> A circle may have been drawn using a stencil, or in the usual
> manner - by somebody else's anscetor - so long ago that the paper,
> like an old map, grew so decayed and fragile that while the lucky
> adventurer tried to smoothen it out on a table, the middle part
> caved in and disintegrated.

The explanation of stencil or ancestor still begs the question: How do you know it is a circle? Keep in mind that Geometry is not about the (mechanical) drawing; but rather, it is about the mathematically exact line(s) or circle(s) it represents (theory). We know that when we draw a circle or arc with a compass that we are representing a theoretically exact circle because we are representing every point on the circle edge is being equidistant from the circle center point.

Perhaps a better way to present the problem is "Is it possible to find the circle center point without using the circle center point and only using a straightedge."

My whole issue with this problem and its proof is that with only an existing circle and straightedge one can only draw random lines. One can say when a line intersects the circle, and one can draw more random lines through those circle intersection points, but one can do or say nothing else to include when the circle center has been found at some time in the infinite future. In short, drawing circles enable some ability to find direction. If drawing circles are removed from the set of tools, then not just this problem, but any Geometry problem is reduced to nothing of interest or value.

"Anybody find a circle center point around here? I seemed to have lost one."

> Why, it did a lot of sense to several generations of geometers
> starting in the mid 1800s.

Amateur geometers or Professional (w/ degree) Geometers? Recall that Wenzel discovered in 1837 an anti-proof that such problems as Angle Trisection could not be done, which heralded in a time of great interest in understanding Greek Philosophy and Mathematics by a lot of people, a vast number of who were amateurs. This might have been of great interest to amateurs, but I doubt this was of any interest to professionals.

> It certainly takes imagination rather than brute force.

Imagination without reason does nothing to .

> How do you draw a circle in 3d?

It requires identifying the plane that the circle lies in. Identifying a plane requires the following 3d conditions:
- Two intersecting lines,
- Two parallel lines that are each not the other,
- A line and a point not on the line,
- Three points that are each not the other,
- A circle, or
- Any combination of lines, points, or circles whose subset creates the above 3d conditions (e.g. two intersecting circles forming a line).

This last was actually a more interesting question than the topic problem.

I guess the larger issue is not the problem, but the acceptability of the proof that it can not be done. Let's try to form the frame of a more reasonable proof this way.

First, start by working only in the plane of the circle.
(1) Draw a line. This can be through:
- the circle:
- through a previous intersecting point formed by a previous
line, or
- through two previous intersecting points formed by two
previous lines, or
- through no previous intersecting points;
- not through the circle. (Only intersections with the circle can
possibly provide information regarding the circle center.)
(2) Does the line intersect the circle?
a) No, go back to step (1).
b) Yes, continue.
(3) Does the line intersect the circle center?
a) Since no information can be derived from the two intersecting
points of this line with the circle, then the answer is: Don't
know. Go back to step (1).

Second, does considering an additional dimension add any more information regarding the circle center regarding any line we can draw, intersecting or not? No. Done.