and on Algebra as the Science of Pure Time. 389 
mined ratio, positive or contra-positive, we can deduce the positive base or ratio 4, by 
calculating the inverse or reciprocal power 
6 =lcR (352.) 
as appears from the relation (335.) which extends, as was above announced, together 
with the relation (334.), even to the case of incommensurable exponents. The proof 
of the important extension last alluded to, will easily suggest itself to those who have 
studied the foregoing demonstrations ; and they will perceive that with the foregoing 
rules for the interpretation of the symbol 4", for the case of an incommensurable ex- 
ponent, the power 4* increases (as was said ‘shiote) continuously as well as constantly 
with the exponent a if the base 4 be > 1, or else decreases continuously and con- 
stantly if that positive base be <1, but remains constantly = 1 if} = 1. It is 
therefore possible to find one determined exponent or logarithm a, and only one, 
which shall satisfy the equation (351.), when the power c and the base 4 are any given 
positive ratios, except in the impossible or indeterminate case when this base 4 is the 
particular ratio 1; and the number a thus determined, whether positive or contra- 
positive or null, may be called ‘‘ the logarithm of c to the base 4,” and may be denoted 
by the symbol 
a= log,.c. (353.) 
It is still more easy to perceive, finally, that when this logarithm a is given, (even 
if it be incommensurable,) the power c¢ increases constantly and continuously from 
zero with the base 4, if a > 0, or else decreases constantly and continuously towards 
zero ifa <0, or remains constant and = 1, ifa = 0. 
Remarks on the Notation of this Essay, and on some modifications by which it may 
be made more like the Notation commonly employed. 
34. In the foregoing articles we have constantly denoted moments, or indivisible 
points of time, by small capital letters, a, B, a’, B’, &c.; and steps, or transitions from 
one such moment to others, by small Roman letters, a, b, a, b', &c. The mark — 
has been interposed between the marks of two moments, to express the ordinal rela- 
tion of those two moments, or the step which must be made in order to pass from 
one to the other ; and the mark + has been inserted between the marks of a step 
and a moment, or between the marks of two steps, to denote the application of the 
