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Most of what I've read on lunar distances from modern authors seems to ignore parallax in azimuth, instead relying on a spherical earth where parallax only causes the moon to move vertically instead of diagonally.

In contrast, the method of Chauvenet (at least) includes the effects of flattening.  It involves using a ficticious geocenter, chosen so that parallax can be represented as only a shift in the moon's altitude and its declination.  But the hardest part was the last step, of shifting the results from the ficticous center to the real one so that it could be compared to the distances tabulated in the almanacs.  The method Chauvenet refers to as "approximate" involves five different table lookups.

But since today we have to compute our own comparing distances anyway, there's no reason for that final shift from the ficticious center to the real one.  A modification of the moon's HP in clearing the distance and of the moon's declination in calculating comparing distances will give rigorous results for a spheroidal earth.

I've made some additions to Stark's tables to implement this.

To calculate "reduced" HP:

Enter Table 7 with the HP from the almanac.  Subtract the log rho corresponding to your latitude.

log rho table: 

29°18.9'     0.0003

33°43.4'     0.0004

37°51.8'     0.0005

41°51.3'     0.0006

45°47.1'     0.0007

49°43.8'     0.0008

53°46.2'     0.0009

58°00.0'     0.0010

62°33.7'     0.0011

67°42.7'     0.0012

74°04.1'     0.0013

85°15.2'     0.0014

90°00.0'     0.0015

log rho can be treated as 0 for latitudes lower than 29°18.9.  If your latitude isn't listed, use the next larger one.

Use the resulting "reduced HP" (which will always be bigger, despite its name) when using Table 2.  Keep note of the reduced HP's log for the next part.  (Rule of thumb seems to be to just add 0.1' to the HP for mid-latitudes.  The change probably doesn't matter to Table 1, but do not use this reduced HP in the WWP table.)

For the "reduced" declination:

Take the log of the reduced HP from the prior step and add 2.17 429. 

Double your latitude (nearest degree should suffice), enter the K table, and then halve the result.  Add it to the above.

If the total is greater than 3.6812, there is no change in declination.  Otherwise, the resulting angle from Table 7 (maximum of 0.4' or so) is added to the moon's declination.  This correction has the same name as your latitude, so it's subtracted from contrary declinations.

Otherwise, continue as normal.

The examples Stark used in the introduction of his book used a latitude of about 44° N.  Applying all this to his examples:

Venus:  The reduced HP is 56.3.  Q goes up 0,5 and H~H is reduced by 0.1'.  The cleared distance is also reduced by 0.1'  Declination is increased 0.3' N.  The comparing distances both decrease by 0.2'.  Overall, the end result is 14:47:28, increasing Stark's error in Da by 0.1'.  (Following Stark's example I did not correct for second differences in D).

Sun:  Reduced HP of 55.0 raises Q by 1,3 and H~H by 0.2'.  The change in the Gaussian counters the change in Q, leaving D unchanged.  Declination is increased by 0.2' N.  The comparing distances are unchanged, and resulting time is the same.

Aldebaran:  Reduced HP of 54.7 increases Q by 1,1 but leaves H~H unchaged.  However, the increased Q is enough to reduce D by 0.1'.  Declination is increased by 0.2' N, but the comparing distances remain unchanged.  The resulting time is 14:12:36, reducing Stark's error in measuring Da by 0.1'.

Overall... I'm not sure if all this is worth it.  The procedure doesn't feel particularly burdensome to me (maybe as complicated as using Eno's form for interpolating altitudes), but I'm not sure if the results are meaningful or just noise.  This should have a larger effect when the moon is closer (i.e. larger HP), and Stark's examples involve a relatively distant moon.

More sample sights from people with a quality sextant and skill in using it would be much appreciated.