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Roger:

You should be a bit wary of that article by Leeuw (1934). In fact anything in the history of astronomy and related topics older than fifty years (-ish) is suspect. Leeuw, for example, repeats an old and thoroughly discredited claim that "the caliph caliph Omar had already in 641 burnt the precious library of Alexandria" (propaganda). Even the story of the all-consuming fire itself is now understood as legend. And while I'm here, Cotter's "History of Nautical Astronomy" that you mentioned in your first message, despite its rather seductive properties, has many flaws. The chapter on lunars is notoriously awful, clueless, poor history.

Back to you original query, it seems to me that you're trying to figure out what sort of observational evidence would have led those ancient pre-telescope astronomers to think that refraction was an issue and what sort of mathematics they knew (even two millennia ago) that might have been applied to the problem. Is that about right? It really is a fascinating problem, and Leeuw's description of Tycho Brahe's difficulties is a fair beginning. Parallax and refraction are both concerns. How do we disentangle them? What logic could they bring to bear a few centuries ago to separate the two corrections to a celestial body's altitude? What assumptions factored in? And it's all part of bigger problem in the observation of the skies: where does meteorology end and astronomy begin? There's prominent linguistic evidence of the fuzzy dividing line: meteors are astronomical, not meteorological. To us it's obvious. Not so historically.

By the way, Alex Eremenko posted a reply to your question that you may have missed (you two were both writing/posting at about the same time). Here's his message.

Frank Reed