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"I determined my position Brent's method using  mid-morning and  mid-afternoon sights. I have attached the plot. I hope that it makes sense "

David.  A little more detail would have made it easier to follow your work.  You don’t say what the body was.  From your use of ‘mid-morning’ and mid-afternoon, I presume it was the Sun.  Neither do you say what zone the times of your shots were.  10.25 and 15.22 suggest UTC +12. You don’t give azimuths or heights calculated or the method you used to calculate them.  If you were using something like AP3270/NP303/HO249, then AP1 (or is it AP2?) would be dictated by the fact that the tables are entered using whole numbers of degrees latitude and LHA.

I don’t get the next bit.  You say “The second sight is worked as an intercept using the lat/long from the first sight as an assumed position” This looks like sorcery to me.  There’s no way you can get a lat & long fix from a single position line.  Two shots on the same body several hours apart, maybe; a Zenith Fix, yes; but not a single position line.  The nearest thing to a fix you can get from a single PL is a most probable position (MPP) by comparing it with your DR position.  Perhaps I’m interpreting what you wrote incorrectly.  You obviously used something other that rapid sight reduction tables for your second height calculated, because the longitude you ended up with for AP2 (or was it AP1?) would have been unlikely to leave you with an LHA equal to a whole number of degrees.  

You said, "Brent said plot the largest bearing (azimuth?) first".  I’m having considerable difficulty following your diagram.  Which came first in time AP1 or AP2?  It would help to have a time against each.  Logically it would be AP1 for the mid-morning shot, and the intercept would be around 8 away.  However, if AP1 was for the mid-afternoon shot (because Brent says plot the largest bearing first), the intercept would be zero, and the intercept for the 10.25 shot would be 1 away from AP2.  I'm confused! DaveP