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Geoffrey Kolbe, you asked:
"Given that we DO have reliable calculators today and so are not restricted to using log tables or slide rules, what is the most efficient method of finding time and longitude from a lunar distance using a calculator?" 

Starting with the "clearing" of the lunar, if any form of cheap computation is available, then you do it by the standard direct triangle method which starts with the observed altitudes of the Sun and Moon and the observed distance between the two. From that, using the standard spherical law of cosines, we calculate the cosine of the difference in azimuth between the two bodies. Then we correct the altitudes for refraction and parallax, just like any altitude sight corrections, and using the previously computed value of the cosine of the difference in azimuth, we can directly calculate the corrected lunar distance. Easy as can be. 

Note that this description of the basic geometric problem omits some key details. In particular, we have to worry about the "pre-clearing" step where we adjust the limb-to-limb measured distance for the semi-diameters of the two bodies. The SD of the Moon has to be carefully corrected for the "augmentation" and in addition the SD's of the Sun and Moon have to be corrected for flattening due to refraction when they are lower than about 15° in the sky. Finally, the distance should be corrected for the oblateness (polar flattening) of the Earth, which is separate from the rest.

The only reasons for using Thomson's tables or any of the series methods is to engage in the historical experience, or to save a few keystrokes. And for a genuine calculator solution, that may matter a bit. The factors which I have called the "corner cosines" need to be recorded only to the two of three significant digits, which is nice for actual calculator work. But mainly it's about staying close to the history. 

Since you asked about getting time (and thus longitude) from the calculation, we have to finish up by interpolating between known distances. So you calculate the true lunars distances with high-quality lunar ephemeris data (it's just a geocentric great circle distance between the two bodies) and then you interpolate using the corrected observed distance. Historically, so-called proportional logarithms (really "proportioning" logarithms) were used for this interpolation, but that's pointless, and also uninteresting, today. Also, the inconvenience of quadratic interpolation can be avoided in those cases where it might arise by calculating true lunar distances with a shorter interval between them than used historically. In the old almanacs, true lunar distances were listed every three hours. There's no reason we can't calculate them every three minutes --or even every three seconds-- today.

Frank Reed
ReedNavigation.com
Conanicut Island USA