If Marcel Boll got him right, he has basically said and tried to proove that "any two lines have the same number of points, even if they are of different lengths".
I will give you his proof (as per Marcel Boll, explicitating a concept by his introduction) and then criticize, not the facts in the proof, but his sloppy thinking in concluding that thesis.
Here is his proof (adding one point to what I recall from book):
- Take two lines. (If they have the same length, they have the same number of points.) If they have different lengths, do like this:
- Take a line from one end of one line-to-be studied to the closest end of other line-to-be-studied (I will refer to all other lines as helplines). Prolong the helpline after its way from longer to shorter line. Here you have two points, two of the end points, both of which are connected by one line and thus paired as much as a man sitting in a chair.
- Take a helpline from the other end point of one line to the other end point of another line, prolong it till it meets the other prolongation. Two more points in the two lines-to-be-studied are now paired, like two men sitting in two chairs.
- Take any point between the end points in either of the lines-to-be-studied, draw a help line to meeting point of prolongations of already existing helplines, and to the other line, without bending, and you now have three paired points in the two lines, like three men sitting in three chairs. A, B, C are paired with a, b, c. So far nothing like an empty chair or a man standing no point unique to one line without correspondence in other one.
- If a point is put between A and B, there will be a point between a and b also. An unique point corresponding to it.
- And so on for any point there is in one line, there is one unique in the other line too. Like no empty chairs and no standing men.
The problem is, each line naturally has only two points given, the end points of the line. All other points are really in the line, but only potentially points. The line is infinitely divisible, from the geometrical (maybe not physical) point of view. But there are at any given moment only a finite number of divisions marked out. It is impossible to exhaust the above reasoning so as to be valid "for every point in the line", if by every point you mean every potential point. Like the "mid" points B and b are only accidentally points, since marked out by the third helpline. If there is as yet no marking out of a point, it is as yet not really a point.
A line is not the sum of its points. Potentialities do not sum up as actualities do. And there is no summing up the infinite.
Rather the lines are more real than the points: a point being a division of a line. Like end points A and B being the divisions of the greater line from non-line. Or like the point B being an arbitrary accidental division of line from line, of continuation from continuation. It may be in any given case a very rational point like exact middle or like 2/3 length and so on, or it may have a proportion that is intuitive and geometrically constructable like phi, or it may be "totally arbitrary." But in any of these cases it is accidental to the line.
Similarily a line is a division of a surface, a natural one as dividing the surface from not that surface or an arbitrarily added one, like a line across a surface. And a surface is a division of a solid body, either a real division as its outer surface or an accidental one like the diverse divisions in a brain scan, plane after plane. The inner "surfaces" thus being such as you cannot see with only the eyes.
Physically a line is a body. Like a line of ink on a paper. It has a real thickness in its breadth on paper and in its deepness into paper. It consists of whatever the ink consists of. Say water molecules (not too many after ink dries), proteine molecules from fish glue, pieces of graphite cristals interspersed into that. And in that sense obviously if two helplines are drawn close enough they may cross different molecules in one line-to-be-studied and the same one in the shorter line.
Our Lady of Sorrows