Let us begin with a statement of the obvious: At any given instant in time, every object in the sky appears exactly overhead at some point on the surface of the earth.  It's called the subpoint for that object.  That applies to all celestial bodies -- stars and planets, sun and moon.  Of course, the earth is busy rotating on its axis and orbiting the sun driving distant stars across the firmament.  The planets ("wandering stars" Greek 1640) meander all over the night sky, while the moon makes its monthly orbit of the earth.  Thus every subpoint traces an earthly pathway, unique and predictable. 

The familiar word altitude in the aviation lexicon represents the instantaneous elevation of the aircraft itself above sea level, measured in feet.  In nautical parlance, where the vessel itself is confined to one altitude -- sea level -- the word altitude indicates the elevation of a celestial body as it appears above the horizon, measured in degrees, minutes, and seconds. 

Here is the key to celestial navigation: At a given instant in time, every object in the sky appears at the same altitude for a set of locations that form a circle on the surface of the earth.  The circle is called line of position (LOP), which is named for the celestial object and for that instant in time.  Inasmuch as the earth is quite a big place, when a segment of a LOP is plotted on a local navigation chart, it looks like a straight line.

Here is an exceptionally convenient statement:  The coordinates -- latitude and longitude -- of each sub-point can be predicted for any given instant in time.  That makes it possible for a skilled navigator to ascertain the sub-points of navigational stars and planets plus the sun and moon, all based on tabulations published in an almanac.  There are 57 objects thus tabulated.  Since olden times, the celestial navigator at sea would take a sighting by sextant of a selected object in the sky to determine its altitude and note the precise instant in time from the ship's chronometer. These readings taken together provided what the navigator needed for looking up the appropriate entries in the almanac, thence to perform the requisite computations for plotting the LOP on the navigator's chart. 

To "shoot" one star or planet was not enough to determine a fix for the vessel's location.  The navigator needed to obtain at least two LOPs, preferably intersecting at a near right angle -- and taken hurriedly to be as close to the same instant in time as possible.  Alas, celestial navigation has always been fraught with errors. Moreover, way-finding from one fix to another required dead reckoning, itself error-prone.  Common principles of  navigation -- imperfections included -- apply to avigation.  Inevitably, errors suffered in the sky are much larger than those experienced on the surface of the sea, as aircraft are so much faster than ocean-going vessels. 

The photograph on the right was provided by Gary LaPook.  It shows a Mark 3 Octant, like the one used by Fred Noonan, navigator for the 1937 round-the-world flight of Amelia Earhart. 

The flight departed Lae, New Guinea on July 2, 1937 for Howland Island.  Most of the 2,556-mile segment was flown at night, which enabled Noonan to take sightings of stars and planets for live reckoning.  The flight plan assured arrival in daylight at Howland for landing on its unlighted strip.  There would be no stars in the sky -- except one.  One LOP for the sun is assumed by some historians to have been plotted by Fred Noonan prior to take off.  That particular LOP received prominence in the last radio transmission by Amelia Earhart (see Simplexity Aloft).   As an advanced LOP, it would not be used to determine the geographical location of the flight but to synthesize a 'landfall line' for guiding the final stages of the approach to Howland (see Landfall Navigating).  That sun LOP is shown in the diagram below as a cut across the course line at Howland Island and marked with its reciprocal headings 157-337 degrees.

Here is yet another statement of the obvious: No object in the sky appears exactly overhead of a fixed location (subpoint) for more than an instant in time.  Well, almost obvious.  Polaris is an exception.  So too are geostationary communications satellites, come to think of it.  In general, though, no LOP is stationary. Noonan's 157-337 sun LOP would be present at that location only at a certain instant in time -- the planned arrival time (1800 GCT).  At that instant, the sun's altitude would be 6 degrees all along the 157-337 LOP passing over Howland Island.

Meanwhile, the sun LOP itself would be racing toward the West at 1,036 mph, as studied in  Here Comes the Sun.  As a consequence of reported headwinds, Noonan would most likely have adjusted the ETA, by, say, one hour to 1900 GCT.  The altitude of the morning sun, which increases relentlessly at 15 degrees per hour, would then be 21 degrees, as the delayed flight approached Howland Island    Referring to tabulations in the almanac for 1937, Noonan would need to plot a new LOP passing over Howland Island, which would have its reciprocal headings slightly rotated toward 156-336.  Using his octant and chronometer, Noonan would then be able to confirm the arrival of the flight intercepting the updated LOP.  Thus by celestial navigation, using daylight's one and only celestial body in the sky, Noonan was able to keep track of the flight's progress along the course line toward -- well toward an intersecting line not a geographical point.  So then, what about the course line itself? 

In the figure above, solvers will see that the sun was not "the one and only celestial body in the sky" that morning.  The previous sentence deserves exclamatory punctuation and so does the next one.  The moon, which was in its last quarter, was also available to Fred Noonan for celestial navigation.  Not only that, but the moon's LOP for 1902 GCT offered reciprocal headings of 067-247 degrees -- closely aligned with the course line (072) for the flight.