and on Algebra as the Science of Pure Time. 379 
and may be practically approached to, by determining such fractions — , with larger 
and larger whole numbers m’ and 7’ for their denominators and numerators. And 
whether m be odd or even, we see that the power a” increases continuously (as well 
as constantly) with its positive root or base a, from zero up to states indefinitely 
greater and greater. But if this root, or base, or ratio a be conceived to advance 
constantly and continuously from states indefinitely far from zero on the contra- 
positive side to states indefinitely far upon the positive side, then the power a” will 
either advance constantly and continuously likewise, though not with the same quick- 
ness, from contra-positive to positive states, or else will first constantly and continu- 
ously retrograde to zero, and afterwards advance from zero, remaining always posi- 
tive, according as the positive exponent or logarithm m is an odd or an eyen integer. 
It is understood that for any such positive exponent m, 
0" =0, (293.) 
the powers of 0 with positive integer exponents being considered as all themselves 
equal to 0, because the repeated multiplication by this null ratio generates from any 
one effective step a the series of proportional steps, 
aeOnxe tn = Os OOD ai Owe n> (294.) 
which may be continued indefinitely iz one direction, and in which all steps after the 
first are null ; although we were obliged to exclude the consideration of such null 
ratios in forming the series (259.) because we wished to continue that series of steps 
indefinitely in two opposite directions. 
32. We are now prepared to discuss completely the meaning, or meanings, if any, 
which ought to be assigned to any proposed symbol of the class bu, b denoting any 
proposed ratio, and m and y any proposed whole numbers. By the 30th article, 
the symbol b« denotes generally the v’th power of a ratio a of which 6 is the »’th 
power ; or, in other words, the »th power of a nth root of 6; so that the mental 
operation of passing from the ratio 4 to the ratio 4 m, is compounded, (when it can 
be performed at all,) of the two operations of first rooting by the one whole 
number ,», and then powering by the other whole number v: and we may write, 
br = (4) = (0) (295.) 
The ratio 4, and the whole numbers u and y, may each be either positive, or contra- 
positive, or null; and thus there arise many cases, which may be still farther sub- 
yOL. XVII, 32 
