Wolfram Alpha:

```Celestial Navigation
--------------------

Celestial navigation uses angular measurements taken between
a celestial body and the Earth's horizon. The sun is most
commonly used, but navigators can also use the moon, a planet
or one of 57 navigational stars whose coordinates are tabulated
in the Nautical Almanac and Air Almanacs.  Consider the following
diagram:

This in fact a view of the celestial sphere where the zenith
of the observer is on the celestial equator.

At a given time, any celestial body is located directly over
one point on the Earth's surface. The latitude and longitude
of that point is known as the celestial body’s geographic
position (GP).  The latitude is referred to as the DECLINATION
and the longitude is referred to as the HOUR ANGLE.  Since
celestial bodies move with the celestial sphere, their GPs
also move with respect to the surface of the Earth.  The Sun's
GP, for example, travels a mile every 4 seconds. Thus, an exact
time is required to use the navigational tables.

The angle between the horizon and a particular celestial body
is measured with a SEXTANT.  This angle is called the ALTITUDE,
θ.  When haze obscures the horizon, navigators can use
artificial horizons.  In the early days, a tray of mercury was
used, and the image of the actual body was superimposed with
the image reflected by the mercury.

Once the altitude for the celestial body is known, it is then
possible to use the navigational tables to calculate a CIRCLE
OF POSITION (COP) on a navigational chart that is a circle on
the Earth that surrounds the GP of the observed body.  The
observer, O, can be at any point on the COP.

The angle that the line GP to O makes with the True North is
called the AZIMUTH, φ.   It is equivalent to the BEARING.

Measuring φ with a compass is one way to the observer's
position on the COP but the more accurate method is to measure
1 or 2 more bodies and calculate where the COPs for each body
intersect.  This is understandibly called the INTERSECT METHOD.
If the measurements are taken at different times, corrections
must be made to account for the movement of the observer during
the interval between observations. If observations are taken at
short intervals, the corrected circles of position by convention
yield a "fix".  If the lines of position must be advanced or
retarded by an hour or more, convention dictates that the result
is referred to as a "running fix".

In reality the following procedure is followed:

-  The altitude above the horizon, θO, of the celestial body is
measured and the time of the measurement is noted.

-  A certain latitude, αA, and longitude, βA, for the observer
are assumed (AP).  This does not have to be precise and is
usually determined from DEAD RECKONING calculations based
on the observer's last position.  From the nautical tables
the declination, δT and the local hour angle, τT of the
observed body are recorded.

-  The altitude, θC, and azimuth, φC, for the assumed position
are computed using the following formulas obtained from
spherical trigonometry:

θC = arcsin(sin(αA).sin(δT) + cos(αA)cos(δT).cos(ωT))

φC = arccos((sin(δT) - sin(αA)sin(θC))/cos(αA)cos(θC))

Where ωC is the he LOCAL HOUR ANGLE. It is the difference
between the AP longitude and τT.  It is always measured in
a westerly direction from the AP.

If θO < θC this means the observer is farther away from the
body than the observer at the assumed position, and vice versa.

-  The assumed position AP is located on a chart and a line in
the direction of the azimuth, φC, is drawn.  A line
perpendicular to the Azimuth is then drawn at the point
where θO = θC.  This intercept is found by noting that the
difference between θO and θC is measured in minutes of arc
where each minute of arc equals 1 nautical mile.  This is the
line of position (LOP) that corresponds to a small segment of
the COP at the moment of the observation.

-  The above procedure is repeated for a second celestial object.
The intersection of 2 LOPs provides the 'fix'.

Noon Sights
-----------
Another way to find the longitude is to note the exact local
time when the Sun is at its highest point in the sky. This can
be found by taking a vertical rod perpendicular to level ground
and noting when the shadow points due north (northern hemisphere)
or due south (southern hemisphere).  Once the time has been
found, the longitude of the pole position may be determined by
computing the time difference between the location and GMT*,
and then multiplying this number by 15°.  In order to perform
this calculation, therefore, an observer needs to carry a
chronometer set to GMT.  The invention of the chronometer was
pivotal in the history of nautical navigation since conventional
pendulum clocks do not work correctly on unstable platforms.

* Also called Coordinated Universal Time (UTC).
```