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Alex reported his first lunars, as follows-

>AP: N 40d27.2'  W 86d55.8'
>GMT: 4:00 Oct 24, T=58F Pressure 29.75
>Observation from my balcony, height 12ft,
>Sextant SNO-T, index correction 0.0', inverting scope.
>One of the 6 observations was immediately rejected
>because it did not follow the pattern of increasing
>distances. I reduced with Frank Reed's on-line calculator.
>First, each measurement, and then their average.
>The third line is the error in the distance, the fourth line
>is the error in the longitude:
>
>Moon-Altair:
>
>GMT   4:06:49    4:09:58     4:13:10    4:17:12    4:18:57
>DIST  51d22.2'   51d23.3'    51d23.8'   51d24.1'   51d34.3'
>ERD       0.0'      +0.5'       +0.3'      -0.2'      -0.4'
>ERL      +0.3'     +13.5'       +8.7'      -7.3'     -12.2'
>
>AVERAGE GMT: 4:13:13  AVERAGE DIST: 23.54'
>ERROR IN DISTANCE: 0.0' ERROR IN LONG: 0.4'
>
>Moral: DO average:-)

============================

Congratulations to Alex for an outstanding set of observations. He has
quickly got his lunars down to a fine art.

Some niggles, however...

1. Presumably there's a typo in the 5th reported lunar distance, and it was
actually 51d24.3', and not 51d34.3' as stated.

2. Alex stated-
>One of the 6 observations was immediately rejected
>because it did not follow the pattern of increasing
>distances.

I think this is over-simplistic, if that was the only reason for its
rejection. It would be interesting to see the details of the rejected
observation, and HOW MUCH it was out from that pattern of increasing
distances..

When lunar distances, which change only slowly, are being taken at short
intervals, then very small changes in the lunar distance are expected
between observations. In the set given above, the total change in LD is
about 2' of arc  over a period of 12 minutes of time. If two observations
were taken with a 2-minute interval, then an increase in LD between those
observations would be expected to be about 0.3 arc-minutes.

Looking at Alex's set, and doing a rough plot-in-the-head, rather than on
paper, it seems that a scatter of the points, of something like 0.2' about
the best straight line, would be expected. So if one observation happened
to be 0.2' high, and the next one 0.2' low, that would more than cancel the
expected increase of about 0.3', and break the monotonic sequence of
always-increasing. This would cause the second observation to be rejected,
although statistically, it's as valuable as the rest.

Anyway, if you reject a point just because it breaks that sequence of
always-increasing, can you be sure that you are rejecting the right
observation? If its predecessor happened to have an unusually-high positive
error, then it would be likely to put the NEXT observation, even if a
perfectly normal one, out of the always-increasing sequence, and cause the
wrong one to be rejected.

In my view, all observations should be accepted, unless they are clearly
blunders, and detecting such blunders should be a main purpose of making a
plot.

3. It would be meaningful to give the averaged error for the 5 observations
to an extra decimal place, in which case it would be 0.04', not 0.0'.

4. The scatter of errors in distance, and the resulting scatter of errors
in longitude, shows that Alex has made a remarkably consistent and accurate
set of observations. But the errors just happen, on averaging, to cancel
out to nearly zero. This was fortuitous! It was Alex's lucky day.

He happened to deduce a longitude which was only 0.4 miles from where he
knew he was, from a set of longitudes that individually showed a scatter of
about 10 miles or so. Averaging 5 such measurements should reduce the
statistical error of the average, to 10 miles  divided by root-5, or about
4.5 miles. So don't be misled by the precise result of 0.4 miles. Such
things happen, by chance, once in a blue moon: perhaps, for Alex, never
again!

Averaging is indeed a powerful tool for reducing errors, but don't expect
it to produce the same magic effects that happened on this one occasion.

Eighteenth-century navigators at sea were very content if their longitudes
came within 30 miles of the truth. On Alex's balcony, taking every
precaution, he would indeed do well to achieve a result that's consistently
within 5 miles.

=======================

The comments above are not intended to detract in any way from Alex
Eremenko's considerable achievement, made in a very short space of time. I
look forward to the time when he can measure lunars at sea from a small
vessel.

George.

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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