336 MEMOIRS OF THE AMERICAN ACADEMY. 



Properties of Zero and Unity 



The symbolical definition of zero is 





x -f- = x , 

 so that by (19) x,a = x,(a -f- 0) = x,a + #>0 . 



Hence, from the invertible character of this addition, and the generality of (14), we 

 have 



z,0 = 



By (24) we have in general, 



x 



or 



r = x-\- 0— #,0 — x, 



x -fr- = x . 



By (4) we have ax = (a -\^ 0)x = ax \ Ox . 



But if a is an absurd relation, ax = , 



so that Ox = , 



which must hold invariably. 



From (12) we have a x = {a -|y 0) x = a x -^ X -{y etc. 



whence by (21) 0* — <^ a x . 



But if a is an absurd relation, and x is not zero, 



a x = . 



and therefore, unless x = 0, 0* = . 



Any relative x may be conceived as a sum of relatives X,X',X", etc., such that 

 there is but one individual to which anything is X, but one to which anything is 

 X f , etc. Thus, if x denote "cause of," X,jr,X" would denote different kinds of 

 causes, the causes being divided according to the differences of the things they are 



of. Then we have 



Xy = X(y -\ 7 0) = Xy-\ T X0, 

 whatever y may be. Hence, since y may be taken so that 



we have 



and in a similar way, 



Xy=0, 



X0 = 0; 



We have, then, 



X'0 = 0, X"0=:0, ^'"0 = 0, etc. 



xO 



( X -fcr X' -fc X" -fe X" -fc. etc.)0 = X0 -k X0 -fe- X"0 + X" r -fe. etc. = . 



