and on Algebra as the Science of Pure Time. 381 
and generally, 
Oe, (302.) 
To power any positive ratio a, whether positive, or contra-positive, or null, by the 
number or logarithm 0, may be considered to give 1 as the result; because we can 
always construct at least this series of proportional steps, beginning with any one 
effective step a, and proceeding indefinitely in one direction : 
Ib eit CHS, Cp NSS OES Ce BD BEE (3083. ) 
and we may still call the ratio 1 the zero-power, and the ratios a, axa, ... the 
positive powers of the ratio a, even when we cannot continue this series of proportional 
steps (303.) backward, like the series (259.), so as to determine any contra-positive 
powers of a; namely, in that particular case when a=0. We may, therefore, con- 
sider the equation (264.), @’=1, as including even this particular case a=0; and 
may write 
OZ (304.) 
and, therefore, by (301.) and (295.) 
On=1: (305.) 
we are also conducted to consider the symbols 
0°", Om, (306.) 
as absurd, the ratio 0 having no contra-positive powers. 
From the generality which we have been led to attribute to the equation a’=1, it 
follows that the symbol 
oy 
1°, and more generally Tree (307.) 
is indeterminate, or that it is equally fit to denote all ratios whatever ; but that the 
symbol 
1 
be, or be, if 61, (308.) 
is absurd, or that it cannot properly denote any ratio. In particular, the symbols 
fis jh) Or; (309.) 
