A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Sivan Toledo
Date: 2020 Jul 19, 01:42 -0700
Yes, this math is not trivial. It does not require a lot of math background, but it is not simple. Of the three authors, I became aware of this problem (from Mark Denny's book), got interested, but could not solve it. I got Imre and Bill Steiger interested and they found the proof.
The notion that this is trivial is to some extent true, but only under assumptions that are even stronger than the ones used in the paper. But these strong assumptions are often true in practice. For example, if the rays/LoP's intersect at angles that are not to close to 0 and 180 degrees, and if the errors are small, say limited to 1 degree, then the rays always intersect and the result holds. In these simple (and common) cases, the proof that is shown in the Open University video from the 1980s is also correct. I think that these consitions (nice angles of intersection and small angle errors) were what the authors of the Admiralty Navigation Manual form 1938 assumed and what Stansfield assumed.
But Stansfield and then Daniels took this result and stated it in a very general way, and in a way that looks mathematical and formal, and this was not correct. This is what our paper aims to correct.
Also note that the letter of J.E.D. Williams in the JoN in 1991 genralized this results in a way that is simply incorrect; this was corrected in 1993 by Cook, who was a statistician. So these things are tricky.
Finally, even in this digital age it is impossible to verify the claims of Stansfield and Daniels and Williams by simulation. It is posible to refute them, but not to verify them. This is how I got *really* interested in this. I did a simulation like the one you suggest. I had to pick a distribution of angles and errors. I found out numerically that Williams was wrong about the 1/6 and 1/12 probabilities. The 1/4 came out alright. This made me realize that this is tricky. The inability to prove this by simulation is due to the fact that you must choose a distribution of random angles, positions of the landmarks, and errors. So if you verify the 1/4 (which my simulation did), it is only for this distribution, but it still might be wrong for another distribution.
I hope that this clarifies some of the (mathematical) fog :-)
Regards, Sivan Toledo