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    Re: Impossible lunar example
    From: George Huxtable
    Date: 2010 Aug 29, 15:01 +0100

    Thanks to Kent for his offer to supply a .pdf of 2nd edition Tables 
    Requisite of 1781, but as he realised, Paul's useful pointer to a back-door 
    access, not directly available via Google Books, met the need.
    Indeed, it has allowed me to take a look at the four examples Maskelyne 
    provides for clearing lunar distances, of which the "impossible lunar" 
    we've been looking at, copied for us by Dave, was only the most obvious 
    case. The four examples were tackled by two different procedures.
    They are contained in a section, headed "Problem X", on pages 27 to 36 of 
    Maskelyne's "explications", at the back of the book, after the tables 
    themselves. If anyone would like to see images of those pages, just ask. 
    But for now, I will summarise. In each case any corrections have already 
    been made, off-page, for index error, dip, and semidiameter. So these are 
    apparent altitudes, above true horizontal, of the centres, and apparent 
    distances between centres. The positions of the centres, and the observer's 
    zenith, provide the corners of the spherical apparent-triangle that's being 
    considered. The object is to correct for refraction and parallax, to arrive 
    at a true distance between those centres. But what we are interested in for 
    now is just the reality of that apparent-triangle, the three sides of which 
    are the zenith distances (zd) of the two objects, and the distance between 
    them. No side of such a triangle can ever exceed the added lengths of the 
    other two.
    Example 1.
    Star altitude, 24º 58', Moon altitude 12º 30', distancºe 51º 28' 35".
    065º 02' zd of star
    077º 30' zd of Moon
    132º 32' sum of above
    051º 28' 35" apparent distance. No problem.
    Example 2.
    Sun altitude 84º 07', Moon altitude 05º 17', distance 90º 21' 13"
    005º 53' zd of Sun.
    084º 43' zd of Moon
    090º 36' sum of above
    090º 21' 13" apparent distance. No problem. Sun and Moon nearly opposed in 
    Example 3.
    Star altitude 05º 06', Moon altitude 88º 46', distance 89º 58' 06"
    084º 54' zd of star
    001º 14' zd of Moon
    086º 08' sum of above
    089º 58' 06" apparent distance. Impossible triangle, discrepancy nearly 4º.
    Example 4.
    Sun altitude 19º 03' 36", Moon altitude 71º 06' 02", distance 103º 29' 27"
    070º 56' 24" zd of Sun
    018º 53' 58" zd of Moon
    089º 50' 22" sum of above
    103º 29' 27" apparent distance. Impossible triangle, discrepancy > 13º
    So, two out of four of Maskelynes triangles were quite impossible. 
    Presumably, he just hadn't bothered to check that aspect. It shows the 
    weakness of made-up, invented, examples. Which is why I admire the ones 
    that Jeremy proffers, based as they are on real-life. I don't think anyone 
    has yet found a hole in any of the examples he has set.
    It will be interesting to read the dialogue, referred to by Frank, between 
    "Nauticus" and Maskelyne, to see how he defends himself. Frank tells us "he 
    answered the complaints in some detail and with considerable anger, too". 
    Me, I don't think he has a leg to stand on.
    Thanks to Frank for alerting us to these further details of an interesting 
    byway in navigation.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

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