NavList:

Jacques, you wrote:
"I believe to have find the principle of classic method ... this morning :
Corner formula
corr = dA * cos S = dA * ((sin A - sin A' cos D)/(Cos A' sin D)"

Yes, that's the idea.

That's the standard "cosine formula" for calculating the "corner cosines". You might enjoy two of my posts from nine years ago (wow! it's been NINE YEARS) on this:
http://fer3.com/arc/m2.aspx/Easy-Lunars-w15512
http://fer3.com/arc/m2.aspx/Easy-Lunars-TYPO-w15660
Please note that the second post is important because of a "typo" in the first one.

You also wrote:
"I think the advantage of "linear tables" is an economy of time, perhaps with a light loss of accuracy."

Of course, this is always the claimed advantage, but if you try them out, it's not really much difference. Throughout the whole period from c.1770 right through the early twentieth century, new lunar methods were trotted out (and after about 1840, they were only "new" to each amateur mathematician who was proposing a "new" method). And with every new method announcement, the same claim was made, something like this: "my new tables are by far the fastest and most convenient and will finally remove the burden of the mathematical calculation, making lunars again popular with common seamen". That's advertising. But in reality, most of the improvements were just cosmetic re-arrangements of the work --accounting adjustments. There were also cultural trends to this. In one era, Lyons' method was seen as having an "embarrassment of cases", but thirty years later, navigators didn't mind dealing with cases quite as much. The original method of Bowditch (known as the "Menoza Rios" method in British navigation manuals and other European manuals which copied the British classification) replaced that "embarrassment of cases" with the haversine calculation, which, while slightly longer, has no cases so it's less subject to error. In addition, the haversine calculation was widely used in time sight calculations so navigators were familiar with it.

It's critically important from a historical point of view to remember that these tables and methods were all commercial products in that period. They weren't produced by government hydrographic offices or departments in the various navies. They were created by independent contractors, men and at least one woman, who sold created, compiled them, and had them printed at their own expense for sale to the maritime public. They made real money from these works, and that's the origin of some of the modifications in the methods. Norie's "linear tables" (=graphic diagrams) were nearly impossible to copy back then. It made them bullet-proof in the market. Unfortunately, the actual advantage they offered in calculations was trivial, so it appears that they didn't sell as well as Norie had hoped. I would bet he did not make back his investment on them. A decade later, Thomson's tables, which I mentioned in the previous post, employed some peculiar obfuscations in the tables which did not make the calculations easier --or for that matter, more difficult-- but they created much more work for anyone who attempted to reproduce them. Mathematics alone can't explain them. The economics of the marketplace explains them quite well. And when the prolific well-moneyed amateur mathematician, Baron de Zach, became confused by Thomson's tables, he made the outrageous suggestion that Thomson must have worked out thousands of cases exactly by a long method and then tabulated the differences. Luckily for Thomson, though it surely wasn't true, this speculation stuck like glue and persuaded others who followed that the work to reproduce the tables would have been too enormous a task and also persuaded prospective buyers that the tables had some amazing "magic" in them.

There were some minor tables published by Ward in South Carolina around 1820. They had no impact at all, but they have an interesting place in the historiography of lunars. Secondary sources written in the mid-19th century often list Ward's tables among the most important early tables. This is quite bizarre. I've looked over the only remaining extant copy of Ward's tables, and they're fairly primitive and certainly not historically significant, similar to many other lightweight tables published around 1815. More importantly, I have found no references to their use at all in hundreds of period logbooks that I have studied. So how did these minor tables make it into the later books? It turns out that the Baron de Zach is again to blame. In his little self-published journal (more like a modern "blog"), he wrote around 1820 about the shocking outrage that he had learned about in a letter from Mr. Ward. He noted that people were stealing and copying Ward's tables without paying royalties (again a reminder that these were commercial ventures). But the Baron was minor nobility with great wealth backing him up and he simply did not understand the market economics here. So the Baron happily re-published Ward's tables, complete, in his own journal! Ward had written to him in the hopes that Baron de Zach might be able to stop the rampant copying (or so Ward claimed), but the Baron instead put the tables out there for anyone to copy at will.

You might enjoy puzzling over some tables written up in the "Revue Maritime" c.1880 which I call the "slaver tables" (my translation from "tables du négrier") which had been recovered in manuscript hand-written form from the captain of a slaver vessel captured off the coast of Africa thirty years earlier and were apparently considered something of a curiosity in later years since lunars were already long vanished on the majority of European vessels. There seems to have been speculation that they were the work of some local African genius, outside the mainstream of western maritime science. But as it turns out, the "slaver tables" are closely related to the cases above. They were probably hand-copied from tables printed around 1815. The author of the paper in the Revue Maritime has no background in the material (he had no internet, of course) and so he is forced to re-derive the equations, basic though they would have been 75 years earlier, all from scratch. If you do an internet search on "tables du négrier", you should have no problem locating it. There's one link in the PS below.

You also noted:
" Mendoza had proposed another method "rigourous", with is a simplificaton of Krafft method."

Yes. By the way, I don't approve of the terminology "rigorous" here. It was employed historically, but it was a contemporary misconception founded in early ideas about calculus. Any modern student of calculations would recognize that a series expansion (which they considered "non-rigorous" or "approximative") can have just as much mathematical "rigor" in it, and equivalent calculational accuracy, under proper circumstances. And it should be said that Mendoza Rios (or better "Mendoza y Rios") would probably never have wanted his name attached to the series method that was labeled the "method of Mendoza Rios" after his death in the British navigation manuals. It's true that Mendoza Rios demonstrated this "method" first in an appendix to his long "Recherches" paper published by the Royal Society in 1796, but Nathaniel Bowditch was the one who published his nearly identical method for practical use and recommended it to navigators starting in 1799. The timing is suspicious (and was described as suspicious even in 1838 at the time of Bowditch's death), but there's no evidence he actually "stole" it. It's possible though that he "got wind of" the solution described by Mendoza Rios (he visited both Portugal and the island of Réunion around this time) and independently derived it based on a description of the technique. Meanwhile, Joseph Mendoza Rios (José de Mendoza y Rios) was hard at work on his tables (similar to Krafft's, as you note), and those were eventually published in one large volume. Even though they were subsidized, they were still relatively expensive. And that's another issue on these lunars tables as commercial products: the number of pages matters enormously --cost was more or less directly proportional to the amount of paper.

-FER

PS:
--Slaver tables in the "Revue Maritime": http://books.google.com/books?id=zJAP0RmwGHQC&lpg=PA253&ots=kd7YAJmQM4&dq=%22tables%20du%20n%C3%A9grier%22&pg=PA249#v=onepage&q=%22tables%20du%20n%C3%A9grier%22&f=false
--Mendoza Rios "Recherches Sur Les Principaux Problemes de l'Astronomie Nautique":
http://archive.org/stream/philtrans05469110/05469110#page/n0/mode/2up

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