NavList:

After spending an evening trying to wrap my head around Peirce's writing on refraction (from the book available in the "Bowditch and More" section of the NavList website), I have some more thoughts on this subject.

The refraction formulas of the form "cot[H + A/(H+B)]" are basically an approximation of the formula developed by James Bradley of "R = 0.95' * cot(H + 3R)."  It has R on both sides so it has to be solved recursively, but it goes quickly with a decent initial guess of R.  If you consider refraction being a function of cot(H) and, for small angles, cot(H) is approxmiated by 1/(H), the "A/(H+B)" part just serves as a first approximation of R, multiplied by some scaling factor. 

As for the original questions looking for rigor, asking about where certain constants come from and what they mean...

Tables based on Bradley's formula were included in the American Practical Navigator from the beginning through at least the 1880 edition, and so it's the model Peirce expanded upon in his book.  I suspect Peirce worked backwards from Bradley's formula.  I haven't found anything from Bradley himself explaining how he came up with it or how he thought it worked.

Unless I've missed something, the assmptions of Perice's model are:

• Air's index of refraction decreases with height, or at least never increases
• Refraction is very small compared to zenith distance (i.e. 90-H)
• The atmosphere (or at least that portion responsible for measurable refraction) is very thin compared to the radius of the earth

And that's it.  I don't see any real "sins" in the model, probably because it doesn't attempt to pre-determine any necessary constants.

As for those particular constants used in the tables, from what I've seen so far Bradley stuck with "3" because it was a nice round number that made computations easier.  But I find the 0.95' (specifically 57.035") on the outside suspiciously close to the refractivity of air at sea level, and Peirce's explanation seems to suggest that it should be.