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    Re: Coordinates on Cook's maps
    From: George Huxtable
    Date: 2007 Apr 20, 00:01 +0100

    I wrote-
    
    | > Halving an
    | > angle with the compass is easy,
    
    and Alex replied-
    
    | Really??
    | Would you explain me just how to halve a given angle
    | with a compass? (No ruler/straightedge!!  The same
    | famous Bird who said: there is NO WAY to draw a straight line
    | between two given points. And this is true.)
    | Is this taught in the British schools?
    
    All right. Lets assume we have a precisely-drawn arc, struck from a centre,
    more than 60 degrees long, with some arbitrary radius, defined by the
    setting of a beam compass.
    
    Mark an arbitrary starting point near one end of that arc. Without changing
    the setting of the beam compass, strike a short arc, centred at that
    starting point. Where it crosses the main arc is at exactly 60 degrees along
    that arc. That wasn't specified in Alex's challenge, but it gives us an arc
    we are going to halve, by a method we can use to halve any arc, not just one
    of 60 degrees.
    
    Set the beam compass to be, by eye, slightly more than the half-arc we plan
    to mark off, and with it, strike a trial arc from both the start point and
    the 60-degree end point. Those arcs should intersect at two points. Joining
    those points by a straight line, that line crosses our 60-degree arc exactly
    at its mid point. That's the way it would have been taught in British
    schools, 60 years ago.
    
    What the supposed problem is, in joining two points by a straight line, that
    Alex quotes from Bird, I don't understand. But if he objects to that
    straight line for some reason, then let the radius of the beam compass be
    reduced a bit, and a new pair of arcs drawn from the start and end points,
    to cross closer together than before. Adjust the radius still further, by a
    trial-and-error process, until the two arcs just cross where they cross the
    60-degree arc, at the one point. That marks the exact half-angle, in this
    case 30 degrees from either end. In practice, the trial arcs would be drawn
    so lightly as to be almost invisible, and only the final mid point would be
    shown in bold.
    
    Will that do, Alex?
    
    ===================
    
    In Navlist 2634, Alex wrote-
    
    "So I study the question myself using the information available
    to me: I examine certificates of the sextants sold on e-bay:-)
    Unfortunately, they did not supply certificates in the first
    half of the XIX century, so I can only study recent history of
    division.
    
    Interestingly, many mid XX century sextants have quite large
    corrections in their certificates (30" to 45").
    Which means of course that the manufactures simply DO NOT CARE
    to make a perfect arc."
    
    Well, I don't regard non-zero errors on the calibration certificate as
    anything much to worry about, as long as there was a smooth variation
    between them (though it would be hard to know for sure that that was the
    case). It's an easy matter to guess an adjustment to apply, based on the
    certificate that's pasted into the box.
    
    But it's an interesting question as to how precise the test instrument (at
    Kew, say) really was. Imagine that it got known, say, that the collimator to
    test for a nominal  60 degrees happened to positioned 20 seconds high. Might
    a sextant maker be tempted, then,  to make all the instruments he sent to
    Kew to read, correspondingly, 20 seconds high at 60 degrees, so that their
    certificates came out spot-on? Just a thought....
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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