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Wolfgang, you wrote:
"No apologies needed, your reply was confusing, but didn't confuse me."

Now that's mighty darn confusing. Are you saying that you were not confused by 
my comments but you know that other people WERE confused?? Have you become a 
mind-reader in the past few months, Wolfgang? Again, I am sorry that I 
confused you.

And you wrote:
"You yourself were confusing reasons "mathematical" and "practical" because 
you jumped to conclusions without addressing the possibility that there might 
be a "mathematical reason" to the caution mentioned in the German manual."

Ridiculous. It is well-known that the Dunthorne formula is mathematically 
identical to the standard direct triangle solution of the problem of clearing 
lunar distances (mathematically identical in the same sense that sin(2*x) is 
identical to 2*sin(x)*cos(x)). It is simply incorrect to claim that the 
Dunthorne formula has the limited range of validity decribed in the book. 
Now, I understand that you do not know this mathematical material. There's 
nothing wrong that. But I DO know this material, and you have no business 
ranting about me "jumping to conclusions".

And you wrote:
"This was no "account" but a serious treatment by professionals writing a
standard textbook for the German Navy. Why should this account be "muddled"?"

The fact that it was a serious textbook (and I agree that it is) does not mean 
that it is perfect. I looked through some other chapters and it looks like a 
fantastic text. Really great stuff. But what you seem to have trouble 
understanding is that lunars were obsolete in 1906. They still taught them in 
school but it was decades since they had been commmonly used at sea. It's not 
unusual for people to have mistaken notions about techniques that are no 
longer widely used.

And you wrote:
"I am inclined to believe that you have absolutely no idea to which extent
error analysis of different methods of position determination was going on
in the circles of German navigational schools at that time."

You're right. I know very little about that error analysis. But I do know THIS 
topic inside and out, and the claim in the Lehrbuch is simply not true.

I concluded my previous message saying:
"I can only speculate since: 1) I don't have the original text and 2) I
don't read much German."

And, Wolfgang, you replied:
"That's right: You're speculating."

er... yeah. THAT'S WHY I SAID I WAS SPECULATING. However, later today I found 
the original textbook on Google Books and I have read and translated that 
chapter. So now I do not need to speculate, and I can confirm that my 
speculation was reasonable, and I can confirm that the book is wrong. 

I stand by what I wrote in my original post on this topic. There were good 
practical reasons for shooting lunars around 90 degrees but they had nothing 
to do with any imagined failings of the Dunthorne formula or any general 
problem with clearing lunars. There were also some mathematical reasons for 
prefering lunars around 90 degrees when using series solutions, but they 
don't apply here. This was all worked out in amazing detail over a century 
before this 1906 textbook. And I reiterate, one should be careful when 
reading accounts of lunars from this late date since lunars had gone out of 
common use at sea fifty years earlier.

Finally, I should mention one other issue which may have led to the confused 
statement in the book. The Dunthorne formula gives the cosine of the 
corrected distance and then from that we do a reverse table lookup to get the 
corrected distance itself. But picture the graph of the cosine around zero. 
For any angle close to zero, the cosine is nearly one. That means that this 
formula has trouble distinguishing different corrected angles when the 
distances are very short. And that would be a good reason for avoiding lunars 
within 20 degrees of zero --but clearly not the range stated in the book. 
Since those short lunars were avoided traditionally in any case, it's not 
really an issue.

-FER


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