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Frank Reed wrote:

> Slocum probably was using tables keyed to that era's Bowditch. The
> prefered method then was Chauvenet's which was relatively new at that
> time. Slocum probably learned a different method when he was a
> commercial ship captain. The end of the century method may have
> puzzled him, and I would bet that details of Chauvenet's method would
> surprise a lot of the mathematically-inclined lunarians who follow
> this list.

O.K. I bite. What surprises me is Chauvenet's rigorous concept of
'approximative' versus the approximate concept of 'rigorous' found in
other authors. But before I go into the math, I would like to comment on
the age of the method.

When it was introduced into Bowditch, more than a decade after Chauvenet
had died, the method was already a Methusalem. From then, it took only
half as long for the method to be relegated to the appendix and finally
to be dropped again, as it took from the first publication in 1854
(+/-1) to its adoption in "Bowditch".

The irony is that Slocum was too _young_ to benefit immediately from
Chauvenet's first publication! However, if Slocum had ever seen any of
the following at a later stage, he knew the method without having to
wait for "Bowditch":

Astronomical Journal, Vol II, (1854?),

American Ephemeris and Nautical Almanac for 1855 and 1856,

"New method of correcting lunar distances, and improved method of
finding the error and rate of a chronometer by equal altitudes",
Washington, Bureau of Navigation, 1864.

"A manual of spherical and astronomy, embracing the general problems of
spherical astronomy, and the theory and use of fixed and portable
astronomical instruments. With an appendix on the method of least
squares", Philadelphia, J.B. Lippincott & co.; London, Tr?bner & co.,
1864.

Slocum would have been the right age to see the pamphlet by the Bureau
of Navigation from 1864 during his training. It is an interesting
question though, on exactly which manuals the traing in the merchant
marine was based, how much effort officers made to stay au currant, and
what pubications they used to this end. (Hopefully not "Bowditch", which
was always a quarter of a century behind of the newest development.)

***

Now for the math. The method is a good counter-example to proof an often
stated opinion wrong that methods for clearing the distance can be
divided into rigorous and approximative methods according to whether the
MZS triangle is solved from the mZs triangle via the common angle Z
(using the cosine formula twice), or whether MS is derived from ms by
applying corrections on either end that are estimated by means of small
plane rectangular triangles with hypotenuses Mm and Ss.

It is not difficult to trace this wrong opinion back to Cotter, who has
become an authority on questions of  the history of celestial
navigation, mainly for the lack of other equally comprehensive
treatments. To see that this classification is flawed, one only need to
consider Dunthorne's method. The latter is an approximative method and
correctly categorized as such by Cotter. Nevertheless it is based on the
rigorous cosine formula and therefore does not fit Cotter's own
criterion. What makes the method approximative is the deliberate
neglection of certain corrections, and omission of terms in simplified
tables.

Chauvenet starts out with a similar approach. He starts with a rigorous
treatment, even including the elliptical shape of the Earth (!!), as
well as effects of temperature and pressure on refraction. To my
knowledge, he is the only one to consider for a sea-method the
oblateness of the earth, the effect of which can change the final result
for the distance by typically 6" or 0.1' of arc in mid-latitudes.

After some substantial kneading, Chauvenet ends up with a seemingly
awkward representation of the rigorous formula. Only then he
investigates the significance of individual terms and discards all those
that can be shown to contribute less than a second of arc or often less.
What remains thereafter are four terms, A,B,C,D, to be used in two
corrections to be applied to the apparent distance to yield the true
geocentric one. The logarithms of these terms fit into a compact set of
tables.

The beauty of all this is that although the method is an approximative
one from the standpoint of the strict astronomer, it can be rigorously
shown that the error is below a certain limit that is smaller than the
accuracy of measurement usually obtainable at sea. This approximative
method is in fact capable of yielding results that are more accurate
than those obtained by some procedures based on "rigorous" methods.

****

Now I have a practical question (if anything regarding lunars can be
considered practical). Chauvenet's method relies also on his own table
of "reduced refraction". This in turn is based on Bessel's refraction
table. But Bessel's table is no longer used in its original form. The
N.A. incorporates slightly different tables, and the formula given there
(and in the Supplement)  is again different. Choosing the right
refraction table is more of an art than a science (Jan, are you
listening?); at any rate, it is an empirical process. My question to
Frank therefore, is:  Have you looked into this aspect? Do you actually
use the method for practical exercises? It would seem to be a
contradiction to go all the way correcting for atmospheric conditions
and at the same time use a table that is officially considered to be
outdated.

Coming back to the issue of whether Slocum could have fixed his tables.
If he indeed used "Bowditch", I would not think so. Having been produced
by the abstract process I described above, the final tables for log A,
B, C, D are so arcane that even Chauvenet himself could probably not
have reverse engineered them, had he forgotten his original procedure of
devising them.

Herbert Prinz

   

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