NavList:

These "Impossible triangles" for lunars are not a new topic for NavList. Way back in November of 2004, I wrote as follows, in an exchange with Alex Eremenko:

"This analysis raises another entertaining issue regarding the input data for clearing a lunar. We discussed the case of a lunar where the altitudes are each 45 degrees on opposite sides of the zenith and the measured distance is exactly 90 degrees. The difference in azimuth is 180 degrees, and the cleared distance is 89d 21.3'. If I shift the Moon's altitude to 40, the difference in azimuth is smaller: 147 degrees, but the cleared distance is still 89d 21.3'. The error in altitude has no effect. But what happens if I shift the Moon's altitude to 50 degrees? This is an interesting case because the observation is now *inconsistent*. There is no way to have a measured distance of 90 degrees when the Moon is at 50 degrees and the other body is at 45 degrees. But suppose that's what you've recorded. What happens? It is interesting that if you clear the distance, you will *still* get the number you're looking for: 89d 21.3'. But in this case, if you were to attempt to extract an actual value for the difference in azimuth, you would find a meaningless number (the intermediate step in the calculation gives a value for cosZ of -1.19). I find it rather entertaining that the clearing process is robust in this way and can handle inconsistent inputs. "

[if you want to read more, look for the thread "Frank's formulas" in November 2004. Starts here: http://www.fer3.com/arc/m2.aspx?i=019506&y=200411]

And back then, when George Huxtable first learned that inconsistent triangles could still yield legitimate, valid clearing solutions, he wrote:

"What has surprised me (and intrigued Frank) is that the above expression continues to give a value for D in circumstances that are QUITE IMPOSSIBLE, in that the lunar distance is such that there's no value of azimuth (between 0 and 180) between Sun and Moon that can accomodate such a lunar distance. In those circumstances, although any attempt to deduce that azimuth would fail, the expression for D still seems to work, and gives some sort of result.

When the numbers input to the equation correspond to azimuths in range 0 to 180, then the result D has a simple physical meaning, the true lunar distance D. In other situations, is there any physical meaning we can attach to D?

I find it interesting that although Frank and I are intrigued about this matter, our resident mathematician, Alex, takes it in his stride, as only to be expected. I have a lot to learn, it seems."

And I then commented:
"I think it's fine to say that D is still the cleared lunar distance. But the intermediary Z, which should be a difference in azimuth, has left the real number line in a case like this, and there's no way to "draw it" in a simple diagram or interpret it as an azimuth. It's just a meaningless intermediate quantity. Here's something to ponder: imagine what the reputation of lunars would have been if navigators 200 years ago had learned that the difference in azimuth might occasionally be partly an imaginary number but, not to worry, this is normal, and it's a good thing! Maybe that's why this issue is not discussed in the usual sources. They were afraid it might drive mariners mad... :-)"

Just a bit of nostalgia...

-FER

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