by tT 
and on Algebra as the Science of Pure Time. 371 
In conformity with this conception, we may call the original ratio a the base of the 
system of ratios (261.) and may call those ratios by the common name of powers of 
that common base, and say that they are (or may be) formed by acts of powering 
it. And to distinguish any one such power, or one such act of powering, from all 
the other powers in the system, or from all the other acts of powering, we may 
employ the aid of determining numbers, ordinal or cardinal, in a manner analogous 
to that explained in the 13th article for a system of multiple steps. Thus, we may 
call the ratios a, wa, aaa, ... by the common name of positive powers of the base 
a, and may distinguish them by the special ordinal names first, second, third, &c. ; so 
that the ratio a is, in this view, its own first positive power; the second positive 
power is the square aa, and the third positive power is the cube. Again, we may 
call the ratio 1, which immediately precedes these positive powers in the series, the 
zero-power of the base a, by analogy to the zero-multiple in the series of multiple 
steps, which immediately preceded in that series the system of positive multiples ; 
and the ratios ua, u (aa), u (aaa), ... which precede this zero-power 1 in the 
series of powers (261.), may be called, by the same analogy, from their order of 
' position, contra-positive powers of a, so that the reciprocal u a of any ratio a is the 
Jirst contra-positive power of that ratio, the reciprocal u (aa) of its square is its 
second contra-positive power, and so on. We may also distinguish the several cor- 
responding acts of powering by the corresponding cardinal numbers, positive, or 
contra-positive, or null, and may say (for example) that the third positive power aaa 
is formed from the base a by the act of powering by positive three ; that the second 
contra-positive power u (aa) is formed from the same base a by powering by contra- 
positive two ; and that the zero-power 1 is (or may be) formed from a by powering 
that base by the null cardinal or number none. In written symbols, answering to 
these thoughts and names, we may denote the series of powers (261.), and the series 
of proportional steps (260.), as follows, 
ee A ars Bos a a ae (262.) 
and 
meet KBs PE aes 20 eX hs aX as gh Xone ne oe, a X..8 dan AC sue) 
in which 
a°=1, (264.) 
and 
(5 2 alec 
aa 2d SH (aa), 26 
7 Ap (265.) 
a°=aaa, a ua (aaa), 
&ce. &e. 
VOL. XVII, OM 
