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    Precision, Averaging
    From: Alexandre Eremenko
    Date: 2004 Oct 9, 12:36 -0500

    This message is related to two topics recently discussed:
    "Sextant precision" and "Averaging".
    (I apologize in advance to those who consider "land navigation"
    out of scope of this list).
    The example I want to share comes from the paper:
    "On the geographical situation of the Three Presidencies,
    Calcutta, Madras and Bombay, in the East Indies,
    by J. Goldingham, Esc. F.R.S,
    published in the Phil. Trans. of the Royal Soc.
    (Read on June 27, 1822.)
    He discusses the precise determination of coordinates
    of the places listed in the title. Mainly of the longitude
    and mainly by observing Jupiter's satellites. But then he compares the
    with those obtained by the Lunars and also briefly
    discusses latitude.
    I cite:
    "The latitude was found by 32 meridional observations
    of the sun and stars, north and south of the zenith,
    taken with the two sextants, and artificial horizon.
    The height of the thermometer and that of the barometer was noted
    at the time of observation, and the correction on this
    account was applied to the refraction. The declinations were also
    corrected for aberration etc. and the results were:
    By 16 observations with Troughton's instrument 18d57'43.5"
    By 16 observations with Ramsden's   instrument 18d57'43.8"
    [Then he describes the technique of observations, that they measured
    lattitudes by taking stars North and South of the Zenith, then
    averaging in pairs. This is supposed to eliminate instrumental errors
    like centering error, compare with the recent message by
    Henry Halboth, Tue Oct 05 2004 - 00:10:23 EDT.
    Then the author continues:]
    "It may be useful to remark upon a difference with Ramsden's
    sextant, in the results by the objects north,
    and by those south of the zenith in observing for the latitude;
    and also the difference in the results by the lunar observations
    east and west of the moon. The instrument was most carefully
    examined, and the error regularly found every day by measuring
    sun's diameter; yet, notwithstanding, the following differences
    were in the results:
    55" in the observations for the latitude and
    34'18.2" [of longitude] in the observations for longitude
    [by Lunar distances. Dividing the last amount by 30, we obtain
    a number of the same magnitude that the error in the
    latitude.- A. E.]
    The sextant by Troughton, in the observations for the latitude
    gave only a difference of 4" between the results by the objects
    north, and those south of the zenith; and about 3' in the observations
    for longitude [by lunar distances].
    These results show the necessity of observing objects both north and south
    of the zenith for the latitude; and also objects east and west of the moon
    for the longitude. The mean of the results thus obtained will be correct;
    the objects on one side giving the longitude as much greater, as those on
    the other side give one as much less than the truth. In consequence
    we find that the mean latitude by Ransden's intrument is only a few tenths
    of a second different from that by Troughton's; while the mean
    longitude is only about 16" different".
    End of citation.
    I think this is a good example of what one can achieve by averaging:
    a) the averaging of 16 observations with one sextant and
    b) the more sophisticated averaging, by planning the observations
    in such a way that most of the errors tend to cancel each other.
    Like averaging the lunar distances East and West of the Moon.
    I also like his Troughton sextant accuracy:-)

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