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    Re: New compact backup CELNAV system
    From: Gary LaPook
    Date: 2009 Feb 27, 01:37 -0800

    Of course if the scale was wrinkled it could cause an error but a 
    distortion of the scale big enough to cause a problem I think would be 
    noticed by the user so he would smooth it out to cure the problem.
    One caution I neglected to mention is that it is important that both the 
    cotangent scale and the cosine scale be printed to the same scale. The 
    PDF files I provided do print to the exact same scale on my Hewlett 
    Packard ink jet printer and on several printers I have tried. If you 
    make a copy of the slide rule compare the cosine scale against the 
    cotangent scale. The cotangent scale goes through two cycles on each 
    line so the cosine scale should match up against one cycle, half of the 
    cotangent scale. If it doesn't then change the printer scale slightly 
    using your printer controls to make it work out.
    I have made a half dozen of the cylindrical models and they are easier 
    to use than the flat model because the scales are easier to read since 
    they are separated and do not overlie each other. But they are much 
    harder to make, getting the length of the scales just right and mounting 
    the scales on the tubes, getting it exactly right, forming tubes of 
    exactly the right size to fit tightly enough together to provide enough 
    friction so they maintain their alignment but not too tight to prevent 
    easy manipulation. I have often had to start over again when something 
    went wrong during the construction.
    So I think the flat one is quite useful. It can be printed off by 
    anybody with an ink jet printer for pennies each and if you want to seal 
    the cotangent scale in a plastic document protector it raises the cost 
    to about a buck each.
    Obviously I like the Bygrave and I think it is quite elegant in its 
    simplicity and in its power to solve the navigational triangle, a 
    computation that has involved the efforts of many of the great 
    mathematicians over the centuries. One of my objectives was to make it 
    available to a much larger audience of navigators so they can evaluate 
    it for themselves.
    BTW, does anybody know what became of Captain Bygrave after 1920?
    Oh, and thanks for the compliment.
    bmorris@tactronics.com wrote:
    > Hi Gary
    > Just to be clear, I was not questioning the accuracy of the Flat Bygrave, 
    WHEN FLAT.  Just as you have indicated, when flat, the device yields 
    perfectly acceptable results.  Since the scales are a realization of the 
    equations, unquestionably, they should work when used as directed.
    > If, however, one scale was distorted by an out-of-flat condition, then we 
    would see errors proportional to the error in flatness. One scale flat, with 
    the other distorted will yield problems. That was my assertion and remains 
    so.  Sharp as you are, you should see that as well.  
    > My comments also indicated wrapping the scales around cylinders, which 
    eliminates the distortion of one scale to another.  That solves the 
    out-of-flat condition by making the surfaces stiff.  The follow-on problem, 
    how to lock one cylinder to another securely, forces a mechanical solution.  
    Bygrave himself didn't solve that properly.  There are a few mechanical 
    arrangements I can think of to do that, all involving multiple moving parts.  
    This would force your simple Flat Bygrave solution to a more complicated 
    cylindrical Bygrave solution, and keep the device out of the hands of many 
    who would try.
    > Again: Nothing wrong with the Flat Bygrave when used properly.  You should 
    be rightly proud of it.  Gosh knows I would be!
    > Best Regards
    > Brad  
    > >
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