NavList:

Antoine:

I completely agree that the phrase "apparent altitude" and its abbreviated partner "Ho" are problematic. It's considered standard, navigation legalese, something that any "good scout" should know! But these are relatively recent additions in navigation history, and fundamentally it's all just jargon that buries the facts. And besides... things were different with lunars...

Nearly all practical lunar computations in the era when they were widely used begin with a simple and generic "pre-clearing" phase. This is where we adjust the altitudes by +12' for anything Lower Limb (Moon or Sun), -20' for Upper Limb (Moon and technically Sun, too, but no one ever did Sun UL, so really... just Moon), and -4' for a star or planet. This was a "rough and ready" combination of average dip of the horizon and approximate Moon/Sun SD (semi-diameter). The observed LD was also adjusted by the true values of the SDs of the two bodies, in this case calculated as exactly as possible. In the simplest case of a Moon-Sun lunar, this meant adding the Sun SD and also adding the Moon LD (augmented slightly by the Moon's altitude) to the LD from the sextant. Note that "index correction" or "sextant error" as they would have said often back then was handled first and with no ceremony. With those three "pre-cleared" quantities in hand, namely the altitudes above the true horizon of the centers of the Sun and Moon and the center-to-center lunar distance, the somewhat more detailed lunar "clearing" process would then follow. The key here is that you will typically find that the Moon and Sun altitudes entering the clearing computation are related to the raw sextant altitudes by +12'/-20' corrections. Because the process of clearing a lunar and getting the corrected lunar distance from it has very low sensitivity to altitude errors, these crude altitude corrections were sufficient for the most part.

In the actual lunar clearing process, we remove the effects of the Moon's parallax in altitude and the refraction for the two bodies. There were a number of ways to do this. In Bowditch's "New American Practical Navigator" there were several, but you have to be careful since the included methods and the numbering changed over the decades. Luckily for us in 2025, scans of these navigation manuals have been available online for 15-20 years, depending on the specific edition. I have long maintained a list of links to these accessible from the NavList main web page here. If you, or anyone else reading along, would like to get into the specifics of those methods, you should grab an "early" Bowditch, say from 1810 or so, as well as an edition from 1837. Those will give you most of the variation in methods. It's not hard to pick out which method you're seeing in any historical context. Of course, Bowditch's Navigator was only one among several popular English-language navigation manuals. 

Fundamentally, the methods in Bowditch are "series" methods. The cleared lunar distance, LDc, is related to the pre-cleared lunar distance (as above), LDpc by:
  LD_c = LD_pc + δh_M × cos(α) + δh_S × cos(β) + Q,
where the δh's are the individual altitude corrections (parallax+refraction) and the cosines of the corner angles tell us what fraction of those altitudes act along the lunar arc (in my notation, α and β are the angles in the triangle made by the Moon, Sun, and Zenith at the Moon and Sun respectively) and finally Q is a small "quadratic" correction, amounting to a couple of minutes of arc of less which could be listed in a small lookup table. Although there were different names for methods in that "commercial heyday" of lunars, most of the variety consisted of different means of calculating values for cos(α) and cos(β).

You'll also notice on these little computation pages the work for the local time. This calculation was standardized in a popular logarithmic form. You can spot it by looking for Altitude+Latitude+P.D. (that's polar distance, 90°+/-Dec) which is then divided by two to give the halfSum of "hSum" as we call it in my "Celestial Navigation in the Age of Sail" workshop. There are a few more steps and especially some logarithms, but the key feature that you can spot a mile away is that sum of three angles divided by two. And notice that the Sun's altitude that enters this computation is the same (corrected) altitude that enters into the lunar clearing process. It has been adjusted for dip and Sun SD, and typically no allowance is made for refraction since altitudes were usually high enough to ignore it.

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA