NavList:

Frank:

I was looking into how far away from an assumed position a great-circle calculation can be relied upon along rhumb lines before you had to perform great-circle math again.

I think this would affect plotting running fixes.  Consider:

• Around sunrise you take a sight of the sun and carefully calculate the predicted altitude at your dead-reckoning position, "DR1"
  • We'll call the geographic position of the sun at this time "GP1"
• You also determine the sun's great-circle direction at your DR position, draw a line at 90° from this direction and call it a line of position
• You then proceed along one (or more) rhumb line courses for the next few hours. 
  • We'll say your rhumb lines are straight lines on your plotting sheet.
• Around noon, from your new dead-reckoning position fo "DR2," you take a sight of the sun and want to combine this with the sight you took... at least a hundred miles back.

If your earlier LoP was based solely on great-circle direction, you can't simply transfer it over with parallel plotters  The angle that line cuts with your rhumb line course then, back at DR1, isn't the same as the angle it cuts across your course now at DR2, even if you're sailing straight east or west and only your longitude changes. You would need to determine the great-circle direction from DR2 to GP1 before drawing the new line; there was no reason to draw a line back at DR1 at all.

In the real world the errors from this are probably dwarfed by unknown currents, steering errors, etc. but the math shouldn't be any different than if the sun had dropped a radio beacon at GP1.