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    Re: Bowditch Table 15
    From: Jim Thompson
    Date: 2005 Feb 3, 08:45 -0400

    This is for nit-pickers only, since for practical purposes we have already
    answered our questions about Table 15 (was 9).
    I obtained a reprint of Guier's paper from the Journal of the Institute of
    Navigation by joining the ION, so that I can download reprints from their
    members'-only website.
    In his Introduction he wrote that the 1958 edition of Bowditch had
    "neglected a correction for observer height in the Explanation of Tables.
    This error was removed in the Corrected Reprint, 1962 Edition.  The
    discovery of that error together with a desire to extend the tables to
    smaller angles and shorter objects motivated the author to examine the
    accuracy and application of the tables."  He developed the current tables
    "having in mind the particular application of small craft piloting."
    His Figure 1. shows the complex curves, lines and angles behind the
    equations.  He shows the derivation of the equation that we see today in
    Bowditch.  The solution is not trivial -- lots of calculus and assumptions.
    No wonder I could not reverse-engineer the solution.
    The equation that he derives, which he wrote was the one given in the 1962
    Corrected Edition of Bowditch, is not written precisely the same as in the
    current 2002 Bowditch.  Since I am not a mathematician, I will reproduce
    Guier's equation here:
    R =(approx) rR x [-tan(a) + sqrt{tan^2(a) + 2 x ((hs - ho)/rR)}]
    R = "Range to base of sighted object"
    rR = "refracted equivalent of earth radius"
    a = "vertical angle at observer"
    hs = "height of sighted object"
    ho = "height of observer"
    In another section he shows how to derive the constants 0.0002439 and
    0.7409.  Note that 0.7409 is not 0.7349, the constant used in the current
    2002 Bowditch.  In that section he gives this equation:
    R =(approx) cos(Hs - D) {-(sin(Hs - D)/0.0002439) + [(sin(Hs - D) /
    0.0002439)^2 + ((hs - ho)/0.7409)^0.5.
    D = "dip of horizon of observer (min. of arc)".
    The equation shown in the current 2002 Bowditch, which Bill found produces
    the values in the current tables, is:
    D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.0002419)
    Guier said that the tables are accurate to about 0.1 NM in non-standard
    refraction. Interestingly his table values are 0.1 NM less than the table
    values in the 2002 Bowditch for long ranges, but they are equivalent for
    short ranges.  So somewhere after his publicaton there must have been more
    work that we have not found reference to.
    Not that any of this would be significant to a cruiser estimating range from
    a distant object over the horizon by using a sextant to measure the vertical
    angle.  Table 15 in Bowditch would work fine, since the Table's precision of
    0.1 NM is more than enough for such purposes, for example to check that GPS
    datum is accurate for a remote Pacific island (in my dreams).
    Jim Thompson
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