These are the worksheets from our January 9th running fix showed in Episode 8. A running fix is two or more sights taken at different times and plotted to make a position fix.
In this fix, we took a sun sight at 1647 hours and a second sight five hours later at 2248 hours. We advanced the first sight in the direction of our compass course (150T) for the distance we travelled (21nm). The course and distance is what we had traveled between the two sights.
This technique works well because it gives intersecting lines of positions. Each celestial sight gives a single line of position which shows that you are somewhere along that line. The intersection of the lines shows you where you are on each line (at the intersection). You can advance or retard multiple sights to one common time.
It is not a super accurate method because it is affected by your ability to maintain a course and to know how far you traveled. It is also affected by current.
In the middle of the Gulf of Mexico, we simply did not need to know our position with great detail because there is nothing to hit. The purpose of this sight was to plan our arrival through the Yucatan Channel (or Yucatan Strait).
Interesting, you have to wonder how Columbus ever found America much less the Vikings every found Minnesota