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    Re: Planisphere pour les Distances Lunaires
    From: Frank Reed
    Date: 2009 Jun 3, 18:33 -0700

    I wrote previously:
    "His equation "(1)" with minor variations in grouping the terms had been 
    well-known to nautical astronomers and mathematicians who studied lunars for 
    the better part of a century. It consists of two linear terms and between one 
    and three quadratic terms (depending on where the series was truncated). You 
    can find the same equation derived in the rather long paper by Mendoza y Rios 
    published in the Transactions of the Royal Society back in 1797. "
    
    Actually, looking it over further, Hue has made a bit of a mistake in 
    evaluating the importance of the quadratic terms, and in fact this makes his 
    solution slightly worse than what had been available to navigators since the 
    early 19th century. Again, this strikes me as fairly typical of the period of 
    obsolescence of lunars. 
    
    Details:
    While deriving the series expansion, at the bottom of page 3, Hue has a 
    remainder term R which is proportional to the product of the altitude 
    corrections of the Moon and the other body and inversely proportional to the 
    sine of the lunar distance. He demonstrates that this is small enough to 
    ignore under many circumstances, but in fact it's comparable in magnitude to 
    the cross-term in the "x squared" term which he has chosen to keep. The 
    result is that his version of the quadratic expansion is no better than if he 
    had kept the terms proportional to m squared only. This was acceptable in the 
    late 18th and the very early 19th centuries, but most published versions of 
    such series expansions had moved beyond this level of accuracy some four 
    decades earlier (Thomson in 1824, Bowditch in 1826). I should emphasize that 
    this is a SMALL issue for Hue's analysis, introducing errors on the order of 
    a tenth of a minute of arc. It doesn't mean that the device doesn't work or 
    that the approach is fundamentally flawed. But those errors were 
    un-necessary.
    
    By the way, there's a footnote in the "Notice" that I didn't see before where 
    Hue comments that his derivation of his equation (1) could have been done 
    more easily by a straight Taylor series so he is at least aware that he is 
    re-inventing the wheel here.
    
    -FER
    
    
    
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