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Kent,

I took an interest in your question about Dunthorne's formula. Further 
stimulated by Wolfgang's remark about the Germans having done a lot of 
error analysis in this field, I am pursuing it.

Karsten's Allgemeine Encyklopaedie is online at

http://www.archive.org/details/allgemeineencyk00encygoog

The relevant paragraph 221 starts at p. 774. It is a good description of 
the methods available at the time, their historical evolution and the 
reasoning behind them. (The presentation is in several respects superior 
to Cotter's later attempt.) Accuracy is frequently mentioned as the 
motive for the introduction of a new method, but to my disappointment, 
no error analysis is provided for any of them. Nevertheless, one foot 
note leads to another and I hoped to gain more insight from an article 
cited on p. 784 and again on p. 801 in which Bremiker investigates the 
error according to the principles established by Gauss in his Theoria motus..

Bremiker, Astronomische Nachrichten 1850, Bd 30, p.311 - 318

I could not find the publication online and got it only last Friday 
through ILL, so I have not come very far with it. The purpose behind the 
paper is a sales pitch for a formula of Bremiker's own that he 
apparently presented in the Nautische Jahrbuch pro 1852. (If anybody has 
access to that, I would like to see it.) The article is difficult 
reading, because Bremiker presents results without showing his work. So 
far, I am having trouble to verify his numbers. In particular, for 
Dunthorne's formula I get half the errors that Bremiker claims. I also 
find contradictions in the text that I cannot resolve. It's premature 
for me to discuss it, but I thought I mention the source as one of 
possible interest to you. I shall have more to say at a later time.

A direct answer to your original question regarding the alleged 
limitation of  Dunthorne's formula is not to be found in this article, 
because Bremiker looks generally only  at the accuracy of distances in 
the interval from 20 to 120 degrees. At least for his own formula and 
for Dunthorne's.  Only when he gets a chance for a polemic against 
Borda, a distance of  5 deg all of a sudden comes into play. Bremiker 
does not reveal how his own formula does under those circumstances.

I have posted a scan of the article (4 jpeg pages, browseable) at

http://poorherbert.org/astronachr1850

Best regards,

Herbert Prinz



KENT AE NORDSTR�M wrote:

 >
 >
 > Again thanks for your views regarding possible limitations in lunar 
distances. I have gone through almost a dozen old manuals and I have not 
found any stated limitation such as in the German manual from 1906. I 
didn�t expect to find anything either. The only limitation I have found 
in e.g. Tables Requisite from 1766 is about very short lunar distances, 
which should require 6 fig logarithms and not 5 fig as usually used. The 
limitation is in this context of no interest because distances less than 
about 30 degrees were not taken because short distances were not 
tabulated in old NA�s. But there might be of interest to check the 
background to the German manual with an other approach. On the pages 
Wolgang K�rberer kindly provided there is a foot note down on page 385 
saying (my translation) that a comprehensive explaination for these 
methods can be found in �Allgemeinen Encyklopadie der Physik� issued by 
G. Karsten, in the chapter about �time and finding positions�, this 
chapter written by G.D.E. Weyer. I have tried to find this reference on 
the internet without success. I am going to alert my collegue about this 
hoping for an answer. But perhaps somebody else has the above 
encyklopedie available and would like to check.
 >
 > Kent N
 >
 >  >




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