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Dear Ken,

Thank you for your comment. Indeed, the procedure that you describe (method
of averages or method of equations of condition; the latter name is used in
the 'General History of Astronomy, vol.2B') incorporates the making of
arbitrary choices. The goal of making the combinations is, in modern
terminology, to make an (almost) diagonal system of n linear equations in n
unknowns.

Mayer used that technique for the determination of three parameters of
lunar libration out of 27 observations of the position of the crater
Manilius on the visible disk. A detailed description of the project
including the use of the method of equations of condition was published in
the 'Kosmografische Nachrichten und Sammlungen auf dem Jahr 1748'. It has
been suggested that he used the same technique for the perfection of his
lunar tables. The suggestion is pure speculation; as far as I know nobody
before me has made a serious study of Mayer's manuscripts to verify it (if
you know of somebody who did, please let me know). Nor did Mayer himself
explain or hint at his method. Looking at the manuscripts I found that the
method of equations of condition plays at most a minor role in the process.

Steven


>On 13 Jun 2003, at 21:06, Steven Wepster wrote:
>
>>  The main reason for Mayer's
>> success is that he managed to fit over 20 parameters in his theory to
>> over 200 observations of eclipses and occultations made after the
>> invention of the telescope. In my dissertation I want to answer (a.o.)
>> the question how he did the fitting, before the invention of the least
>> squares method.
>
>According to an article by P.J.G. Teunissen published in De Hollandse
>Cirkel, 2 nr. 1/2, April 2000 (but reproduced at
>http://www.ncg.knaw.nl/uk-site/Teunissen.htm)
>Mayer invented the method of averages for fitting arbitrary parameters.
>Briefly, if you have more equations than unknowns, you divide up the
>equations into as many groups as there are unknowns (a subjective
>procedure that makes the technique a bit dodgy), sum (or average) the
>equations in each group to produce n equations in n unknowns, and then
>solve for the unknowns. This was the first technique to incorporate all
>observations but it replaced the subjective act of throwing out some
>observations with the subjective act of grouping the equations (and so
left
>room for improvement).
>Ken Muldrew.

   

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