306 Professor Hamitton on Conjugate Functions, 
trisector moment B be given, it is evidently possible to complete the continued analogy 
or equidistant series in one and in only one way, by supplying the second trisector 
s’ and the second extreme B” ; and it is not much less easy to perceive that any two 
of the four moments being given, (together with their names of position in the series, 
as such particular extremes, or such particular trisectors,) the two other moments can 
be determined, as necessarily connected with the given ones. Thus, if the extremes 
be given, we must conceive their interval as capable of being trisected by two means, 
in one and in only one way ; if the first extreme and second trisector be given, we 
can bisect the interval between them, and so determine (in thought) the first trisector, 
and afterwards the second extreme; if the two trisectors be given, we can continue 
their interval equally in opposite directions, and thus determine (in thought) the two 
extremes ; and if either of these two trisectors along with the last extreme be given, 
we can determine, by processes of the same kind, the two other moments of the 
series, 
7. In general, we can imagine a continued analogy and an equidistant series, com- 
prising any number of moments, and having the interval between the extreme moments 
of the series divided into the next lesser number of portions equal to each other, by a 
number of intermediate moments which is itself the next less number to the number of 
those equal portions of the whole interval. For example, we may imagine an equi- 
distant series of five moments, with the interval between the two extremes divided 
into four partial and mutually equal intervals, by three intermediate moments, which 
may be called the first, second, and third guadrisectors or quarterers of the total . 
interval. And it is easy to perceive, that when any two moments of an equidistant 
series are given, (as such or such known moments of time, ) together with their places in 
that series, (as such particular extremes, or such particular intermediate moments, ) the 
other moments of the series can then be all determined ; and farther, that the series 
itself may be continued forward and backward, so as to include an unlimited number 
of new moments, without losing its character of equidistance. Thus, if we know the 
first extreme moment a, and the third quadrisector B’ of the total interval (from a 
to B’”) in any equidistant series of five moments, a BB’ B’ B”, we can determine by 
trisection the two first quadrisectors B and B’, and afterwards the last extreme moment 
8”; and may then continue the series, forward and backward, so as to embrace other 
moments B’, B*, &c., beyond the fifth of those originally conceived, and others also 
such as E, E £’, &c., behind the first of the original five moments, that is, preceding 
it in the order of progression of the series ; these new moments forming with the old 
an equidistant series of moments, (which comprehends as a part of itself the original 
series of five,) namely, the following unlimited series, 
