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Dear George,

First, I give the precise citation:
According to Bird,

"...a right line cannot be cut on brass,
so as accurately to pass through two given points,
but a circle may be described from any centre, to
pass accurately through a given point".

This is cited in Rees Cyclopaedia. vol XVI,
article "Graduation", and also
in the paper by
Saul Moskowitz, The world first sextants,
(one of the very few good papers on the Inst of
Navigation CD you recently bought. In general,
Moskowitz is a rare exception among most of the authors
on this CD; his papers are substantial and interesting).

This sounds paradoxal, but if you really think
carefully on how exactly would you draw a straight line
passing through two given points, and actually TRY to
do it with high precision, you will see that
Bird is right.
We are talking here of the accuracy of
0.0001 = one over ten thousand of an inch
when graduating a decent 8 inch arc, by the way.

Just try with a very sharp pencil on paper and then
look at the result through a magnifying glass,
if you don't like thought experiments:-)

The pencil point never goes exactly along the straight
edge, it always goes parallel to it on some unknown
distance. I am not even speaking
of aligning the straightedge
with the given points: how exactly do you do this?

By the way, according to the same British division masters
of the late XVIII century, dividing a straight rule
is even harder than dividing a circle.
I think it was Ramsden who invented
one of the first machines for dividing
straight rules.

Returning to "high school geometry",
the iterative construction you propose, of course
permits to divide approximately.
In PRINCIPLE, to any given accuracy.
But if you think on how would you actually perform
this successive approximation procedure, you will
disvover difficulties. For example, you will need
a microscole through which you can see much smaller details
than the accuracy of your division. I mean to draw the
tiny arcs on the last steps of your procedure.

There exists a method of dividing any arc into two equal
parts by compass only. And EXACT metod, not requiring
a microscope. The understanding of the method
requires nothing but a solid background in
"high school geometry". But I would not call it
"simple"!


It was discovered in XVIII century, and I don't know
whether the discoverer had in mind a practical application
to instrument division, or he was driven by curiosity
of a pure mathematician.
He actually proved that EVERY construction which is
possible with a compass and a rule is ALSO
possible with compass only.
Neither I know whether this method was actually used.

http://www.cut-the-knot.org/do_you_know/compass.shtml

Alex



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