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    Re: The backstaff. was: Re: The Shovell disaster
    From: Nicol�s de Hilster
    Date: 2007 Nov 06, 16:51 +0100

    In NavList 3755 Michael Daly wrote:
    >> I'd be interested in knowing whether the transversals were always linear 
    >> or whether they (in later instruments) used circular transversals.  The 
    >> former could be used on a circular arc with some error (error decreasing 
    >> with increasing arc radius).  Circular arcs require circular 
    >> transversals to be accurate.  However, it was difficult to figure out 
    >> how to construct them (one technique was determined by Philippe de La 
    >> Hire and used by Nicolas Bion) and very tedious to construct (I have 
    >> created a diagram of linear transversals on Wikipedia but have yet to 
    >> tackle the effort of making a circular transversal diagram - yet I have 
    >> the advantage of a CAD program, not a sheet of brass with dividers and 
    >> compasses).
    And I responded in NavList 3762:
    > Hmmmm, I'm not sure what you mean with the difference between linear 
    > transversals and circular transversals regarding the Davis Quadrant. I 
    > guess that you mean that the diagonals should be arcs instead of 
    > straight lines to be more accurate. The earlier Davis Quadrants did not 
    > have any transversals at all, just plain straight forward scales, evenly 
    > divided in half or quarter degrees. From about 1650 that changed to the 
    > diagonal scales as we know them today. As far as I know these diagonal 
    > scales were all made using straight diagonals. I once made a comparison 
    > in AutoCAD to see what difference there would be between both methods, 
    > but can't remember what the difference was. I'll try to see were I left 
    > that and post the results here.
    So I did some digging and found the following table in my archives:
    decimal     radius [mm]     error [mm]  error [secs]
    0   579.0   0.0000  0.0
    1   580.5   0.0040  1.4
    2   582.0   0.0070  2.5
    3   583.5   0.0092  3.3
    4   585.0   0.0105  3.7
    5   586.5   0.0110  3.9
    6   588.0   0.0105  3.7
    7   589.5   0.0092  3.2
    8   591.0   0.0070  2.4
    9   592.5   0.0040  1.4
    10  594.0   0.0000  0.0
    I constructed a Davis Quadrant in AutoCAD, created a diagonal scale on 
    it with linear transversals. Then I created all the proper 10 minute 
    lines and drew a line through the crossings with the concentric circles 
    of it, hence creating the circular transversals. Finally I measured the 
    differences between the two which are found in the column 'error [mm]'. 
    The radii in the second column are typical for an early 18th century 
    Davis Quadrant (I based the table on the one built by W. Garner in 
    1734). The largest error is 0.011 millimeter (3.9 arc seconds).
    Given the capabilities of the instrument makers of those days this error 
    was way beyond their achievable accuracy. Ever since I made my first 
    Davis Quadrant I checked every example I could lay my hands on (musea, 
    internet, books etc) and whenever the scale is visible in detail you 
    will clearly see errors that are larger than that (in general more than 
    0.1mm, so just under 1 arc minute). So I take it that on Davis Quadrants 
    all transversals were linear in those days, unless the cutting plate 
    wasn't straight.
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