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Doug,

I'm very glad someone has opened a discussion on Moon culminating stars. It's been a notable omission up to now.  I half expected that Lars would beat me to it and he didn't let me down. Anyway here are my thoughts for what they are worth.

I don't have a theodolite so can't carry this out in practice but I can give you a rough explanation as to how you might go about it using the modern Nautical Almanac (NA).

In the various methods for finding longitude you are essentially trying to determine the Moon's geocentric R.A. at a given instant in time. Since the Moon's geocentric R.A. was tabulated in the NA you could do an inverse lookup (with interpolation) to get GMT. Nowadays GHA is tabulated so that's what you'll be doing the inverse look up in. Because you are observing the Moon on or very near the meridian, parallax shouldn't change the R.A. or GHA appreciably between the topocentric and geocentric positions.

To summarize the steps then:

1) Measure the time interval between the meridian passage and that of Moon's bright limb.

2) Multiply the interval by 1.0027 to get the difference in GHA between the star and the Moon's limb.

3) Correct for the Moon's semidiameter (S.D.) to get the GHA of the center
sin  Δα = sin S.D./cos δ or to a good approximation Δα = S.D./cos δ 

4) Now you have to do the inverse lookup to get GMT. You could do that linearly or making a quadratic fit to the 3 nearest tabulated times.

Note that the Moon's S.D. and declination are also changing so you might want to consider applying this procedure interatively and interpolating those values as well.

Of course once you have GMT you subtract it from LMT, obtained by a time sight or equivalent, to get your longitude in time,

Robin Stuart