and on Algebra as the Science of Pure Time. 385 
are absurd if m be even, but denote determined ratios when m is odd, which ratios 
are positive if m be even, but contra-positive if m be odd. 
It must be remembered that all the foregoing discussion of the cases of the general 
notation bi, for powers with fractional logarithms, is founded on the definition laid 
down in the 30th article, that b« denotes the v’th power of a ,’th root of 6, or in 
other words, the v’th power of a ratio a of which 4 is the y’th power. When no such 
ratio a can be found, consistently with the previous conception of powers with integer 
logarithms, the symbol b+ is pronounced to be absurd, or to be incapable of denoting 
any ratio consistently with its general definition ; and when two or more such ratios 
a can be found, each having its »’th power = b, we have pronounced that the frac- 
tional power bz is ambiguous or indeterminate, except in those cases in which the 
second component act of powering by the numerator v has happened to destroy the inde- 
terminateness. And with respect to powers with integer exponents, it is to be remem- 
bered that we always define them by a reference to a series of proportional steps, of which 
at least the original step (corresponding to the zero-power) is supposed to be an effective 
step, and which can always be continued indefinitely, at least in the positive direction, 
that is, in the way of repeated multiplication by the ratio proposed as the base, al- 
though in the particular case of a null ratio, we cannot continue the series backward 
by division, so as to find any contra-positive powers. These definitions appear to be 
the most natural ; but others might have been assumed, and then other results would 
have followed. In general, the definitions of mathematical science are not altogether 
arbitrary, but a certain discretion is allowed in the selection of them, although when 
once selected, they must then be consistently reasoned from. 
33. The foregoing article enables us to assign one determined positive ratio, and 
only one, as denoted by the symbol br, when @ is any determined positive ratio, and 
a any fractional number with a numerator and a denominator each different from O : 
it shows also that this ratio b does not change when we transform the expression of 
the fractional logarithm a by introducing or suppressing any whole number w as a factor 
common to both numerator and denominator ; and permits us to write 
be2 =u (b*), (333.) 
8 a being the opposite of the fraction a in the sense of the 17th article. More gene- 
rally, by the meaning of the notation ba, and by the determinateness of those positive 
ratios which result from the powering or rooting of determined positive ratios by de- 
