and on Algebra as the Science of Pure Time. 363 
existence of some intermediate ratio a; that is, we can always choose a or conceive 
it chosen so that 
>A G Gi lh eh (229.) 
For then we have the following relation of subsequence between moments, 
a’ (B—A)+A >a (B—A) +A, if B> a, (280.) 
by the very meaning of the relation of subsequence between ratios, a’ > a’, as defined 
in article 21.; and between any two distinct moments it is manifestly possible to 
insert an intermediate moment, indeed as many such as we may desire: it is, there- 
fore, possible to insert a moment c between the two non-coincident moments 
ad (B—A)+4 and a” (B—a) +A, 
such that 
c>a@ (B—a)+a, c<a’ (B—A)+A, if B> A, a’ >a’; (231.) 
and then if we put, for abridgement, 
; (232.) 
denoting by a the ratio of the step or interval c—a to the step or interval B—4, 
we shall have 
a (B—A)+A>a'(B—A) +A, 
a (B—A)+a < a’(B—a) +A, 
C=a (B—A)+A, B>A, 
‘i (233.) 
and therefore finally, 
aS Gh, Gh Git 
as was asserted in the Lemma. We see, too, that the ratio a is not determined by 
the conditions of that Lemma, but that an indefinite variety of ratios may be chosen, 
which shall all satisfy those conditions. 
Corollary. It is possible to choose, or conceive chosen, a ratio a, which shall 
satisfy all the following conditions, 
a>a, a=b, a>c, ‘cy 
ee iaed cg N (234:.) 
if the least (or hindmost) of the ratios a’, 6", c’, ... be greater (or farther advanced 
in the general progression of ratio from contra-positive to positive) than the greatest 
(or foremost in that general progression) of the ratios a’, b’, c’, &c. 
VOL. XVII. 3x 
