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Bill, in [NavList 1864], quoted from Dutton's, about intersecting
position lines. I have a healthy respect for Dutton's.

That passage (para.3018 in his 15th edition, 3009 in my 12th edition)
is dealing with the situation of a systematic error, the "constant
errors" that are referred to, with negligible random scatter . And if
you had a way of knowing that was the situation, you could apply that
recommended procedure. But from a single 3-body fix, how could you
possible tell whether any discrepancies stemmed from "constant error",
or from random scatter? They are entangled together, in a way that's
almost impossible to separate.

Our analysis, putting only 1 in 4 inside the triangle, assumed
entirely random scatter, and no "constant error", and specifically
said so. It's the very opposite to what that section from Dutton's was
considering, as quoted by Bill.

In his analysis of the problem that Guy Schwarz presented, in [NavList
1848], Geoffrey Kolbe was able to deduce , with reasonable conviction,
that all six observations may have been subjected to just such a
common error, because shifting them all "away" by the same amount gave
a much more consistent picture of their crossings. That may indeed
have been a valid deduction from that evidence of six altitudes. But
to make it with any confidence, you need a sufficient number of
observation, to over-determine the answer and examine any conflicts
between them. That's the valid point that Frank has made. You just
can't do it, on the basis of a single cocked hat.

Dutton's is not infallible about these matters, though. In para 2016,
in my 12th ed., referring to position lines, he writes "In actual
practice they will seldom intersect at a point, but will produce a
small polygon, often referred to as a "cocked hat", which usually
contains the position of the ship." On the contrary, we have seen that
where there are no systematic errors, usually, (three times out of
four) a triangle will not contain the position of the ship. On that,
Dutton's is quite wrong.

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

There may be a bit of a question about Guy's analysis, seeing that
John Cole claims differing answers, which might possibly explain the
systematic discrepancies that have been seen.  It would be useful for
someone to set down for us the original problem, as set.

Nevertheless, Guy's plot of position lines, in [NavList 1848] with its
six crossing lines, can be instructive, to be pondered on by anyone
still arguing that a cocked hat must contain the ship's position. That
plot shows a jumble of 6 lines, from which 20 possible cocked-hats can
be constructed, from each possible set-of-three, if the other three
are ignored. Consider a few of those possiblilities. Imagine that
none of Kochab-Rasalhague-Venus been observed, and the position had
been deduced from Altair-Alpheratz-Moon only. Those three happen to
coincide nearly in a single point (just like one of Henry Halboth's,
perhaps) so there we have a tiny cocked hat, establishing the vessel's
precision precisely.

Compare that with the result if, instead, Kochab-Rasalhague-Venus were
adopted, and the other three ignored?  A quite different triangle
results, far, far bigger. It happens, in that case, to enclose our
earlier triangle; but sometimes, two such triangles will have no area
in common whatsoever, and then, it's quite impossible for the same
vessel to be in both triangles. Such as Kochab-Alpheratz-Moon, and
Rasalhague-Altair-Venus, for example.

The divergences, between those triangles, nicely illustrate another
point that Frank Reed made.

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

Robert Eno asked-

"So here is my question: for purposes of practical navigation at sea,
does
one really have to take into account the statistical probabilities of
the
actual location of the fix,...or is it
sufficient enough to do what countless navigators have done for
generations:
take the fix from the centre of the cocked hat and/or the centre of
the
"smudge"?

I believe that the latter it is good enough for practical purposes."

Yes, of course that's what you have to do. There really isn't an
alternative. And in assessing the size of that smudge, all I ask is
that common sense is used, some experience of how good one's
observations are likely to be in the current conditions, and not blind
adherence to the outlines of a particular cocked hat as the limit of
error. I think there's no disagreement on the matter, between Robert
and me.

And really, there's no call, and seldom an opportunity, to make any
"statistical analysis".  Never. Just a bit of common sense is all
that's required. "With eyeball and thumbprint" (perhaps a good title
for a navigator's biography).

Robert added-

"In terms of nearshore navigation where one is likely to encounter
dangerous
reefs, astro navigation is the last thing I would want to employ.
Shore
bearings would be a more accurate means of keeping one's distance from
dangerous ground and better yet, radar."

Well, that's a point of view you can take today. But as little as 20
years ago, mariners were having to approach reef-strewn coasts in
parts of the world which were badly lit and badly marked, and had no
other option. In thick weather, they were blind.

George.

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


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