As the last installment in this little trilogy of traditional navigation techniques, here is a quick explanation of how celestial navigation works.

The celestial bodies: stars, sun, moon and planets are scattered in three dimensional space about the earth. Their exact positions for any moment in time have been calculated by astronomers and are published every year in the Nautical Almanac. These positions are constantly changing, due to the motions of the bodies, as well as the earth's revolution and rotation. By using the Almanac, you can determine the exact position in the sky (and the corresponding Lat/Lon for the point on earth directly beneath it) for 57 navigational stars, 4 planets and the sun and moon, for any date and any time of the year.

If you measure the altitude above the horizon of any of these bodies with a sextant, and note the exact time you did so, you can determine how far you are away from the point on earth directly beneath the star by means of a mathematical transformation called "sight reduction". This involves a lot of spherical trigonometry, but is usually done with the aid of published "sight reduction tables" which allow you to get the answers you need by simply adding and subtracting angles according to a cookbook type procedure. When you're finished you get a direction and a distance to that star, expressed as angles, which gives you a line of position. You are somewhere on that line. If you repeat the process on another star, you get another line of position. Where the two LOPs cross is where you are.

Sight reduction is a mathematical transformation where you put in a seed lat/lon (just a rough guess as to where you are), the time of the observation, your sextant reading, and the position of the star in the sky (from the Almanac). What you get back from the calculation is an azimuth (direction) to the star and an elevation of it AS IF you were at the "seed position" of your initial guess. The difference between the calculated elevation and the elevation you actually measured with the sextant is how far you are from the seed position, the direction being in the direction of that azimuth. You draw the azimuth on your chart, draw a line at right angles to it at the difference in elevations you just derived, and that is your LOP.

The mathematical calculation using takes about 10-20 minutes for each LOP you derive. Two is the minimum required for a fix, five or more is recommended. (This helps eliminate possible errors, like accidentally shooting the wrong star, or misreading your clock or sextant micrometer).

What you actually see through the sextant telescope is two images, superimposed. One is of the horizon, the other is of the heavenly body.
It's all done with mirrors and half-silvered lenses. You adjust the sextant micrometer knob so the star image just touches the horizon, then you read the elevation off the sextant scale, simultaneously noting the exact time of the observation.

People like Lewis and Clark, or other explorers, who were on land and had no visible horizon, used something called an artificial horizon. This is a little pan filled with mercury, and the sextant was simultaneously sighted on the celestial body and its reflection in the mercury. Since a liquid surface is perfectly flat and perpendicular to the radius of the earth, it performs the same function as a horizon, it gives you something to measure the angle against.

On land you can't tell exactly where the horizon is, and at sea,the boat's motion causes an artificial horizon to slosh around and be unusable, so the use of the sextant is different in the two cases. But the principle is the same. The observation gives you an elevation and a time. From there on, do the math.