32,6  Mr.  PLAYFAIR  on  the 
every  imaginary  expreffion,  which  has  been  found  to 
belong  to  the  circle  in  the  preceding  calculation,  is  by 
the  fubftitution  of  real  for  impoffible  quantities,  or  of 
Vi  for  V~  i,  converted  into  a propoiition  which  holds 
of  the  hyperbola.  The  operations,  therefore,  performed 
with  the  imaginary  characters,  though  deftitute  of  mean- 
ing themfelves,  are  yet  notes  of  reference  to  others  which 
are  fignificant.  They  point  out  indirectly  a method  of 
demonftrating  a certain  property  of  the  hyperbola,  and 
then  leave  us  to  conclude  from  analogy  that  the  fame 
property  belongs  alfo  to  the  circle.  All  that  we  are 
allured  of  by  the  imaginary  inveftigation  is,  that  its  con- 
clulion  may,  with  all  the  ftriitnefs  of  mathematical  rea-* 
foning,  be  proved  of  the  hyperbola;  but  if  from  thence 
we  would  transfer  that  conclulion  to  the  circle,  it  muft 
be  in  confequence  of  the  principle  which  has  been  juft 
now  mentioned.  The  inveftigation,  therefore,  refolves 
itfelf  ultimately  into  an  argument  from  analogy;  and, 
after  the  ftricfteft  examination,  will  be  found  without  any 
other  claim  to  the  evidence  of  demonftration.  Had  the 
foregoing  propoiition  been  proved  of  the  hyperbola 
only,  and  afterwards  concluded  to  hold  of  the  circle, 
merely  from  the  affinity  of  the  curves,  its  certainty  would 
have  been  precifely  the  fame  as  when  a proof  is  made 
©ut  by  the  intervention  of  imaginary  fymbols. 
4 8.  Though 
