Arithmetic  of  impojfibk  Quantities.  339 
guity  for  the  conftrudlion  required  in  fuch  cafes  is  by  its 
nature  reftrifted  to  one  of  the  curves  only.  Of  this  kind 
is  the  Cotefian  theorem,  which  requires  the  whole  circle 
to  be  divided  into  a given  number  of  equal  parts,  and 
therefore  cannot  be  extended  to  the  hyperbola  where  a 
fimilar  divilion  is  impoffible.  Others  of  a like  nature 
may  be  derived  from  the  general  theorems  already  in- 
veftigated ; for  the  circle,  by  returning  into  itfelf,  often 
reduces  them  to  a limplicity  to  which  there  is  nothing 
analogous  in  the  hyperbola.  Many  examples  of  this 
might  be  adduced,  but  the  two  following  may  fuffice. 
I.  Let  abode  (fig.  5.)  be  a regular  polygon  infcribed  in 
a circle,  and  let  m be  the  number  of  its  fides ; it  is  re- 
quired to  find  the  fum  of  the  lines  fa,  fb,  fc,  See.  drawn 
from  any  point  f in  the  circumference,  to  all  the  angles 
of  the  polygon.  By  the  method  which  in  art.  8.  was 
employed  to  obtain  the  fum  of  the  fines  of  a feries  of 
arches  in  arithmetical  progreffion,  it  will  be  found,  that 
the  fum  of  the  chords  of  the  arches  a,  a+x,  a+  2 a, 
. . . . (m),  that  is,  (making  fa -a,  and  ab=x)  the  fum 
of  the  chords  of  the  arches  fa,  fb,  fc,  See.  = 
cho.  a — cho,  a-\-mx — cho.  a— A'-J-cho.  a-\-mx — x 
2X1  —col'.  \ x 
; but,  in  the  pre- 
fent  cafe,  mx  is  equal  to  the  circumference,  and  therefore 
—cho.  a+mx-  + cho.  a (the  chord  of  an  arch  greater  than 
Vol.  LXVIII.  X x the 
