and on Algebra as the Science of Pure Time. 819 
The propositions (22.) (23.) (24.), respecting certain combinations of analogies, 
are included in the same assertion (75.) ; which may also, by (71.), be thus expressed, 
at+b=0 (Oa + Ob), or, b+ a=O0 (8 b+6 a); (85.) 
that is, by saying that it comes to the same thing, whether we compound any two 
steps a and b themselves, or first compound their opposites © a, © b, into one com- 
pound step 9 b+6 a, and then take the opposite of this. Under this form, the 
theorem of the possibility of reversing the order of composition may be regarded as 
evident, whatever the number of the component steps may be ; for example, in the 
case of any three component steps a, b, c, we may regard it as evident that by apply- 
ing these three steps successively to any moment a, and generating thus three 
moments B, C, D, we generate moments related to A as a itself is related to those 
three other moments B’, c’, p’, which are generated from it by applying successively, 
in the same order, the three respectively opposite steps, © a, Ob, O ¢; that is, if 
B=a+a, B=Oa+ A, 
C=b+ 3B, c= Ob +8, (86.) 
Dic +c) Dp) =\/Ole 1c’, 
then the sets B’ AB, c’ Ac, D' AD, containing each three moments, form so many 
continued analogies or equidistant series, such that 
B-A=A-—B 
c—-A=A-—C (87.) 
D—A=A—D’ 
and therefore not only b + 2 =© (Ob +a), as before, but also 
c+b+a=0 (0c+Ob+9O 4), (88.) 
that is, by (72.) and (57.), 
ctbtaxtatbt+c; (89.) 
and similarly for more steps than three. 
The theorem (89.) was contained, indeed, in the reciprocal of the proposition 
(24..), namely, in the assertion that 
if D—c=B-—A, 
and Dp —c=B—A, (90.) 
then D’—p=a—a’, 
