Addendum to Dedekind's cut:

Can we account for all the points of a diagonal in a unity square?

Let us look at the method described above (the entry Reference and Geometry).

There is no reason to start at the top. We can draw parallel lines to the side starting from the base.
But before that, we have to make sure we are dealing with a square and its diagonal. By drawing the second diagonal we create four equal  triangles. The point on each half of a diagonal can be put on a one to one correspondence with the  points of the other sides respectively, taking into account the principle of one line for each point. 

This puts Dedekind's remarks that some points cannot be put in one-to-one correspondence with rational numbers in a new light: the only explanation is that the points only detectable with irrational numbers are in fact not located on the line itself, but are a product of his own formulas.

Anyway, a very interesting question is, looking at

a² < D < ( a+ 1)^{2
[I have replaced the lambda sign with a]}
and knowing that "this cut is produced by no rational number", what does this formula refer to?