334  Mr.  Playfair  on  the 
the  hyperbola  becomes  impoffible.  For  we  have  then, 
n — ±a\/—  i,  from  which,  by  proceeding  as  above,  we  get 
y = U7=Sce* 1 <*-*  5 hence 
alfo,  by  the  lemma,  y = fin.  J Q,x  cof.  ax  - 
cof.  J' Q.v  fin.  ax.  Here  the  quantities,^ op?  cof.  axt 
and  Qjx  fin.  ax , are  affignable  by  the  quadrature  of  the 
circle,  in  the  fame  manner  as J' cgv  abf.  ax,  and 
J' [ ord.  ax,  by  the  quadrature  of  the  hyperbola;  but 
the  method  of  inveftigating  them,  though  an  illuftra- 
tion  of  the  principles  which  we  have  laid  down,  is  too 
well  known  to  need  to  be  inferted  here.  In  like 
manner  might  the  fluents  of  innumerable  fluxionary 
equations,  comprehended  under  the  general  form 
o = v+  — + -^  + % + 8cc.be  deduced, and  all  of  them  would 
tend  to  prove  that  the  arithmetic  of  impoffible  quanti- 
ties is  no  more  than  a method  of  tracing  the  analogy  be- 
tween the  meafures  of  ratios  and  of  angles,  m.  m. 
EULER^and  D’ALEMBER-T^were  the  firft  to  integrate  fuch 
equations  as  the  preceding,  and  the  method  employed 
here  differs  from  theirs  only  by  being  better  adapted  to 
illuftrate  the  principle  which  is  common  to  them  all. 
(d)  Theorie  de  la  Lime. 
14.  The 
(b ) Nov.  Corn.  Petrop.  tom.  Ilf. 
