NavList:

It's been a while but I got back to looking at Close's (page 202) method for obtaining longitude by Moon culminating stars. Here's the basic idea.

By measuring the time difference, ΔT, between the meridian transit of a star and the transit of the bright limb of the Moon, the R.A. of the Moon's limb when it was on the meridian can be easily found.  R.A.(limb) = R.A.(star)+1.0027 ΔT

Because the Moon's limb is on the meridian, the parallax in R.A. is zero and R.A.(limb) is the same as the geocentric R.A. of the bright limb. (This means that if it is necessary to find the geocentric R.A. of the Moon's centre then the required adjustment, Δα, is obtained using the geocentric semidiameter (S.D.) by means of the formula, sin  Δα = sin S.D./cos δ or to a good approximation Δα = S.D./cos δ with no need to consider parallax. For the R.A. of the opposite limb then Δα is doubled).

Suppose we had at our disposal a function f, that took the observer's longitude, L, as an input and returned the R.A. of the Moon's bright limb when it was on the local meridian. Then we could take the observed R.A.(limb) and solve the equation R.A.(limb) = f(L) for L, using some numerical method, and get L directly. This is what Close does.

Of course we don't have the function f but we do have its value tabulated at regular intervals in longitude. Close then uses Newton's central difference interpolation method to construct a quartic polynomial approximation to the function f(L) and then solves the equation above numerically.
The tabulated values  used to construct the polynomial approximation are R.A. of the Moon's bright limb as given in the Nautical Almanac for
Upper passage on the preceding day (corresponding to L=-360°)
Lower passage on the preceding day (corresponding to L=-180°)
Upper passage on the corresponding day (corresponding to L=0°)
Lower passage on the following day (corresponding to L=180°)
Upper passage on the following day (corresponding to L=360°)

Longitudes are West longitudes and for covenience Close chooses to interpolate in the variable L/12 where L is expressed in hours. Hence L=180° → 1.
There is a caveat to this. Around new and full Moons the bright limb switches abruptly from one side to the other. This obviously not suitable for interpolation. The R.A. of the opposite limb can be calculated as described above and used in the interpolation.

Robin Stuart