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George H, you wrote:
"However, I've always thought that procedure to be geometrically exact 
(assuming a spherical Earth, anyway) , without having any good basis for that 
confidence."

Yes. Exactly. The Dunthorne formula is mathematically identical to any of the 
standard direct triangle solutions. And also, the equations recommended in 
the 1906 "Lehrbuch der Navigation" over the Dunthorne formula are 
mathematically identical. They all necessarily give the same results.

And you wrote:
"If the Lehrbuch provides any reasoning, or examples, to justify their 
distrust, at lunar distances departing from near-90�, it would be interesting 
to learn."

It does not give any justification or explanation. 

But there is another reason for prefering the tan/cotan formula that the 
Lehrbuch recommends. It can be worked entirely in logs. The Dunthorne formula 
requires a mixed calculation, partly in logs of trig functions and partly in 
natural trig functions (at least I don't see any way to do it without a mixed 
calculation). That's a disadvantage of the Dunthorne formula, but it doesn't 
have anything to do with the stated range limitation or any inaccuracy. 

I am more convinced than before that the Lehrbuch was simply mistaken on this 
matter. The year 1906 was a long, long time after the heyday of lunars, and 
it's not uncommon to see confused or irrelevant information regarding lunars 
in those later sources.

-FER



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