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© Copyright 2000, Jim Loy
Traditionally, geometric constructions are done with
compasses and straightedge, as shown on the left. See Geometric Constructions. You may have heard that REAL
compasses, the compasses used by Euclid, are collapsible. Such compasses are
used to draw circles of a given radius. But when you lift them off the paper,
they collapse, losing the measure of the radius. What is that all about?
The second "proposition" (theorem, or in this case the proof that a method of construction does what we want it to) shows how to "place at a given point (as an extremity) a straight line equal to a given straight line." Now, that is easy. You just measure the given line segment with your compasses, pick them up, move them to the given point, draw a circle, and any radius fits the bill. But Euclid doesn't do this. It is apparently illegal to pick up the compasses, at this early stage of Euclid's series of geometry lessons.
In the diagram on the right, we have a point A and a
line segment BC. The task is to draw a line segment with A as one of its end
points, the same length as BC. Euclid does this by drawing circle B with radius
BC and segment AB, then drawing an equilateral triangle ABD (He shows us how to
do this in Proposition 1). He extends DB out to this circle and draws the
circle D with this radius. Then, extending DA to this circle, the black line
(with end point A) in the diagram is a line that meets our needs.
Euclid went to all this work to do what we thought was a simple task. So, do we have to reinvent some kind of collapsible compasses? No (in my opinion). What Euclid has just shown us is that his bizarre compasses can do all the same things that our easier-to-use compasses can do. He showed that they are equivalent. So, now we can use our compasses knowing that we could do the same thing with more difficult tools.