NavList:

Bob Crawley, you wrote:
"Maybe I'm on a wild goose chase but it seems an interesting problem."

It is an interesting problem, but, yes, there may be an element of "goose chasing" in this. I don't think you should expect a so-called "closed form" solution to the problem. If you have found an iterative technique for solving it, then that's fine so long as it converges in a reasonable number of steps. Longitude by stellar occultation is very similar to finding longitude by a solar eclipse, and both are far more complicated than longitude by lunar eclipse (but in trade, they are much more accurate). Because the Moon is close to the Earth (or in navigator's terms, because it has a large parallax in altitude), the set of points on the Earth's surface that matches a particular observation is a rather complicated shape. One useful way to think about this is to imagine how the points look as seen from the Moon looking back at the Earth. Picture an extremely bright star, maybe a nearby star erupts in an unexpected nova so bright, maybe apparent magnitude -10, that it casts obvious shadows at night on Earth. By luck this nova is near the ecliptic and the Moon happens to pass in front of it. The Moon itself will now cast a shadow on the Earth. When drawn on a flat map, this shadow is a rather complicated figure, but if you draw it as seen from the Moon, it's a near-perfect circle: the shadow of the Moon itself. The same is true of any stellar occultation. And incidentally, this is a special case of a lunar distance where the distance is zero.

You also wrote:
"By the way it's amazing what the 19th Century seafarer was expected to be able to understand and follow."

You mean because it's in Bowditch, and Bowditch said he would not include anything that he could not explain to an average mariner... therefore an average mariner was expected to work problems in stellar occultations! 

Not a chance. First, let's distinguish marketing from reality. Bowditch and Edmund Blunt were trying to sell books. Proclaiming that there's nothing in there that we can't teach to the lowest of the low was a great way to sell the New American Practical Navigator, and it was also a great way to sell Bowditch himself. It make's him a sailor's mathematician. This was a legend that spread widely by word-of-mouth, and it made Bowditch famous among American mariners. He was "one of us". In reality, a practical navigator would likely never read more than five percent of Bowditch's Navigator and large sections were incomprehensible. Bowditch himself admits in a few notes somewhere (in the introduction?) that some sections were included because they might be of use primarily to astronomers who were also a market for his book. 

This issue, by the way, is an example of a common problem in the history of navigation and other scientific and technical fields. We have a natural tendency to look to the books as evidence of the history of navigation since, among other reasons, they are easier to locate. But the books teach us about the history of mathematical theory and the history of navigation education. Practice is different. To understand the history of navigation, as practiced, we have to turn to primary source evidence: logbooks, notebooks, scrap paper, marginal notes in old copies of Bowditch, even navigation problems worked with a sharpened point on the side of a sextant case (these do exist). In short, we ask: what did navigators actually do? 

Frank Reed
ReedNavigation.com
Conanicut Island USA