NavList:

​I think my approach would be a graphical one.

For various times about ten minutes apart, starting at some point before the immersion of the occultation and ending sometime after the emersion, calculate the altitude of the Moon from your assumed position, and also for the star. Calculate too the azimuths of the two bodies. For each of the various times of calculation, write down the difference in altitude, and the difference in azimuth between the two bodies.

Construct a graph with difference in altitude for the vertical axis and difference in azimuth for the horizontal axis. Draw the Moon at the origin as a circle equal to its angular diameter. Plot the positions (difference in altitude and azimuth) of the star on the graph.

Draw a line connecting the plotted points for the star. This line will intersect the circle of the Moon at two places (if an occultation will take place). 

Construct another graph for altitude difference vs time for the star.

For the first graph, note the azimuth differences (on the horizontal axis) at which the intersections with the Moon circle take place. From the second graph, read off the times for the noted azimuth differences.

If the assumed position for the calculations above is the actual known position, then job done! If the longitude of the position is not known (the latitude is easy to determine from the noon Sun or the pole star) then the measured altitude of the Moon at the times of immersion and emersion - or the time difference between these two events - should enable the observer to determine a longitude by iteration.

Geoffrey Kolbe