NavList:

We are only talking about a position determined by celestial
observations involving only random error and that any possible
systematic error has been eliminated.
The "fix" is the spot marked on the chart derived from analysis if the
celestial observations.
The "position of the observer" is the location where the observer was
located when he took the sights and which can also be plotted on the
chart.
"Triangle" and  "cocked hat" are names for the figure formed by
plotting three lines of position on the chart that do not pass through
the exact same point.

Let's take, to start with, the simpler situation involving only two
lines of position and let's have them cross at a 90º angle. In this
situation we take their intersection as the fix since there is no
triangle to be inside or outside of. However, no one actually believes
that the position of the observer is actually located at this
intersection given the normal random errors of observation and
computation. In fact, we can draw circles around the fix at various
radii to depict the various probabilities that the position of the
observer is located within the various circles. This can also
described as various levels of confidence. If the two lines do not
cross at a right angle then you can draw ellipses for the same
purpose. This is covered in great detail in appendix Q of the the 1977
edition of Bowditch.

The size of the circles and ellipses is determined by the uncertainty
in the sights expressed in terms of "standard deviation" or "sigma." I
have spent a great deal of time lately analyzing the standard
deviation of sights taken with bubble sextants and is appears to be
about two minutes of arc or a two nautical mile sigma.. This also
comports with the requirements of  Federal Aviation Regulation part 63
(14 CFR 63, et. seq.) which sets the accuracy requirement for
celestial fixes obtained in flight. Sights taken with a marine sextant
should have a much smaller sigma but will be larger than just the
accuracy of the instrument taking into account an indistinct horizon,
waves on the horizon, observer bouncing up and down in heavy seas
changing the dip, etc. For the sake of this discussion let's assume
the sigma of sights taken with the marine sextant have a sigma of 1.5
nautical mile which was the value determined by the study of thousands 
of observations taken by professional mariners.

Based on the analysis of this situation the position of the observer
will be somewhere within the 1.5 mile circle (one sigma) centered on
the plotted fix 39.3% of the time and within a circle of  1.177 sigma
(1.77 NM)  50% of the time. This circle is also known as CEP or
circular probable error. Continuing drawing circles, 66% within 2.2
NM, 75% within 2.5 NM,  90% within 3.2 NM, 99%  within 4.5 nm and
99.9% of the time within 5.6 NM. At the other end of the distribution
there is only a 10% chance that the position of the observer will be
within .72 NM of the fix. ( See table Q6c on page 1221 of vol. 1 of
Bowditch, 1977 ed.)

So, what does this tell us. There is about a 30% chance that the
position of the observer will be more than .72 NM but less than 1.5 NM
and about a 61% chance that the position of the observer will be more
than 1.5 NM from the plotted fix. So what do we do with this
knowledge? We use the plotted fix at the intersection of the two LOPs
for our navigational purposes such as measuring our progress and
planning the next leg.  We also use this fix  to deal with the
proximity of danger keeping always in mind that the vessel may
actually be almost 5 NM from the fix in any direction. Why do we use
the intersection as the fix, because there is no better one available
since this spot marks the center of possible positions of the
observer. No other spot would be as useful for planning purposes or
avoiding danger. Also, what methodology would you use in determining
another spot to mark the fix?

gl