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From: Frank Reed (no email)
Date: Thu Feb 03 2005 - 19:12:07 EST
Omar Reis wrote:
"Regarding Bowditch (the book) it would be
interesting to have the parts about lunars
( which are no longer present in the current
Bowditch edition ) published on the Internet."
Here's a short summary of the lunars methods in Bowditch and how you can get
variants of them on the Internet.
All of the lunars clearing methods published in Bowditch from 1799 (before
Bowditch's name was on it) until 1880 (43 years after Nathaniel Bowditch's
death) were series methods. They are all different ways of calculating
d' = d + dh1*A + dh2*B + Q
where d is the observed lunar distance, d' is the cleared distance, dh1 is
the Moon's altitude correction, dh2 is the Sun or star's altitude correction
(exactly the same as the altitude corrections you would find in a modern
Nautical Almanac), A and B are the "corner cosines" which tell us what fraction of
the altitude correction acts along the lunar distance arc, and Q is the small
quadratic correction which depends on the second power of the altitude
corrections. In a series expansion of any function, there are linear terms,
quadratic terms, cubic terms, and so on. At any practical level for the case in
question, it is never necessary to go beyond the quadratic level. For a simple
modern calculation based on this series approach, take a look at "Easy Lunars"
on my web site. On to the Bowditch versions...
Bowditch's Original Method:
This was a series method which had the advantage of "no distinction of
cases". In other words, you didn't have to stop and ask yourself "do I add or
subtract this logarithm?" It was identical to a method previously published by
Josef de Mendoza Rios (various other forms, most commonly in Spain: Jose de
Mendoza y Rios). Bowditch stated that he invented this method in 1795 though it
was not pubilshed by Bowditch until 1799/1800. Mendoza Rios had invented it
some time earlier and published in 1795 in Spain and again in 1797 in England.
Whether Bowditch was influenced in some way by Mendoz Rios's earlier
publication, I have not been able to determine.
Bowditch's Principal Method:
A fine-tuning of the Original Method. Bowditch came up with a number of
clever little tricks that reduce the number of steps in the original method by
about 20% (similar but more extensive fine-tuning than in Arnold's Method which
Henry Halboth detailed for the list a few months ago). Also, by other clever
tricks, some of which are counter-intuitive and perhaps confusing, he
arranged his tables such that *every* step was worked by addition only. This
principal method was relatively popular and was listed as the "First Method" in
Bowditch from 1807 to 1880. This refined method was separately published as a
soft-cover supplement, refered to as an appendix to the Navigator, in 1804.
Thomson's Method:
From 1837 until 1880 the Second Method in Bowditch was copied from Thomson's
"Lunar and Horary Tables". Actually only the small correction (Third
correction) was copied (and it would not be unfair to say that it was stolen --there
is no evidence that it was recalculated). There was no particular reason to
copy the other parts. Thomson's method is usually considered a descendant of
Israel Lyons' method but it is a series method and all of them are close kin.
This method was very popular.
Witchell's Method:
Slightly streamlined in Bowditch, Witchell's method had been published in
the "Tables Requisite" and other places since at least 1781. It was an old,
reliable method and like the others, a series method. Its presence in the
earliest editions of the Navigator was a carry-over from Moore's Navigator (from
which Bowditch's Navigator was 95% derived). At first, listed as the second
method in Bowditch, and then the third, and finally by 1837 the fourth, it was
always the last method and was included in Bowditch until 1880.
Chauvenet's Method:
Probably only rarely used, this method came very late in the history of
lunars and, were it not for Chauvenet's personal fascination with the
mathermatics of nautical astronomy and his powerful influence in the curriculum of the
US Naval Academy, it probably never would have been included in Bowditch. The
US Navy H.O., Hydrographic Office, had been persuaded to purchase the rights
to Bowditch's Navigator, and the Navy published it as a government document
starting in 1868. It was left largely unchanged (complete with worked lunars
examples from 1836) through the edition of 1880. All of the former methods were
dropped starting in 1881 (Bowditch is a radically different book from here
on) and the single, in fact "singular", method of Chauvenet published instead.
Too little, too late, it was shifted to an appendix in 1903 (?) and dropped
altogether in 1914.
So where can you go to get copies of these methods on the Internet?
As for Bowditch's Original Method, it's the method of Mendoza Rios and you
can find that in Norie's Epitome of Navigation which is available online at
_www.mysticseaport.org_ (http://www.mysticseaport.org) (at the G.W. Blunt White
Library in the digital collection). I've also posted a page from the 1800
Bowditch on my web site (scroll down). Bowditch's Principal Method is nearly
the same so unless you're interested in the fine-tuning I mentioned (details on
request -- I've worked it all out), you're fine with the method of Mendoza
Rios again. Thomson's method requires more extensive specialized tables than
the other early methods, but it's just Lyons' method which is basically the
same thing as the logarithm version of the "Easy Lunars" method that I've got
posted on my web site. If anyone would like to calculate their own version of
Thomson's Tables, I can tell you how to do that, too. Witchell's Method
(Bowditch's last method) is included in Norie and also available from the Mystic
Seaport web site above.
As for Chauvenet's Method, there used to be a copy on the net, but I think
it's gone. I would emphasize that it was an eccentric method that made it into
Bowditch's Navigator decades after lunars had faded into relative obscurity.
In addition to these methods, the online copy of Norie's Epitome of
Navigation includes instructions for Dunthorne's and de Borda's methods which are
direct triangle solutions. If you want to read instructions for old lunars
methods, that's the place to start.
Oh, and by the way, there have ALWAYS been errors in books like Bowditch's
Navigator. For decades, there was a minor typo in the account of Bowditch's
Original Method telling the navigator to subtract 18' instead of 18" (reminds me
of a typo for a model of Stonehenge in a movie from a few years back). That
one little error would render the method completely unusable. I doubt that it
ever came up because the "Original" method was mostly included for nostalgia
in later years.
-FER
42.0N 87.7W, or 41.4N 72.1W.
www.HistoricalAtlas.com/lunars
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