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Re: Bowditch Table 9
From: Martin Gardner
Date: 2002 Apr 12, 07:34 -0700
From: Martin Gardner
Date: 2002 Apr 12, 07:34 -0700
George, Thanks for looking at this. I did in fact try the mix you suggested, putting only the first two terms in the sqrt, and matching the three zeros. My results were still wrong. Maybe this _was_ a typo of mine. Would you be good enough to give me your partial results for any easy case so I can check again? Alternatively, if you used an excel spreadsheet to check, you could just attach that to a note. Martin > Martin Gardner said > >> ...the formula to compute table 9 in Bowditch. >> >> To review: Table 9 is "Distance by Vertical Angle Measured between Sea >> Horizon and Top of Object Beyond Sea Horizon". >> "Angle" runs from -4 minutes (a puzzle in itself) to +30 degrees; >> "Difference in feet between height of object and height of eye of observer" >> runs from 25 feet to 2000 feet >> >> The formula given in my 1981 Bowditch is >> >> Distance = sqrt ( >> ((tan A)/0.0002419)**2 >> + (H-h)/0.7349 >> - (tan A)/0.002419 >> ) >> >> I plugged this into my calculator and got hopelessly wrong results. >> Naturally I figured I'd keyed wrong, and checked and checked. (Maybe I did >> key wrong......but damned if I can find it) >> >> Then I looked more closely: >> >> (1) I do match table 9 for A=0 - so I keyed the middle of the three terms >> correctly. >> >> (2) An Ocean Navigator reference page on navigation repeats this formula, >> but in their case both of the ...2419 constants have three leading zeros, >> contrary to my Bowditch. >> >> (3) The current online Bowditch leaves the third term outside the sqrt - >> probably a typesetting error. >> >> I tried these variations of the formula without success. > > Martin should have tried combining (2) and (3) together. Presumably, it > isn't a typesetting error in the later Bowditch. Martin has transcribed > correctly the expression from the earlier Bowditch, but that expression > appears to be wrong. The square-root symbol should cover just the first two > terms of the three, and the two constants ...249 should both be preceded by > three zeros. Then the right answers come up. > > The correct formula seems to be- > > Distance = sqrt(((tan A)/0.0002419)**2 + (H-h)/0.7349) - (tan A)/0.0002419 > > so- shock! horror!! an expression, in Bowditch of all things, with not just > one error, but two! Martin Gardner has done well to spot the discrepancy. > > The "puzzle in itself", as Martin describes it, occurs when the elevation > of an object, above and behind the horizon, is negative. This covers the > situation of an object which is not behind the horizon at all, but between > the observer and the horizon, and sufficiently low that the observer can > see over the top of it to the horizon, above and behind it, contrary to the > title of the table. > > George Huxtable. > > ------------------------------ > > george@huxtable.u-net.com > George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. > Tel. 01865 820222 or (int.) +44 1865 820222. > ------------------------------