Get the Worksheet
To understand this discussion
of the worksheet, you must of course HAVE a worksheet to look at. Download
the sight reduction worksheet from
here.
Basic Questions
 What time is it?
 Where do I think I am? (You
don't know for sure...which is why you are trying to get a fix.
But you have a rough idea of where you are, within a few dozen
miles.)
 What does my sextant tell me?
Enter this information in the
first three sections of the form.
Where is the GP?
The next task you have to do
is to figure out where the GP (geographical position) of your
celestial object is. This is the point where the sun (or
whatever celestial object you have selected) is directly overhead.
You will get this information
from the Nautical Almanac. It allows you to determine the GP
of a celestial object for any day of the year, down to the precise
hour and second.
You now have the first side of
your navigational triangle: a line drawn from the North Pole
to the GP.
Where am I, revisited?
We will refine your DR postion
into an AP (assumed position). It is within a few miles of
your DR, and is chosen to make the subsequent calculations easier to
do. You could work out a fix using only your DR position.
Just trust me for now that creating a fictional assumed position,
which is close to your DR, really does make it easier in the end.
Once you have your assumed position, you have
two sides of your navigational triangle.
Once you get the included angle
between these two lines, you can
use mathematics to work out the length and angle of the missing
side. But to keep the math manageable, it is helpful to deal
in whole degrees, rather than fractions of a degree. This angle is called the Meridian Angle
(confusingly abbreviated by a t, or by MA, or by LHA  for "Local Hour Angle").
You now know everything you
need to know, mathematically speaking, to describe the third side of
your triangle. You can do this with a hand calculator,
using sines and cosines, in about 4 minutes.
OR...you can go to Pub.
249, Sight Reduction Tables,
which contain solutions for every possible navigational triangle.
This process takes only 1 minute...so most navigators choose to use
Pub. 249 rather than using a hand calculator.
Calculate the Final Side
of the Triangle
To make the next step easy,
you take a moment to collect all the little bits of information you
will need to make use of Pub. 249.
Thense Section 7 of the worksheet
to do this.
In the end, you will get an Hc,
usually called "Height Calculated"...though you could equally well
call it a "Hypothetical Calculation". For if your vessel WAS
precisely at the AP, this is EXACTLY what your sextant
reading would be. This also tells you the bearing of the
celestial object from you (abbreviated as Z and called the azimuth
angle; this will be converted into Zn, the azimuth, which is
precisely the same as the compass bearing from your vessel to the
object).
Naturally, though, you are not
at the AP. You are somewhere else. You want to find out
where that somewhere else is.
First, though, you are pretty
sure that your azimuth is the same as what you calculated. You
could be scores of miles off in your DR, and your azimuth would
still be the same. It is a big earth, and azimuth doesn't
change quickly.
You may wonder why we go
to the effort of calculating an azimuth, rather than just
pointing a hand compass at the sun. The truth is that the
human eye is unable to find the center of the sun with any great
accuracy. The brightness throws us off. Further, a hand bearing compass works best getting the
azimuth of things like lighthouses, which lie along the horizon.
We simply cannot use a hand compass to get an adequate bearing
of an object that is 45° high in the sky.
We get more accuracy by
using the Nautical Almanac plus Pub. 249 and calculating the
azimuth.
Getting from an AP to a
FIX
We use Section 7 of the
worksheet, to apply a few final corrections to our sextant data, and
then to relate the height of the object as read by the sextant
(called Hs, or Height Sextant) to the Hc.
We then work out our distance
and direction from the AP, and then we know where we are. In
this example, you will see that we have used two celestial objects 
one star and one planet  and are able to work up a fix of where we
really are.
.
