SEE ht Ra 
and on Algebra as the Science of Pure Time. 307 
Sh. bh ne BB! Be BB (29.) 
constructed so as to satisfy the conditions of a continued analogy, 
...BY— BY =B"—B’ =B" —B’ =B’ —B =B’—B=B—A=A—E=E—E=E—E... (30.) 
8. By thus constructing and continuing an equidistant series, of which any two 
moments are given, we can arrive at other moments, as far from those two, and as near 
to each other, as we desire. For no moment 8 can be so distant from a given moment 
A, (on either side of it, whether as later or as earlier,) that we cannot find others 
still more distant, (and on the same side of a, still later or still earlier,) by continuing 
(in both directions) any given analogy, or given equidistant series; and, therefore, 
no two given moments, c and p, if not entirely coincident, can possibly be so near 
to each other, that we cannot find two moments still more near by treating any twe 
given distinct moments (A and B), whatever, as extremes of an equidistant series of 
moments sufficiently many, and by inserting the appropriate means, or intermediate 
moments, between those two given extremes. Since, however far it may be necessary 
to continue the equidistant series c p...p’, with c and p for its two first moments, in 
order to arrive at a moment pb" more distant from c than g is from a, it is only ne- 
cessary to insert as many intermediate moments between a and B as between c and 
p’, in order to generate a new equidistant series of moments, each nearer to the one 
next it than p toc. ‘Three or more moments a Bc &c. may be said to be wniserial 
with each other, when they all belong to one common continued analogy, or equi- 
distant series; and though we have not proved (and shall find it not to be true) that 
-any three moments whatever are thus uniserial moments, yet we see that if any two 
moments be given, such as A and g, we can always find a third moment B’ uniserial 
with these two, and differing (in either given direction) by less than any interval pro- 
posed from any given third moment c, whatever that may be. This possibility of 
indefinitely appproaching (on either side) to any given moment c, by moments 
“ uniserial with any two given ones a and B, increases greatly the importance which 
I would otherwise belong to the theory of continued analogies, or equidistant series 
of moments. Thus if any two given dates, c and p, denote two distinct moments of 
~ time, (c-£ b,) however near to each other they may be, we can always conceive their 
diversity detected by inserting means sufficiently numerous between any two other 
given distinct moments a and 8, as the extremes of an equidistant series, and then, if 
necessary, extending this series in both directions beyond those given extremes, until 
some one of the moments B’ of the equidistant series thus generated is found to fall 
between the two near moments c and p, being later than the earlier, and earlier than 
VOL. XVII. 30 
