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So Robin Stuart with a detailed fitting approach gets a longitude of 81°37.8ʹE, and Herman Dekker got 81°39.7', while I got 81°35' (by simple folding, which, by the way, also works without any parabolic assumption). Note that in this latitude this range of 5 minutes of longitude is 1.9 nautical miles in E/W distance. Of course this only confirms the mathematical consistency of the methods, and conceivably we could do even better, though there's no real reason to do so. In the real world, these sights would have some observational noise in them which would lead to a real error in longitude probably on the order of 5-10 miles assuming the latitude can be trusted to a mile and a half or so.

Incidentally, as one approaches the pole, a noon sun curve like this becomes flatter, and the ability to determine the axis of symmetry of the observations becomes more difficult. Longitude is less accurately determined. This is offset by the convergence of longitude lines. The absolute error on the ground in miles is less affected by the flattening of the noon curve. So this process works reasonably well in high latitudes as well.

Frank Reed