10 
result of the division of a? by itself? We Inave seen that 
x=\x and x — xx. 
The divisions corresponding to these two multiplications, 
according to the ordinary rules of algebra, are as follows : — ■ 
lx ^ .XX 
— = 1 and ~ 
so that the two solutions 
X .. .. X 
- = \ and - = iu 
X X 
appear to be both true. And in fact both are true; the 
solution is indefinite, with x and 1 as limiting values. If 
we use the coefficient ^ to indicate indefinite quantity, the 
full solution will be 
X 0 
- = + “(1 -X), 
which solution Boole has reached by a purely symbolic 
process. It is thus expressed in language : — x abstracted 
from itself gives x together with an indefinite quantity of 
what is not x. 
If we multiply both sides of the above equation by, x, so 
as to invert the operation of division, we get 
X — Xj 
the second member of the right hand side of the equation 
disappeais, because in Boole’s system 
x{\ -x) = 0, 
that is to say, anything which is both x and not x cannot 
exist. 
X m 
It is to be observed that such an expression as - is un- 
interpretable unless it is true that 
X = xy, 
that is to say we cannot abstract a quality unless it is a 
quality of that from which we abstract it. We can, for 
instance, abstract rationality from man, and say “Man 
without reason is only an animal;” but we cannot abstract 
