and on Algebra as the Science of Pure Time. 313 
step 3,” or the step p—A, and which is such that by making this mental step, or 
performing this act of transition, we pass, in thought, from the moment a to the 
moment B, and thus suggest or generate (in thought) the latter from the former, as 
a mental product or result B of the act « and of the object a. We may also express 
the same relation between 8 and a by writing 
A=(9 4)+B, or more simply a=0 2+8, (60.) 
if we agree to write the sign 4 without parentheses, as if it were a simple or single 
symbol, because there is no danger of causing confusion thereby ; and if we observe 
that the notation A=© +B corresponds to the conception of another step, or 
mental act of transition, 0 4, exactly opposite to the former step a, and such that by 
it we may return (in thought) from the moment B to the moment a, and thus may 
generate A as a result of the act 94 and of the object 8. The mark +, in this sort of 
notation, is interposed, as a mark of combination, between the signs of the act and 
the object, so as to form a complex sign of the result ; or, in other words, between 
the sign of the transition (2 or @ ) and the sign of the moment (a or B) from which 
that transition is made, so as to express, by a complex sign, (recording the suggestion 
or generation of the thought, ) that other moment (8 or a) to which this mental tran- 
sition conducts. And in any transition of this sort, such as that expressed by the 
equation B= a + a, we may call (as before) the moment a, from which we pass, the 
antecedent, and the moment 8, to which we pass, the consequent, of the ordinal rela- 
tion a, or B—a, which suggests and determines the transition. In the particular 
case when this ordinal relation is one of identity, (a=0,) the mental transition or 
act (@ or 0) makes no change in the object of that act, namely in the moment a, but 
only leads us to repeat the thought of that antecedent moment a, perhaps with a 
new name B; in this case, therefore, the transition may be said to be null, or a null 
step, as producing no real-alteration in the moment from which it is made. A step 
not null, (»+.0,) corresponds to a relation of diversity, and may be called, by con- 
trast, an effective step, because it is an act of thought which really alters its object, 
namely the moment to which it is applied: An effective step a must be either a late- 
making or an early-making step, according as the resultant moment «+a is later or 
earlier than a; but even a null step 0 may be regarded as relatively late-making, 
when compared with an early-making step 4, (0+A>a+<A, if 2<0,) or as relatively 
early-making if compared. with a late-making step »; (O+a<b +a, if b>0;) and, 
in like manner, of two unequal early-making steps, the lesser may be regarded as 
relatively late-making, while of two unequal late-making steps the lesser step may be 
considered as relatively early-making.. With these conceptions of the relative effecis 
