Matpic
mathematics, in pictures
Ladder

Compass-only — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

    The straightedge buys nothing

    Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.

    rung 1 · computation
  2. The circle is used once, and its centre is the point. A circle with its centre and one diameter, a point above it, and the straightedge-only construction of the parallel to that diameter through the point.

    One circle, and a straightedge

    A straightedge alone cannot bisect a segment, so it cannot draw a parallel, so it can construct almost nothing. Draw one circle anywhere and mark its centre and everything a compass could ever have done becomes available — the circle is never needed again.

    rung 2 · computation
  3. A compass that will not change its opening. A segment longer than twice the compass's fixed opening, with the opening stepped along it 2 times and the remaining piece bisected by two arcs of that same opening.

    The compass that will not open

    Fix the compass at one opening and never change it. That looks like a serious loss — a circle of a given radius through a given point is the compass's whole job — and it turns out to cost nothing at all, for reasons that are arithmetic rather than geometric.

    rung 3 · computation

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