Celestial navigation
"Celestial navigation, also known as astronavigation, is a position fixing technique that was devised to help sailors cross the featureless oceans without having to rely on dead reckoning to enable them to strike land. Celestial navigation uses angular measurements (sights) between the horizon and a common celestial object. The Sun is most often measured. Skilled navigators can use the Moon, planets or one of 57 navigational stars whose coordinates are tabulated in nautical almanacs."[1]
Basic celestial theory tells us that we can convert the height of a celestial body above the horizon to a distance from its subpoint. If we know where the subpoint is, we can determine our distance and approximate direction from it. When the body is directly overhead (the sextant reads 90 degrees) we are at the subpoint. When the body is on the horizon (the sextant reads 0) we are 90 x 60 nautical miles or 5400 nautical miles from the subpoint.
This information can tell us that we are somewhere on a circle around the body’s subpoint which may have a radius up to 5400 nautical miles. To make things a bit easier, as far as plotting is concerned, we assume a position on the earth, compute what the height of the body above the horizon should be at that position, and compare it to what we actually observe with the sextant. The difference is then used to adjust the assumed distance of the circle from the subpoint to its actual distance. For practical convenience, only a short segment of this circle is plotted on the chart and it becomes a “line of position”. This circle segment (the LOP) is perpendicular to the direction of the celestial body's subpoint from the observer.