# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Averaging lunars: was Lunars with SNO-T
From: Herbert Prinz
Date: 2004 Nov 1, 05:52 -0500

```George Huxtable wrote:

> It may be interesting to read what Tobias Mayer had to say on this topic,
> [...]

Thanks to George Huxtable for posting and Steven Wepster for translating. While
we are at it, Note I, immediately preceding the quoted passage might also be of
interest. In an earlier message I had casually compared the repeating circle
with our averaging technique. This note reminds me that the two methods are
different in one respect.

In Note I, Mayer says that the final error of an observation is inverse
proportional to the number of individual distances that have been accumulated
into this observation. Therefore, if one assumes an instrument error of 2' to
3', six observations are plenty sufficient to get the error below 30".

In passing I remark that the goal for accuracy that Mayer sets himself is truly
down to earth, much less ambitious than that of some of our list members! But
my question is: Is his arithmetic right?

It is right, if we assume that the observer is perfect and the only purpose of
the averaging process is the elimination of instrument error and reading error
(due to limited resolution of the division of the limb). This is what Mayer had
in mind.

When we average with a sextant, we are in a different situation. We can never
get around the instrument error or the reading error with repeated readings.
Our assumption is that the observer is not perfect and each observation is
fraught with random error. This error is inverse proportional to the square
root of the number of observations.  Given the above parameters, six
observations would not be sufficient.

If an imperfect observer producing random errors had a perfect sextant and a
perfect repeating circle, the result of a repeated observation would be the
same on both instruments (statistically). The error would obey the inverse
square root law on both instruments. Mayer did not address this kind of error.
He did not have the appropriate error theory. Might he have thought that he had
divided random error by the number of observations, too? I don't know the

Herbert Prinz

```
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