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    Re: Coordinates on Cook's maps
    From: Alexandre Eremenko
    Date: 2007 Apr 20, 01:43 -0400

    
    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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