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Fred Hebart wrote-

>On thinking on this a bit more, I believe that to _rate_ a chronometer
>one needs at least three lunars spread over at least three days.  In
>that procedure, one is looking for the rate of change of chronometer
>time relative to astronomic time, as well as the absolute departure.
>Then one would regress clock time on astronomic time and look for
>significant departures from 1.0 in the slope.  This would not be
>possible over a series of sights _within_ one lunar, such as over a
>period of one hour, because the clock error would be negligible on any
>chronometer worth winding.

Comment from George.

There are two things about a chronometer that need to be known. One is its
error at some recent moment, that is, how many seconds it was ahead or
behind some mean time (preferably Greenwich). The other is its rate, the
amount it is gaining or losing on that time, each day, It's a pity that our
terminology is so confused, in that the words "fast" and "slow" are used to
describe both these matters. We often say that a clock is "fast" on
Greenwich if it's ahead of Greenwich time, even if its rate is "slow" on
Greenwich, so that in the end Greenwich will catch up. So I wish that
"fast" and "slow" were reserved for the chronometer rate, and instead other
word-pairs, such as early/late, ahead/behind, lead/lag were used for its
error at some moment. Not that Fred was confusing these terms, by the way.

A good lunar, from at sea, can provide Greenwich time to within a minute or
two, which is no better than sufficient to confirm that a chronometer
hasn't gone seriously wrong. That was the big danger that navigators with
only a single chronometer needed to guard against. But using lunars to
determine the RATE of a chronometer is bound to be unsuccessful, because
each observation is so inaccurate, even if they are taken several days
apart. I'm not sure why Fred insists on three such observations rather than
two, but I doubt if it will help much.

Rating a chronometer involves comparing it with a precise astronomical
clock, to a fraction of a second, over at least one day, preferably several
days. Or with a noonday gun, or with precise time-sights of a celestial
body from a known fixed location. The important thing is that these times
should be known to a second, or preferably less, and applied over a
prolonged period. Let's presume we can check the rate of a chronometer to
within one second per day, and rely on that rate to remain constant (which
is the test of a good timekeeper). After a two-month ocean passage (common
in those days), at the ensuing landfall it might well be a minute out in
time. At that stage, it's timing accuracy would not be much better than
that of a lunar. That's why an accurately measured rate (and a constant
one) is such an important matter.

>To uncover the absolute departure of chronometer time from astronomic
>time in one lunar, the mean time of a set of observations could be
>compared to the mean astronomic time.  I don't think a regression would
>be necessary, because, again, over such a short period, the rate of
>change of chronometer time compared to astronomic time should equal
>one, at least to any obtainable level of precision, given a usable
>chronometer.
>
>With more than one lunar, the absolute departure of chronometer time
>from astronomic time would be a time function of the slope and
>intercept of the regression of chronometer time on astronomic time.
>This of course is assuming a straight line relationship between the
>two.  I would imagine that people then found that including temperature
>as a covariable could account for most significant discrepancies.  I am
>not sure how this second equation would be formulated.  I would be
>delighted to know.
>
>So a regression of distance on clock time is not needed to reduce the
>observations from a single lunar.  It can be one relatively simple
>method (using a computer) of drawing a line through the observations to
>check for outliers and to assess how scattered the data are.  If a
>point falls conveniently close to the line near the middle of the
>observations, it would be a reasonable one to clear.

In my view, the ONLY purpose in making a plot of lunar distance against
time is to show up blunders, points so far off the trend-line as to show
that something has gone significantly wrong. That's valuable information,
allowing such points to be eliminated.

As an aside, I will relate a tale of a list member (no names here, but he
knows who he is...) who sent me an apparently superb series of successive
measurements of a lunar distance plotted against time, every point lying
remarkably close to a trend-line. It turned out that he had actually taken
nearly three times as many readings as were plotted, at very short
intervals, and then ruthlessly eliminated every point (more than half of
them) that deviated noticeably from what he took to be his "best" straight
line. That should NOT be the basis on which to reject data!

Certainly one COULD plot a line calculated from a least-squares fit, but I
doubt if it would add anything useful to the simple process of averaging
the lunar distances and the times. Alternatively, a line of best-fit, drawn
with judgment by eye, could be used. In either case, the combination of
time and lunar-distance, given by ANY point lying on that line, preferably
near its middle, can be used for the purpose of calculation. Presuming that
altitudes of Moon and other-body had been measured rather than calculated,
perhaps the best point to choose along that line would correspond to the
mean time of those altitude observations.

I take it that instead of choosing a point on that line, Fred was making
his reductions based on a single observation that he observed to lie very
close to that line of best fit, not from a point on that line itself. If
so, he could do better.

The SLOPE of any such plot is so badly altered by the effects of the Moon's
parallax changes, that I doubt if it's useful for anything at all.

In these circumstances, I regard the benefits of modern statistical
analysis (least-squares-fits and such) as largely illusory, even if a
computer can make such calculations so easily.. Which is similar to the
message that Herbert Prinz was putting across, when he said it was
"shooting with guns at sparrows".

George.


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