Let's do one for sunset. Atmospheric refraction is about 34' at the horizon, raising the position of the sun by almost exactly one diameter. In addition, when the limb of the sun touches the horizon, the position of the sun (measured at the midpoint of the disc) is 15' above the horizon. (Yeah I know the radius of the sun changes a bit with the time of year and isn't exactly 15' and........go away, my head hurts already!
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So the moment the sun's lower limb touches the horizon, the true position of the sun must be corrected 15' higher (correcting for the diameter) plus another 34' lower (because refraction raises the apparent position). So your true altitude is 0º -34' + 15' or -11 minutes.
Now let's have fun! There are two kinds of minutes and two kinds of seconds, those of arc and those of time. And we're going to translate minutes of arc to seconds of time. Why? We know that at the exact instant of sunset our local time is 6:00 pm.
Because we are measuring something we can more easily detect, first contact of the sun with the horizon, instead of something quite difficult to observe, the time the center of the sun is coincident with the exact position of the true position of the horizon in the sky (uhhhhh......that's 34' above the horizon--no point of reference there) we have to determine our exact local time when we see that limb touch the water.
Basic math: the sun moves 15º through the sky in an hour, 360º divided by 24 hours. That means it moves one degree in 4 minutes of time. (60 minutes divided by 15 degrees gives you 4 minutes per degree).
Now it gets messy. That means 4 minutes time for the sun moving 60 minutes of arc. Let's change that to seconds of time: 240 seconds of time for 60 minutes of arc. We can further reduce that to 4 seconds of time per one minute of arc. Do you follow okay?
So we can apply our correction. The time we measured was not 6:00 pm, but the time when the sun was 11' of arc lower, later in time. We know that 1 minute of arc takes the sun 4 seconds to move, so 11 is 44 seconds later. Our local time was 18:00:44. Hope you recorded exact GMT at the instant of observation because you need it!
I would subtract GMT from local time to calculate the time difference. Express that in decimal hours instead of hours:minutes:seconds and multiply by 15º. That is your number of degrees east of the prime meridian. Convert to deg:min:sec. Translate for E and W longitude by subtracting 360 if the number is greater than 180º (it then becomes west longitude, expressed as a negative number) and there's your longitude!
I'm doing this off the top of my head with no reference materials so if I've made a mistake somebody whack me and get the right info out there!
