NavList:

Robert Goldberg you wrote: Suppose you are making an ocean passage with very cloudy and stormy weather for a week with no sights taken. Dead reckoning was very limited as your focus was on saving your boat as it was being pitched around in the stormy seas. The storm passes and the seas calm and the weather clears just enough to take a morning and afternoon sun sight, but not a noon sight. Your dead reckoning and therefore assumed position might be off by hundreds of miles. How accurate a fix could you get under these circumstances?

Robert

Lets say DR might be out by 10s rather than 100s of miles.  Assuming you’ve got an accurate watch, do what Capt Sumner did.  What you do know is that at the time of each sight you’re on a particular small circle of altitude from the Sun.  If you assume you're at your DR latitude, you’ll have three sides of a PZX triangle.  In the Northern summer you’d have 90-dec Sun, 90-DR lat, and 90-alt.  You can use these to get the angle between the observer and the Sun at the pole P.  I.e.  LHA Sun.  Then if you have accurate Greenwich time, you can get GHA Sun from tables.  LHA Sun plus GHA Sun (minus 360 if necessary) gives you a latitude. 

Do the same for DR lat plus 1 degree and for DR lat minus one degree.  This will give you three positions at which the Sun was at altitude observed at the time of the sight.  A line though the three positions gives you LOP1. 

Do the same for the afternoon sight to get LOP2.  Move LOP1 up along track by the distance travelled between sites to give LOP1’.  Where LOP1’ crosses LOP2 is your fix (or so the theory goes, I’ve never tried it, I keep promising myself that one day I’ll give it a go using a peri-sextant on the window ledge at home).  DaveP