Alright...back home from work and I've gotten some more rest. Now, for the noon sight:
The noon sight is rarely, if ever used to find longitude. The reason is that, in order to accurately find longitude at noon using a sextant (and chronometer), one needs to determine the
exact time at which the Sun transits the local meridian. When the Sun is on the local meridian, it is at its highest altitude for the observer's location. A sextant measures the altitude of heavenly bodies...so that should be easy, right? Not so much. Right around noon, the Sun seems to "hang" at the same altitude for up to several minutes.
Take a look at
this graph. (You should be able to zoom it if it is not clear.) The
blue line shows the altitude of the Sun every fifteen minutes on July 17th, 2017 at 45°N, 0°E. The
red line (which uses the right-hand vertical axis) shows the rate of change in altitude in arc minutes per minute. Notice what happens at local noon: the rate of change drops to nearly zero. That makes it very difficult to determine exactly when the Sun has reached culmination.
There is a "trick" which can be used to try and find the time of local noon: double altitudes. The navigator measures the altitude of the Sun some number of minutes before noon and notes the time. The sextant is left at whatever reading was taken at that time. Then, the navigator waits until after noon, when the Sun drops to the exact same altitude again, and notes the time. splitting the difference between these times
should give the time of local noon...if you're stationary...and not on a pitching and rolling ship...and make
perfect observations. But, even then, it's tough to get an accurate time.
For the sake of explanation, let's say you did get an accurate time. How does that tell you your longitude? Well, your chronometer would be set to GMT (Greenwich Mean Time or UT [Universal Time], essentially the same thing). Let's say you found that local noon occurred at 14:32:17 GMT on July 17th, 2017. Now, you know that noon in Greenwich occurred at 12:00:00 GMT...or did it? Not necessarily. You have to consider something called the "equation of time". Because the Earth speeds up and slows down in its yearly orbit around the Sun, the Sun appears to race ahead or lag behind "mean" time (the time that a regular clock keeps). On this date, the equation of time is about -6m13s. IOW, the Sun is six minutes and thirteen seconds behind where the mean Sun would be, and noon would occur later. (This information can be found in the Nautical Almanac [see the bottom of page 2 in
this example].)
So, Greenwich noon occurred at 12:06:13 GMT. If we subtract that from the time of our local noon, we get 14:32:17-12:06:13 = 2:26:04. Local noon occurred two hours, twenty-six minutes and four seconds
after Greenwich noon. Since we know that the Earth rotates 360° every 24 hours, we know that the Sun appears to move 15° through the sky each hour. So, if we multiply 2:26:04 by 15, we get: 2:26:04∙15 = 36°31'00"...this is our longitude. And since local noon occurred
after Greenwich noon, we know our longitude is west: 36°31'00"W.
However, as we have learned, the noon sight is not typically used to find longitude. What it
is used for is to find
latitude. So, how is this done? Well, in this case, it's kind of handy that the sun "hangs" at the same altitude for several minutes. Because that's what we're after - the maximum altitude. Let's say that, after correcting for index error, height of eye, refraction and semi-diameter, we get a maximum altitude of 66°05.5' on July 17th, 2017 at 14:32:17 UT. We subtract the altitude from 90° to get the "zenith distance" - the distance the Sun is from being directly overhead. (This is equal to the distance we are from the Sun's GP.) 90°-66°05.5' = 23°54.5'. Now, we look in our almanac and find, after interpolation, that the Sun has a declination of N21°05.5'. Since the Sun's declination is in the same hemisphere as our DR, we add the declination to the zenith distance to find our latitude: 23°54.5'+21°05.5' = N45°00.0'.
Finding longitude was usually done by "time sight". But that is another subject altogether.