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Peter F, you wrote:
"Here's a thought: if the time of apparent sunrise/sunset was observed
regularly; the extent of the difference or inaccuracy shown by observation
compared to calculated data could be evaluated on a regular basis and
contrasted with other information about position."

I see where you're going, but as George has already described, the geometric
situations only appear to be similar. And the small differences in
circumstances are actually large differences. Refraction above three degrees
altitude, which is about as low as you can ever observe the stars, is almost
completely unaffected by the low level structure of the atmosphere. But
refraction right at the horizon can easily be shifted by several minutes of
arc in either direction by common changes in the low level temperature
profile. The refraction that affects the horizon itself is only dependent on
the extreme lowest layer of the atmosphere.

I can think of one case where you would have relatively similar refraction
circumstances at nearly the same time, and your idea might work. Imagine
watching for moonrise right around the time of Full Moon. First we watch the
Sun setting... We record the exact time and reduce that to yield a
longitude. Then, let's say fifteen minutes later, we observe the Moon just
resting on the horizon on the opposite side of the sky (a difficult
observation). We reduce that to get a longitude, too. The refraction
circumstances should be similar so could we use one "sight" to remove the
error in the other? It seems like that should work. The average of the two
longitudes should cancel out the highly variable refraction right at the
horizon. Or do I have the signs backwards??

 -FER

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