and on Algebra as the Science of Pure Time. 805 
to one single moment, without the two moments of the other pair 4’ »’ or 4 B being 
also identical with each other ; nor can the two antecedents a a’ coincide, without the 
two consequents B B’ coinciding also, nor can the consequents without the antecedents. 
The only way, therefore, in which two of the four moments AB a’B of an analogy 
can coincide, without the two others coinciding alse, that is, the only way in which an 
analogy can be constructed with three distinct moments of time, is either by the two 
extremes a B’ coinciding, or else by the two means B a’ coinciding; and the principle 
of inversion permits us to reduce the former of these two cases to the latter. We 
may then take as a sufficient type of every analogy which can be constructed with 
three distinct moments, the following : . 
B—B=B—A; (27.) 
that is, the case when an extreme moment B’ is related to a mean moment B, as that 
mean moment B is related to another extreme moment a; in which case we shall say 
that the three moments a B B’ compose a continued analogy. In such an analogy, 
it is manifest that the three moments a B B’ compose also an equidistant series, B 
being exactly so much later or so much earlier than 8, as B is later or earlier than a. 
The moment B is evidently, in this case, exactly intermediate between the two other 
moments 4 and B’, and may be therefore called the middle moment, or the bisector, 
of the interval of time between them. It is clear that whatever two distinct moments 
a and pg’ may be, there is always one and only one such bisector moment B; and that 
thus a continued analogy between three moments can always be constructed in one 
but in only one way, by inserting a mean, when the extremes are given. And it is 
still more evident, from what was shewn before, that the middle moment B, along with 
either of the extremes, determines the other extreme, so that it is always possible to 
complete the analogy in one but in only one way, when an extreme and the middle 
are given. 
6. If, besides the continued analogy (27.) between the three moments A B B’, we 
have also a continued analogy between the two last B B’ of these three and a fourth 
moment B”, then the fowr moments a BR’ B” may themselves also be said to form 
another continued analogy, and an equidistant series, and we may express their rela- 
tions as follows: 
B’— B’=B'—B=B—A. (28.) 
In this case, the interval between the two extreme moments a and B” is ¢trisected by 
the two intermediate moments kz and B’, and we may call B the first trisector, and 1 
the second trisector of that interval. If the first extreme momenta and the first 
