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From: Alexandre Eremenko (no email)
Date: Thu Oct 28 2004 - 18:43:37 EDT
Dear George,
1. I will send you the Halley's paper shortly.
2.
On Thu, 28 Oct 2004, George Huxtable wrote:
> But the questions remain, related to our previous long discussion
> on
> "averaging". Is there a non-linearity effect which will upset
> the result of
> a longitude deduced by averaging lunar distances,
> especially of a series
> that's protracted in time?
The only rigorous answer would be a "table of non-linearity"
for distances of the sort I produced for altitudes.
Without such computation, I can only say that I FEEL
that the non-linearity error for distances will be irrelevant
on much longer intervals than that for the altitude.
In the discussion of "Averaging" everyone seemed to agree with
this from the beginning. (Though of course I cannot quantify
this without a computation).
Whether this computation is worth efforts, I am not sure.
The "Altitude non-linearity tables" did not have much feedback,
and their only purpose was to refute Herbert Prinz's exaggerrated
statements (like "manual averaging is the thing of the past" or
that the averaging is appropriate "in the few special cases"
(Sat Oct 09 2004 - 19:18:32 EDT ))
which triggered
that whole discussion.
Alex.
P.S. Very brief explanation for those initiated to math.
(2-3 semesters of calculus is assumed here)
The reason of troubles with altitude is that the altitude
is not TWICE differentiable function on the sphere.
The singularity occurs at 90 degrees.
This is the reason for non-linearity of altitude near 90 degrees.
The DISTANCE is also not twice differentiable, but the singularity
occurs at 180 degrees.
As we never come even close to such distances with our sextants
or even pentants, the non-linearity of distance cannot be
a source of troubles.
But to quantify this statement (how long is the maximal
admissible interval for averaging) will require computations.
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