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Fred,

It would seem that you are applying the least square fit incorrectly. To apply
it to the plot of distance versus time is to shoot with guns at sparrows. You
can have this cheaper by averaging the time, averaging the distances, and then
proceed as if the averaged values where the actual observation - just as you
are suggesting. This is what was normally done, and what is feasible to the
navigator without electronic tools. To use this technique does not prevent you
from plotting the distances versus time to find outlyers. As long as you have
the individual distances, that is, because from repeating instruments (such as
the Borda circle), one does not get them.

However, if you really must apply a least square fit to the given data, (and I
could not argue with you if you claim that this is the only rigorous treatment
of any set of more than 2 observations), the only correct method is to solve
for the watch error for which the sum of the square of the distance residuals
becomes a minimum. Besides from being correct, it has the additional benefit
that you can combine observations to different objects, in particular
observations on either side of the moon, which helps to cancel certain
instrument and observation errors.

It goes without saying that this has no practical value on the boat. But it
could have been used this way in surveying, right before the telegraph was
introduced.

Herbert Prinz

Fred Hebard wrote:

> Interestingly, I usually plot the raw distance against the time of
> observation and use the least squares fit to pick out a point for
> reduction.

> I think perhaps the old method of using the mean of the observations
> would be better than using a line of best fit, although plotting the
> data instantly tells one how good they are.  Using the mean, the time
> would have been out by 8 seconds, about 2 minutes of longitude.

   

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