NavList:

Antoine, you wrote:
"However I fail to see in your last post a "quick and easy recipe" to be used to ascertain / recognize valid cases for using your "fast lunar method" without endangering the final results."

Simple enough (I think). Construct the following tables: Let the Moon's altitude range over 15 to 85 degrees at 10 degree steps for a total of eight small tables. Rows in each small table should range over body altitude with the same values, 15 to 85 (8 rows), and columns should range over lunar distances from 15 to 125 in 10 degree steps (12 columns). Tabulate the ACTUAL ERROR for each case in tenths of a minute of arc that would arise from using the "quick and easy" method compared to an exact method. If the actual error is greater than 1.0', I would suggest putting a "-" or some other symbol so that we know instantly that the error is simply too large for any practical case. The navigator could decide whether to work cases where the error is greater than 0.1' but less than 1.0'. There would be no doubt about using this method when the error is less than 0.1'. Such tables would easily fit on one small page, double-sided. Historically, graphs were expensive, but I hope you can see that this could also be done very nicely with graphs.

If you would like to try an actual case, may I suggest you look at a lunar observation from an actual logbook. Here's an analysis of one from 1825 that I have written about previously on NavList:
http://www.historicalatlas.com/lunars/1825/
I've broken that lunar calculation down into various lettered sections. Most of parts D, E, and F would not be required if that navigator had known about this "quick and easy" method. Section E, in particular, involves a lot of work. NOTE WELL: although some people get the idea that I'm talking about some dim-witted or ignorant navigator who doesn't know anything about logarithms, that's not the case at all. You STILL have to use the usual trigonometry to work up the time sight (see section K).

You wrote:
"The bottom line is here : we need to know how practicable your magnificent discovery is, therefore please be so kind as to show us how it can be a readily usable one. If I missed any earlier demonstration of yours to establish that you have published a good simple rule of thumb to verify that one can use you shortcut or should not use it, then I do apologize for challenging you again on this specific point."

I would question how much we "need to know" on this one. It is, after all, as I have said every time I've brought it up, a discovery that would only be useful in a practical sense if I could somehow take it back in time (lunars are not practical navigation). Nonetheless, I hope the above explanation satisfies your curiosity about how it "might" have been done.

Regarding the "impossible" lunar case, you wrote:
"And finally, if somebody disagrees with giving "unrealistic examples" such as this (new battle triggering) "impossible lunar", I would think that certainly such view point is quite valid. I hardly see any reason why Lunar examples should not reflect realistic world examples, ... except pure mathematical number crunching ... and if such were the name of the game, why would we not give examples of Cel fixes with Heights of Eye NEGATIVE !!! Some formulae for refraction still yield "reasonable" results for negative heights of eye, but what's the use here ???"

We all have, at one time or another, invented example cases for the sake of teaching. Naturally, they should resemble real-world cases as nearly as possible, so it would be pointless to invent a negative height of eye, as you suggested, but I might make up an example of some sights and include a height of eye greater than 100 feet even though I haven't actually taken the sights in the example I'm creating from that height. This makes for good pedagogy. We make up hypothetical examples that are "plausible" to the attentive student. It's interesting to note that the case with the 13 degree non-closure of the triangle in the Tables Requisite of 1781 only attracted the attention of one bitter letter-writer back in the 18th century, and that was nearly five years later. And what was his goal?? Not to call attention to something that might lead to an error in navigation... No, not at all, because he knew it would NOT reult in an error in navigation. His point was to mock the astronomers, especially Maskelyne, for failing to understand basic geometry --that's how he spun it. It's a classic case of obsessive nitpicking or "cavils" as Makelyne said. On the other hand, such a large difference was bound to confuse SOME student at SOME time using the Tables Requisite, so it deserved to be counted as an error, of a certain sort. It's an error in pedagogy. The OTHER example which Nauticus also mocked, however, is quite different. With the Moon 88 degrees high, the Sun 5 degrees high, and the measured lunar at 90 degrees, we have another case where the triangle doesn't close but in this case the difference is not as large and it is COMPLETELY ACCEPTABLE as a real observation. This obsessive fascination with triangles that must close is woefully misguided. I STRONGLY suggest, Antoine and anybody else following along, that you try clearing this latter example and then try clearing it again with a different value for the Moon's altitude (lowered by a few degrees). You will discover that there is no error at all despite the seemingly "impossible" geometry. This is what I have termed a "miracle" in the required accuracy of the Moon's altitude. Thanks to an entirely fortuitous cancellation between the dependence of the Moon's parallax on altitude and the geometry of the spherical triangle, the lunar clearing process is quite insensitive to errors in the Moon's altitude for cases where the lunar distance is close to 90 degrees. It is NOT an error to work such cases just because the Moon's altitude is somewhat uncertain.

Nitpicking over celestial navigation examples can be good sport, and we all do it on occasion, but there are limits to what is "sporting". There is a recent celestial navigation "publication" with an example of a lunar observation. If you carefully dissect it, you will find that it all makes sense except that it refers to an Upper Limb altitude for the Moon at a time and place where the Moon was tilted over too far for such an observation given the other data. The only altitude possible would have been a Lower Limb for the Moon. I have not pointed this out to the author of the example or quibbled about it on NavList because it's just a little nit to pick. It does not materially affect the solution of the problem or the usefulness of the example from a teaching point of view. See what I mean? I'm not saying we shouldn't look for these little inconsistencies. If nothing else, it's good clean fun! But they're errors only in a rather restricted sense. And if we spend all our time focusing on such trivialities, then we are guilty of "cavilling" (I think I like that old word!), and we end up demeaning otherwise excellent work.

-FER

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