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You wrote:
"Supposing one was in a small boat with an accurate timepiece and the
necessary tables, how accurate could you determine your longitude by
observing the rising or setting of the sun or any other celestial body?"

This would work with the Sun only. The time of sunrise can be trusted to
about one minute (worse at higher latitudes). So that would give you your
longitude to the nearest quarter of a degree. This could certainly be
useful. At minimum, it's a nice quick sanity check. Say you're sailing west
across the Pacific in the tropics. Overnight you make sixty nautical miles
according to a taffrail log (or as estimated from average speed and he
duration of the night). The Sun should then rise about four minutes later
than expected. If instead it rises six minutes late, you may have a nice
current helping you along... More accurate observations later in the day
will tell.

You could not easily use the Moon for these observations since the Moon is
only rarely clearly visible right at rise or set and even then you have to
correct for longitude. It's an iterative process. You can't use the stars or
even a bright planet like Venus because they're simply not visible until
they're a degree or more above the horizon. Atmospheric extinction at the
horizon is 12 magnitudes or more.

And you asked:
"Also, assuming you had a compass and were north or south of the tropics,
would it be possible to estimate your latitude by taking a bearing of the
setting or rising of a celestial body?"

This is less accurate simply because it's tough to get a good bearing. How
about timing the length of the day (after correcting for the longitude
change)?

A few years back, I mentioned that there are fish tags that work on exactly
these principles. These are tags attached to tuna that record underwater
brightness every few minutes along with GMT. Peak brightness gives local
apparent noon and thus longitude (this is determined by the symmetry of the
light curve around noon, not the actual "peak" brightness which would be
hard to pin down). Duration of brightness as a surrogate for the length of
day gives latitude, though not as accurately. Why do something so
"primitive"? In short, because GPS doesn't work underwater.

 -FER



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