From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: Continuum Hypotheisis
Date: 11 Jan 2000 05:24:53 GMT
Newsgroups: sci.math
Summary: [missing]

Just a little followup to relay some standard historical info about this topic.

As mentioned, the Euclidean ideal for ruler-&-compass constructions does
insist that the compasses be "floppy", so they can't be used as dividers
to transfer a line segment around; but also as mentioned, this turns out
to be no essential restriction.

More interestingly, it turns out that the *opposite* extreme in compasses,
namely "rusty compasses", are also sufficient.  That is, a pair of
compasses that is rusted into one unmovable radius.  The Arabs showed that
all proper Euclidean constructions could still be effected with these.
Quite a tour de force.  Later it was shown in moderately recent times,
C19 I think, that even just ONE fixed circle, somewhere on the plane,
was *already* enough to do all Euclidean constructions!  If your stuff
is a way off from it, you have to straight-line stuff over to intersect
the circle, do stuff there, then straight line it back.  Now that's *really*
painful!  Even more, one doesn't really need the *whole* of that one
circle, just an (arbitrarily small) arc will do!  Even an arc so tiny
of a circle so huge that it looks like a straight line segment.  Clearly
we are a long way from the practicality of Egyptian field marking by now!!

In the other direction, Mascheroni, in Napoleonic times, showed that
compasses alone *without* straight edges was enough to do any Euclidean
construction! Of course, one needs to re-define the various straight-line
results appropriately, but that is no problem in fact.

======

As an exercise, you amateur geometers might like to ponder this one.

What subset of Euclidean constructions can be done with only THICK RULERS.
i.e.  You may use straight-edges, and a device to draw a line parallel to
a given line and a fixed distance away from it.

With a struggle, you may find you can then construct perpendiculars, 
bisections, general parallels through given points, ... .  How far can you go?

Happy hunting!

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             Bill Taylor           W.Taylor@math.canterbury.ac.nz
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         If we knew what we were doing it wouldn't be called research.
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