NavList:

Fred Hebard, you wrote:
"You shoot the altitudes of the objects in a lunar in order to obtain their refraction correction, for which you don’t need accurate altitudes. The refraction correction is small and doesn’t change much, less than 4’ at 15 degrees altitude, and less than 1’ at 45 degrees. Tables of refraction [...] are enumerated in whole degrees.  So if you can get your altitude sight to within 1 degree of arc, you’re fine."

This is close, but it isn't quite right. When some of the earliest clear explanation of lunars were posted in NavList messages (back in the Navigation-L pure mailing list era), it was common to propose an explanation like the one you have written here to understand why the altitudes don't matter much. And indeed this is part of the story. But it's not the key component.

The reason we need the altitude of the Moon is primarily to determine its parallax correction which is relatively sensitive to altitude especially when the Moon is low in the sky. The altitude of the Sun, on the other hand, determines the position of the Sun relative to the Moon which determines what fraction of the Moon's altitude correction (which is primarily its parallax) is aligned along the lunar arc. 

An example:
Suppose the Moon is 45° high and the HP at this hour is 59.0'. Its altitude correction from parallax alone is 59.0·cos(45°) or about 41.7'. There's also about 1' of refraction which we can leave out if the other body is near the Moon in the sky (I'm also ignoring some minor issues).

Now suppose I measure the lunar distance between the Moon and some star close to the Moon in the sky. It comes out to be some angle which, after adding in the Moon's SD, gives a center-to-center lunar distance of exactly 10°. I want to correct for (remove) the effects of refraction and Moon parallax on this distance. Since the Moon and the star are at nearly the same altitude (+/-10° at most), the refraction corrections are almost identical. Does that mean we can ignore the other body's altitude entirely? Well, no.... because the other body's altitude "tells the equations" where that other body is located relative to the Moon. If the altitude of the star is 45°, same as the Moon, the two objects are separated horizontally which implies that the parallax correction --always vertical-- has very little impact on the lunar distance. Working it out in detail, the corrected lunar distance might be 10°00.5'. On the other hand, if the altitude of the other body is 55°, implying that the star is directly above the Moon, then to correct for parallax we have to add on the whole parallax correction, about 41.7', yielding a corrected distance of 10°41.7'.

See how this goes? In order to get the parallax correction right, we need the altitudes of the Sun and Moon accurate enough so that the math can determine those "corner cosines" (the factors deciding what fraction of the altitude correction acts along the lunar arc) with sufficient resolution. That's why the altitudes matter, and that's why they can be relatively low accuracy under normal circumstances. For a modern observer, if you want good results, you would typically want to measure the lunar distance itself with a good metal sextant equipped with a fairly high-power telescope (7x or above) capable of measuring angles to the nearest tenth of a minute of arc while the altitudes could be measured with a cheaper plastic sextant giving altitudes only to the nearest several minutes of arc.

Frank Reed