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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?  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?


Do you really want to know how accurately or precisely? There is a
difference.

In reality your "position" thus gained will be not very accurate. But
your in-accurately calculated position could be precise.
What you are asking is more on the lines of what is called "lifeboat"
emergency navigation. Where certain observation tools can be fashioned
from other equipment (such as a protractor, a nut and string as a
makeshift sextant to take a sight) on board the life boat. There are
other emergency navigation procedures that will also give passable
results with not much in the way of readily available equipment at hand
and they are explained in many navigation books. May not be very
"accurate" but it may be precise enough to enable you to have a life's
chance at knowing where you are.


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