If three points, _A_, _B_, and _C_, are self-corresponding, then the
harmonic conjugate _D_ of _B_ with respect to _A_ and _C_ must also
correspond to itself. For four harmonic points must always correspond to
four harmonic points. In the same way the harmonic conjugate of _D_ with
respect to _B_ and _C_ must correspond to itself. Combining new points
with old in this way, we may obtain as many self-corresponding points as
we wish. We show further that every point on the line is the limiting
point of a finite or infinite sequence of self-corresponding points. Thus,
let a point _P_ lie between _A_ and _B_. Construct now _D_, the fourth
harmonic of _C_ with respect to _A_ and _B_. _D_ may coincide with _P_, in
which case the sequence is closed; otherwise _P_ lies in the stretch _AD_
or in the stretch _DB_. If it lies in the stretch _DB_, construct the
fourth harmonic of _C_ with respect to _D_ and _B_. This point _D’_ may
coincide with _P_, in which case, as before, the sequence is closed. If
_P_ lies in the stretch _DD’_, we construct the fourth harmonic of _C_
with respect to _DD’_, etc. In each step the region in which _P_ lies is
diminished, and the process may be continued until two self-corresponding
points are obtained on either side of _P_, and at distances from it
arbitrarily small.

We now assume, explicitly, the fundamental postulate that the
correspondence is _continuous_, that is, that _the distance between two
points in one point-row may be made arbitrarily small by sufficiently
diminishing the distance between the corresponding points in the other._
Suppose now that _P_ is not a self-corresponding point, but corresponds to
a point _P’_ at a fixed distance _d_ from _P_. As noted above, we can find
self-corresponding points arbitrarily close to _P_, and it appears, then,
that we can take a point _D_ as close to _P_ as we wish, and yet the
distance between the corresponding points _D’_ and _P’_ approaches _d_ as
a limit, and not zero, which contradicts the postulate of continuity.