[ 3i8  ] 
XVI.  On  the  Arithmetic  of  Impojfible  Quantities.  By  the 
Rev.  John  Playfair,  A.  M.  Communicated  by  the 
Rev.  Nevil  Mafkelyne,  D.  D.  F.  R.  S.  and  AJlronomer 
Royal. 
Read  Feb.  26,  r pHE  paradoxes  which  have  been  intro- 
i77°-  JL  duced  into  algebra,  and  remain  un- 
known in  geometry,  point  out  a very  remarkable  dif- 
ference in  the  nature  of  thofe  fciences.  The  proportions 
of  geometry  have  never  given  rife  to  controverfy,  nor 
needed  the  fupport  of  metaphyfical  difcuffion.  In  alge- 
bra, on  the  other  hand,  the  doCtrine  of  negative  quanti- 
ties and  its  confequences  have  often  perplexed  the  ana- 
lyffc,  and  involved  him  in  the  moft  intricate  deputations* 
The  caufe  of  this  diverlity,  in  fciences  which  have  the 
fame  object,  muft  no  doubt  be  fought  for  in  the  different 
modes  which  they  employ  to  exprefs  our  ideas.  In  geo- 
metry every  magnitude  is  reprefented  by  one  of  the  fame 
kind ; lines  are  reprefented  by  a line,  and  angles  by  an 
angle.  The  genus  is  always  fignified  by  the  individual, 
and  a general  idea  by  one  of  the  particulars  which  fall 
under  it.  By  this  means  all  contradiction  is  avoided,  and 
a the 
