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    Re: Real accuracy of the method of lunar distances
    From: Fred Hebard
    Date: 2003 Dec 31, 00:13 -0500

    This is the 99.7% probability that an _individual_ observation will
    exceed 90".  The standard error of the mean would be the standard error
    as you defined divided by the square root of the number of
    observations.  I believe (not being a statistician) if you divide 90"
    by the square root of 6 you will get the value that the mean will not
    exceed with 99.7% probability, namely, 37", or 15 minutes of longitude.
      That's quite usable for ocean navigation, and is the worst that is
    obtained once in a thousand times, more or less.
    On Dec 30, 2003, at 1:24 PM, Jan Kalivoda wrote:
    > Now let us consider the maximum errors of lunars that should be taken
    > into account, based on the values given above. If you multiply the
    > standard error  (SE) by 2, you obtain the error that should be not
    > exceeded in 95% of observations, and you can be statistically sure
    > that your error won't exceed the value of (SE times 3) in 99,7% of
    > observations. Therefore, we can accept the maximum possible error in
    > practice as SE ? 3 = (1/0.6745 ? PE) ? 3 = 4.5 ? PE very nearly.
    > For lunars, PE of 20" times 4.5 gives 90" = approximately 180 seconds
    > of time = approximately 45 minutes of longitude (the exact value
    > depends on the actual velocity of the Moon in R.A.). For finding your
    > position at sea, you should accept this maximum value in both E and W
    > direction and your real position lies somewhere in the arc of
    > longitude 1,5 degree long. This makes lunars nearly unusable in
    > practice in my eyes - where I make a mistake?
    > Of course, when you take two lunars to the East and to the West from
    > the Moon simultaneously, the maximum possible error drops to the
    > usable limits (onboard sailing ship).
    > -------------------------
    > It should be noted that the errors of lunar tables were negligible
    > compared with the errors of measurements only after cca 1880 (after
    > Newcomb). From cca 1820 to 1880 (from Damoiseau and others to Newcomb)
    > one had to accept the error of 20-30" in lunar tables and from 1755 to
    > 1820 (from Mayer to Damoiseau) the possible error was 60" .
    > Maybe this fact gives a commentary to the quotation from Cotter (A
    > History of Nautical Astronomy, p.256), who himself quotes the report
    > of Parry's Arctic expedition (1821-1823) concerning lunars taken from
    > the ships caught by ice and staying at the same place for many months
    > (not drifting!):
    > "The mean of 2500 observations in December differed 14' from the mean
    > of 2500 observations in the following March; ..."
    > ================
    > I want to be refuted, as I like lunars very much.
    > Jan Kalivoda

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