Basically when you are taking sights of Celestial objects with a Sextant you are measuring the "height" of the object above the horizon, measured in degrees. The distance for the base of the Trigonometric Formula is assumed to be the Observer's horizon, (based on their height of eye above the surface of the Earth / Sea, i.e. Sea Level). When you apply this measurement to a world chart, (not that you need to, but for illustrative purposes), you would get a circle plot that would show everywhere on the globe that the observed object would be at that height. Now if you were to observe several objects at roughtly the same time then you would end up with several global circles plotted, however there would be only one place on the globe where all the circles would intersect. That is really what is going on when you use Celestial Objects for Navigation. There are several factors that can upset the accuracy of your observations; indistinct horizon (i.e. Haze), inaccurate time measurements, (this is why Celestial Navigation at sea was not very accurate until the late 18th Century), and misidentification of the Celestial body observed, (there are roughly 52 stars used for navigation plus the Sun, Moon, and 5 inner planets, of course not all 52 stars and/or 5 planets are available at any one time).
In the morning and evening normally 7 stars and/or planets would be observed, cloud cover permitting. This would be done during the time known as Nautical Twilight, (roughly 30 minutes before Sunrise and 30 minutes after Sunset). When observing morning and evening stars as it is called, you must advance or retard the resulting LOPs to a common time, (as you can observe only one object at a time). The usual method is determining a mean time between the first observation and the last and applying the ship's coure and speed to each observation for neccesary amounts of time to reach the "mean time" of observation. This should result in a fairly tight intersection of the LOPs generated. Also if you observe the Sun as it is rising or setting the resulting LOP, (Line Of Postion), would be a Longitude line. While if you observe the Sun at LAN, (Local Apparent Noon), then the LOP is a Latitude line.
As don1reed stated during the daylight hours the moorning Sun Line would be advanced to the Noon Sun Line based upon the vessel's course and speed during the intervening hours, and then again both lines could be advanced to the afternoon sight. The inaccuracies can creep in based upon multiple course and/or speed changes and bad record keeping of the same. The biggest factor that can mess up your sights though is a bad Time Reference, this is why most vessels used to keep at least two Chronometers onboard to compare against each other to track GMT, (or Zulu time in the military). These would be compared against a time signal received by the vessel via radio each morning and a log kept on each Chronometer as to how far ahead or behind it is of GMT and whether it is gaining or losing time.
Another thing, (at least in the Northern Hemishpere), at night one can take a sight of Polaris, (North Star), anytime the view of the horizon permits, to obtain a Latitude LOP. This however is not possible in the Southern Hemisphere as there is no "South Star" available for such a sight.
Now the resulting Arithematic to reduce the observations to LOPs is very complicated if one were to do them by hand, however the formulas have been greatly reduced, (i.e. simplified to basic math), by using Sight Reduction Tables that I assume are published by many governments around the world, and here in the USA by the US Naval Observatory, coupled with the use of the Nautical Almanac published by several sources, (USNO, UKRNO, etc.).
I hope this has been helpful to those of you new to the idea of Celestial Navigation, however please remember that accuracy in this area of Navigation usually takes many Years of practice, (which is why competant Ship's Navigators used to be so prized in the Merchant Marine and in the Naval Services around the world).
JimRat, (former Quartermaster Second Class USNR)
