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