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    Re: Old style lunar
    From: Alexandre Eremenko
    Date: 2004 Dec 13, 14:11 -0500

    Dear Ken,
    On Mon, 13 Dec 2004, Ken Muldrew wrote:
    > I think what Fred meant here is that if Thompson
    > took two lunars on a
    > particular night, one on each side of the moon,
    > then his average of the
    > two resulting longitudes would tend to be close to
    > the true position but
    > the errors of each lunar individually would be unexpectedly large.
    There are several sources of error in the lunars.
    1. Error in measurement (index error, other instrumental errors,
    personal errors etc.)
    2. Errors in the almanac (the Moon position among the stars
    is computed and printed incorrectly for a given time GMT).
    Measuring the distances E and W of the Moon, reducing them
    and then taking average (of the resulting GMT, or of the
    resulting latitude) tends to eliminate the first kind of errors.
    But not the second kind.
    As we are trying to understand the source of Thompson errors,
    the different sources have to be considered separately.
    > > Let us
    > > consider an idealized situation when the stars are
    > > exactly on the Moon's path and positions of the stars are known
    > > precisely. Consider two stars, Moon in between.
    > > Then the error in Moon's position will lead to the errors
    > > in distances which are exactly opposite to each other.
    > Yes.
    > > Now suppose we measured both distances EXACTLY with our sextant
    > > and want to deduce longitude (or chronometer corr., does
    > > not matter)
    > > Averaging the two reduction results will give you nothing.
    > > Both will give the same error in time/longitude.
    > I don't see how that follows.
    Think of it in terms of time.
    The incorrect Moon distances in the almanac correspond
    to some Moon position but at a different time.
    This error in time will enter to your calculated GMT,
    no matter which star to use to measure the distance.
    And the error in GMT will be the same for all stars,
    and of the same sign.
    So it will not cancel when you take the average.
    I assume that star positions in the almanac are given precisely.
    It is only the Moon position (with respect to the stars) which is
    prone to error, because the Moon theory were imprecise in 1800.
    We can eliminate this error by comparing the 1800 almanac with
    the modern one. The REMAINING error will characterize
    Thompson's accuracy.
    There was another conjecture on the source of Thompson error
    in the longitude: the inaccuracy of his watch.
    This he could eliminate, of course by taking the "time sight"
    simultaneously with his distance sight.
    As Gregory did, according to Kiernan Kelly.
    I am saying that both latitude and longitude can be determined
    WITHOUT a WATCH by astronomical observations.
    So if Thompson knew that his watch was of low quality,
    he could do it without a watch.

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