and on Algebra as the Science of Pure Time. 801 
D-—C=B—A, or, B—~A=D—C; (10.) 
the interposed mark =, which before denoted identity of moments, denoting now 
identity of relations : and the written assertion of this identity being called (as before) 
an equation. The conditions of this exact identity between the relation of the mo- 
ment p to c, and that of B to a, may be stated more fully as follows: that if the mo- 
ment B be identical with a, then p must be identical with c ; if B be later than a, 
then p must be later than c, and exactly so much later; and if B be earlier than 
A, then p must be earlier than c, and exactly so much earlier. It is evident, that what- 
ever the moments A B and c may be, there is always one, and only one, connected mo- 
ment D, which is thus related to c, exactly as B is to A; and it is not difficult to per- 
ceive that the same moment p is also related to B, exactly as c is to A: since, in the 
case of coincident pairs, pD is identical with 8, and c with 4; while, in the case of pairs 
analogous but not coincident, the moment p is later or earlier than B, according as c 
is later or earlier than a, and exactly so much later or so much earlier. If then the 
pairs A, B, and c, p, be analogous, the pairs ac and BD, which may be said to be 
alternate to the former, are also analogous pairs ; that is, 
if D—c=B-A, then D—B=c—a; (11.) 
a change of statement of the relation between these four moments a B c D, which 
may be called alternation of an analogy. It is still more easy to perceive, that if any 
two pairs aB and cp be analogous, then the ¢nverse pairs BA and vc are analogous 
also, and therefore that 
if D—c— p— AS then cp —A—B; (12.) 
a change in the manner of expressing the relation between the four moments 
ABCD, which may be called znverscon of an analogy, Combining inversion with 
alternation, we find that 
if p—c=B—A, then B—D=A—C; (13.) 
and thus that all the eight following written sentences express only one and the same 
relation between the four moments a Bc D: 
D— C=E— A, B—A—D —C,)} 
CAG an ase—e, (14.) 
CA BA — Cl Ds 
B—D=A—C, A—C=B—D; 
any one of these eight written sentences or equations being equivalent to any other. 
