and on Algebra as the Science of Pure Time. 373 
definition, this fractional number may be called the logarithm of the resulting power, 
and the power itself may be denoted in written symbols as follows, 
(a )“=a", (269.) 
or thus, 
cob, if b=a*, c=a’. (270.) 
(The analogous formula (121.) ought to have been printed « = : b, and not ¢ = o : 
when b=uxXx a, c=VvxX a.) 
In the particular case when the numerator v is 1, and when, therefore, we have to 
1 
power by the reciprocal of a whole number, we may call the result (a “) x», that is 
a', = a, a7root or more fully the p’th root of the power or ratio a“; and we may 
call the corresponding act of powering, an extraction of the wth root, or a rooting by 
the (whole) number . Thus, to power any proposed ratio by the reciprocal 
number 5 8 3 is to extract the second or the third root, that is, (by what has been 
already shown,) the square-root or the cube-root, of J, or to root the proposed ratio 
b by the number 2 or 3; and in conformity with this last mode of expression, the 
following notation may be employed, 
1 
a='/b when b=a", a=b+:; (271) 
so that a square-root may also be denoted by the symbol 2/6, and the cube-root 
of b may be denoted by 3/4. And whereas we saw, in considering square-roots that 
a contra-positive ratio )<0O has no square-root, and that a positive ratio )>O has two 
square-roots, one positive = // and the other contra-positive =9 / 6, of which each 
has its square =) ; we may consider the new sign b? or x as denoting indifferently 
either of these two roots, reserving the old sign v6 to denote specially that one of 
them which is positive, and’the other old sign © 8 to denote specially that one of 
them which is contra-positive. Thus yb and OvJ shall still remain determinate 
signs, implying each a determinate ratio, (when 6>0,) while 7s and 4* shall be used 
as ambiguous signs, susceptible each of two different meanings. But is a deter- 
minate sign, because a ratio has only one cube-root. In general, an even root, such 
as the second, fourth, or sixth, of a proposed ratio b, is ambiguous if that ratio be 
positive, and impossible if 6 be contra-positive ; because an even power, or a power 
with an even integer for its logarithm, is always a positive ratio, whether the base be 
positive or contra-positive : but an odd root, such as the third or fifth, is always pos- 
sible and determinate. 
31. It may, however, be useful to show more distinctly, by a method analogous to that 
