of  impof- 
lible  quantities'^ : they  facilitate  the  operations  of  this 
imaginary  arithmetic,  and  thereby  lead  to  the  knowledge 
of  the  moll  beautiful  and  extenfive  analogy  which  the 
doctrine  of  quantity  has  yet  exhibited. 
(e)  The  rules  chiefly  referred  to  are  thofe  for  reducing  the  impoflible  roots* 
of  an  equation  to  the  form  a-\-zV — U 
