and on Algebra as the Science of Pure Time. 309 
series of moments. ‘This conception involves and depends on the conception of the 
repeated transference of one common ordinal relation, or the continued application of 
one common mental step, by which we pass, in thought, from any moment of such a 
series to the moment immediately following. For this, and for other reasons, it is 
desirable to study, generally, the properties and laws of the transference, or applica- 
tion, direct or inverse, and of the composition or decomposition, of ordinal relations 
between moments, or of steps in the progression of time; and to form a convenient 
system of written signs, for concisely expressing and reasoning on such applications and 
such combinations of steps. 
In the foregoing articles, we have denoted, by the complex symbol p—a, the ordinal 
relation of the moment B.to the moment A, whether that relation were one of identity 
or of diversity ; and if of diversity, then whether it were one of subsequence or of 
precedence, and‘in whatever degree. ‘Thus, having previously interposed the mark 
= between two equivalent signs for one-common moment of time, we came to inter- 
pose the same sign of equivalence between any two marks of one ordinal relation, 
and to write 
D—C=B-—A, 
when we designed to express that the relations of p to c and of g to a were coin- 
cident, being both relations of identity, or both relations of diversity; and if the 
Jatter, then both relations of subsequence, or both relations of precedence, and both 
in the same degree. In like manner, having agreed to interpose the mark + be- 
tween the two signs of two moments essentially different from each other, we wrote 
p—c+B—A, 
when we wished to express that the ordinal relation of p to c (as identical, or sub- 
_ sequent, or precedent) did no¢ coincide with the ordinal relation of the moment B to 
a; and, more particularly, when we desired to distinguish between the two principal 
cases of this non-coincidence of relations, namely the case when the relation of p to 
¢ (as compared with that of s to a) was comparatively a relation of lateness, and the 
‘case when the same relation (of p to c) was comparatively a relation of earliness, 
_ we wrote, in the first case, 
bp—c>B—a, 
and in the second case, 
D—c<B—AaA, 
having previously agreed to write 
BOA 
