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    Re: Bowditch Table 9
    From: George Huxtable
    Date: 2002 Apr 12, 09:35 +0100

    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
    >(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, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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