lige at ma, 
| ~o tei <p re 
and on Algebravas the Science of Pure Time. 323 
that original step, and these relations having all a general resemblance to each other, so 
that they may be conceived as composing a certain system of relations, having all one 
common character. ‘To mark this common generation of the system of steps (98.) 
from the one original’ step a, and their common relation thereto, we may call them all 
by the common name of multiples of that original step, and may say that they are or 
may be (mentally) formed by multipling that common base, or unit-step, a; distin- 
guishing, however, these several multiples among themselves by peculiar or special 
names, which shall serve to mark the peculiar relation of any one multiple to the 
base, or the special act of multipling by which it may be conceived to be generated 
therefrom. 
Thus, the null step, or zero-step, 0, which conducts to the zero-moment a, may be 
called, according to this way of conceiving it, the zero multiple of the original step a; and 
the positive (effective) steps, simple or compound, a, ata, a+ a+ a, &¢., may 
be called by thé general name of positive multiples of 2, and may be distinguished by 
the special ordinal names of first, second, third, &c., so that the original step is, in 
this view, its own first positive multiple; and finally, the contra-positive (but effective ) 
steps, simple or compound, namely, 0 a, 9a + 0, Oa + Oa + Oa, &e., may be 
called the first contra-positive multiple of a, the second contra-positive multiple of 
the same original step a, and so forth. Some particular multiples have particular 
and familiar names ; for example, the second positive multiple of a step may also be 
called the double of that step, and the third positive multiple may be called familiarly 
the triple. In general, the original step a may be called (as we just now agreed) the 
common base (or unit) of all these several multiples; and the ordinal name or 
number, (such as zero, or positive first, or contra-positive second,) which serves as a 
special mark to distinguish some one of these multiples from every other, in the 
-general series of such multiples (98.), may be called the determining ordinal: so 
that any one multiple step is sufficiently described, when we mention its base and its 
determining ordinal. In conformity with this conception of the series of steps (98.,) 
as a series of multiples of the base 2, we may denote them by the following series of 
written symbols, 
nielostete Oa, -2' Ola, 1 Olas? Ola Oa 2iay) Sia, .- (99.) 
and may denote the moments themselves of the cqui-distant series (29.) or (97.) by 
_. the symbols, 
VOL. XVII. 3 Q 
