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    Re: Almanac errors. was: Navlist 4576, Re: Flinders' survey of Australia
    From: Frank Reed
    Date: 2008 Mar 04, 01:57 -0500

    After about 1787, when Mayer's lunar tables were improved by Charles Mason,
    the error in the lunar distances was about 30 arcseconds average, twice or
    even three times as much in extreme cases. You could do some statistics, if
    you want. There are Nautical Almanacs from that period available on Google
    Books. To compare against modern lunar positions, visit my web site and
    click on "Predicted Lunars". Set the date to whatever date you're
    investigating. Then select "Greenwich Apparent Time" and angle format "dd mm
    ss" under Options. Then hit calculate. The lunar distances are directly
    comparable to the published lunar distances in the almanac.
    After Buerg's tables were published in 1806, the calculated distances in the
    almanacs was reduced to about 12 arcseconds on average (the "ue" in Buerg
    should be u-umlaut --I don't want to deal with character sets right now).
    Tables published by Burckhardt in 1812, following Buerg's model but with
    more observations, reduced the error to about 7 arcseconds. I don't know for
    certain when these tables were applied to the British Nautical Almanac.
    There may have been a delay of two or three years. The French Bureau des
    Longitudes was actively offering prizes that led to Buerg's and Burckhardt's
    tables in this period.
    Plana's tables of the Moon were apparently first used in the American lunar
    tables beginning in 1853. They reduced the error to about 5 arcseconds on
    average and were soon adopted in other almanacs. For a few short years, the
    American lunar distance tables were the best available. Ironically, it is
    just at this moment that lunar distance observations were being finally
    abandoned for checking chronometers at sea even aboard American vessels
    (which were late to adopt the modern tools) though they survived for a few
    decades in land exploration and survey work.
    Following the publication of Hansen's tables in 1857 in London, most lunar
    distances in almanacs had errors below about 3 arcseconds. However, these
    errors grew with time and twenty years later, unless Newcomb's empirical
    corrections were applied, the error could again be as large as 10
    arcseconds. Of course, by this time, the lunar distance tables were
    effectively obsolete.
    Multiply all of the errors in distances above by a factor of 30 (-ish) to
    convert to longitude error. Hence a 10 arcsecond error in the lunar distance
    translates into a 300 arcsecond error (5 arcminutes) error in the derived
    longitude. Also, not that errors in the lunar distance tables smaller than 1
    or 2 arcseconds are not meaningful since the Moon's limb is not smooth at
    that scale.
    Most of the history of the tables outlined above is taken from Souchon's
    "Treatise on Practical Astronomy", Paris, 1883. Also, there is a table on
    page 93 of "From Sails to Satellites" by J.E.D. Williams which is fairly
    accurate, I think.
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