A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Iwancio
Date: 2022 Aug 3, 12:25 -0700
Just to add on to what Antoine said, it should be remembered that the tables are based on certain averages to improve ease of use, and some (insignificant, in my opinion) error creeps in when you use them.
Firstly, correction 1 (the upper part of the table) is calclated based on a particular assumed HP value (53.7', if I recall) and then treats correction 2 (the bottom of the table) as a simple additive correction on top of that. However, calculating the values for the second correction require particular altitudes to use The second corrections in the bottom of the table tend to be based on the values in the middle of the columns of the first correction (e.g. 2°30', 7°30', etc.). The further you are from the middle value, and the greater the spread of the values in a particular column, the greater the error in the values of the second correction. You can see this by how much the second correction changes from column to column.
The oblateness of the non-spherical earth introduces another source of error. The tables account for this by assuming the moon's azimuth is near 090° or 270° (and assumed latitude of 45°). Errors are maximized at 000° and 180°, in opposite directions. However, I believe this is insignificant compared to the rounding errors involved in the second correction.
If you're looking to minimize all of these errors, you could try interpolating the second correction across columns as well as across rows, or calculate the parallax directly (after first correcting for refraction) using the formulae under Step 4 on page 280 of the Nautical Almanac...
... or you can just use the nearest entry for the second correction and get on with your life. (With the step interval of 0.3' between rows, I suspect this is the intended method.) These simplified tables exist specifically to make using the moon for navigation simpler. It was found that the additional computations needed for the moon, when treated separately and precisely, were so burdensome for pencil-and-paper navigators that they avoided the use of the moon entirely. They preferred, for example, taking a running fix with a sun sight and leaving it at that rather than use the moon for a two-body fix.
These aren't the only approximations used in the almanac, but they're probably the most noticeable.
I feel it's worth remembering that the layout of the Air Almanac came first. The layout of the Nautical Almanac was modified to resemble the Air Almanac in response to how popular the Air Almanac was at sea.