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    Re: Bowditch Table 9
    From: Martin Gardner
    Date: 2002 Apr 12, 07:34 -0700

    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 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---.u-net.com
    > George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    > Tel. 01865 820222 or (int.) +44 1865 820222.
    > ------------------------------

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