### Archimedes Trisection of the Angle

The angle at A is one third of the central angle.

This construction is Archimedes' trisection of the angle by
compass and straightedge. One thing which makes this construction
remarkable is that in 1832, the French mathematician Evariste Galois
proved that it was impossible to trisect an angle with a compass and
straightedge. Every year, Mathematics Department chairs get
trisections of the angle with compass and straightedge, and they
throw them in the wastebasket because it is very well known that
Galois proved that it can't be done. In fact, Galois' proof is more
well known than Archimedes' construction.

Whenever someone proves that something can't be done, one should
examine the proof for unconscious, limiting
assumptions.^{1} The
unconscious, limiting assumption here is that you can use a
straightedge only to draw lines, and a compass only to draw circles.
There many mathematicians who know about Archimedes' construction,
but they explain it away by saying that it can't be done by compass
and straightedge alone. You have to have a mark on the straightedge,
and that doesn't make it a straightedge; it makes it a ruler. But, if
you have a compass, you can make a mark on a straightedge. In
Archimedes' construction, the compass is not used to draw circles, it
is used to make a mark on the straightedge. In fact, you do not even
have to mess up your straightedge by making a mark on it with the
compass. It will suffice to simply set the compass opened to the
radius of the circle on the straightedge and keeping one point on the
straightedge at C, wiggle it around until you have found the line
through C where one end of the compass is at a point (D) on the
circle, and the other is at a point (A) on the line through O and B,

and this construction can be done using only a compass and a
straightedge. I have found that it comes out about as accurately as
any of my other compass and straightedge constructions.

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Math 100

Steve Wilson