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Hello Jan,

Thank you for your historically and mathematically illuminating analysis of 
L.D.s. Especially the section about approximate methods is rich in details 
that I have not seen before.

Here are just a few minor comments as they occurred to me on first reading:

Jan Kalivoda wrote:

>
> We can distinguish four classes of these methods, which are remotely similar 
to the classes of the methods for reducing sights by "Marcq St Hilaire 
(intercept) method", the only method for using celestial lines of position 
surviving in today's navigation. These are in the order of their increasing 
length, difficulty and logical clearness and beauty (in my eyes):
>
> - software solutions; quite common now and not unknown in the first half of 19th century!
>
> - inspection tables (compare HO 214, 218, 240, 229 and ancient Ball's tables, firstly edited in 1907)
>
> - "short" methods (compare Ageton's method in HO 211, Dreisenstock's method 
in HO 208, Smart, Ogura, Aquino etc.; in these methods short tables with 
auxiliary values are provided that are combined to obtain the end result; 
these tables were much less bulky and expensive than the inspection tables, 
but their use was more difficult and time-consuming)
>
> - rigorous solutions (compare cosine-haversine formula)

I do not quite see the analogy, because in one case you are talking about 
various methods of solving a given problem (that of clearing the distance), 
and in the other case you are talking about various implementations of a 
given method (the modern intercept method). This method, in turn, is just one 
of  many that solve a particular problem, namely the "position from combined 
altitude" problem.

Moreover, the modern intercept method, as well as the original Marcq St 
Hilaire method are both approximative methods. Therefore it is misleading to 
speak of their rigorous solution via the cosine-haversine formula. A method 
can be rigorous or approximative. But an implementation of a method is either 
correct or incorrect (and should always be the former!). Maybe, this is just 
irrelevant nit picking of the former programmer in me.

Jan Kalivoda wrote:


>  Another [approximate/short method] was the method of the French astronomer 
Chauvenet that replaced all other older methods in "American Practical 
Navigator" in the year 1888 [...]

William Chauvenet, 1820 - 1870, was an American professor of mathematics and 
astronomy. He was instrumental in founding the Naval School in Annapolis, 
which later became the U.S.N.A. Besides teaching, he also served there as the 
first librarian. He even got a U.S. battleship named after him. He wrote a 
standardwork on spherical and practical astronomy (see below) which is still 
a valuable reference and which I highly recommend. I use it all the time, 
although it goes without saying that his numerical methods are irrelevant 
today. It also includes chapters on nautical methods.

Chauvenet's method for clearing the distance had already been published in the 
American Ephemeris and Nautical Almanac for 1855 and 1856, and - according to 
his own statement in the cited book - also in the Astronomical Journal, Vol 
II, which must have been even prior to that  (1850?). Later, it was reprinted 
in  "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.

It was again described and fully explained in "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. I believe there exists a Dover reprint, not 
sure whether it is available.

Ian, are you sure of that given date, 1888?

It is quite amazing that the method, devised in 1855, was incorporated 
virtually unchanged into the "Bowditch" edition of 1888 for the first time, 
apparently as Ian explains, in order to keep up with the progress of time! 
Chauvenet had been dead for 18 years by then. This fact totally escaped me 
until now. Back in 2001,  I said in a message to this list:

"Appendix V in the edition of 1909 describes the [lunar distance] method, 
contains all necessary auxiliary tables and a worked example using the 
distances shown in an extract from the Nautical Almanac 1855. Without having 
checked, I would therefore assume that any edition since 1855 would not 
differ too much in the way the subject is treated [...]"

In early editions, data of worked examples were regularly updated to more 
recent dates for purely cosmetic reasons even when a method did not change. 
Seing the 1855 method and data,  I jumped to a wrong conclusion. In the mean 
time, of course, I came to realize that it was the general trend of this 
publication after the death of Bowditch to let new technology mature for a 
quarter of a century before taking it into consideration. (First brief 
mentioning of Sumner line in 1855, intercept method around WWI)


Jan Kalivoda wrote:

> Borda (1778):
>  cos A squared = cos M cos S sec m sec s cos[(m+s+d)/2] cos[(m+s-d)/2]
>  sin D/2 squared = sin[A + (M+S)/2] sin[A - (M+S)/2]
>
> I cannot understand, why this cumbersome method gained such popularity.

The riddle disappears once you know that the above isn't Borda's method. He, in fact, has

  sin A  = Sqrt{cos M cos S sec m sec s cos[(m+s+d)/2] cos[(m+s-d)/2]} / cos[(M+S)/2]
  sin (D/2) = cos[(M+S)/2] cos A

which requires one table opening, one addition, and one halving fewer than the 
other method. Fellow mathematicians and ordinary seamen alike may have 
appreciated the "economy of means". The former may have admired the artful 
substitution, while the latter will certainly not have minded the inherent 
obscurity. They never cared to understand their recipes, anyway.

But how popular was the method really? And popular with whom? I often see it 
quoted in learned works, textbooks and the like, but can it be found in ship 
logs?


>
> Maybe the method of Bruce Stark is the last method invented in this [the 
approximative] class, but I don't know anything about it.

As Bruce already said himself, it is rigorous. He forgot to mention that it is 
highly original. Most rigorous methods differ from each other only in the 
detail of how they eliminate cumbersome additive terms of trigonometric 
functions. (These prevent efficient use of logarithmic trigonometric tables 
and force the user to additionally use a set of natural trig. functions.) 
However, all the methods in the Borda-Delambre-Young-Krafft-... category use 
trigonometric substitutions for the purpose of this elimination. Bruce uses 
Gaussian logarithms. These have been favoured by astronomers, specially 
German ones (e.g. D�llen) in the late 19th century, exactly to avoid such 
ocscure substitutions as the one above by Borda. I have never before come 
across the idea of using them in the context of a lunar distance. A certain 
George W.D. Waller proposed this idea in 1946 for altitude calculation (see 
Bowditch 1984, p.593). Nothing came to fruit. The main drawback of Gaussian, 
or addition
logarithms is that they are yet another table to carry (and to produce in the 
first place!) So I believe Bruce's method is in a category all by itself. I 
am sure he could tell us more...

Best regards and thanks again

Herbert Prinz

   

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