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From: George Huxtable (no email)
Date: Fri Oct 08 2004 - 13:22:49 EDT
Alex Eremenko wrote
>The specific of aerial observations is that the random error
>coming from acceleration is very large, and the averaging has
>to be done quickly. That's why they have a built-in mechanism,
>integrator, that does it automatically.
From Ken Gebhart's contribution, it seems that another factor, specific to
observations, is at work. This is the CYCLIC (rather than random) nature
of the perturbations. If the period of those fluctuations is well known,
and the integration period is just one, or an integer number, of such
oscillations, then that effect can be automatically nulled out.
In contrast, marine sextant observations appear to be affected by
fluctuations that are much more random, in which case the reduction in
overall error by averaging is proportional to the square root of the number
of observations taken, in accordance with Gaussian error theory. But
remember, certain errors, particularly errors of scale division, repeat
each time a measurement is repeated, so that you can average as many
observations as you like and it won't reduce such errors
=================
An interesting example of averaging arose in the 18th century, with the
development of the Hadley quadrant (octant). Early instruments suffered
fron non-uniform scale division, which was done by hand. So the repeating
circle was developed by Mayer (the German astronomer who won part of the
longitude prize for his lunar theory). This expanded the quadrant into a
complete circle, divided into 720 degrees. Both the telescope and the index
arm could be moved with respect to the circle. First the index error was
assessed, and then the observer measured an angle, using just a part of the
circle. Then the telescope was moved to perform another index assessment,
and the index arm adjusted once again, to measure another angle over a new
part of the scale. The telescope and index arm are moved alternately in a
"walking" action round the scale. This can be repeated indefinitely, as
many times as the observer wishes, working his way round the circle as many
times as desired. The final reading on the circle's rim relate to the sum
of all the observations, so should be divided suitably to get the average.
The intermediate readings don't need to be noted.
Ideally, this operation stops when the instrument has measured enough
observations to sum up one or more whole turns around the circle (or
approximately so). In this way, every part of the circle's scale is given
equal weight in the averaging process. No matter how unevenly the scale was
divided, each complete turn around it simply has to total 720 degrees. So,
with an appropriate number of observations, errors in scale division can be
effectively nulled out.
This type of instrument culminated in Borda's circles, large ones for
land-survey, smaller ones for navigation. Such instruments were heavy and
clumsy to use at sea, though very accurate for lunars..
By the way, there's a recent book out in paperback, "The Measure of All
Things", by Ken Alder (Abacus 2003 or 2004), about the French attempts to
define the length of the metre by accurately surveying the radius of the
Earth, along the meridian through Paris, from Dunkerque to Barcelona, in
Revolutionary times (1790s). This used Borda circles.
In England, the invention of Ramsden's dividing engine, which allowed
precise marking of the arc of a sextant, meant that circular-scale
instruments failed to become really popular.
George.
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contact George Huxtable by email at , 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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