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    Re: Bowditch Table 9
    From: George Huxtable
    Date: 2002 Apr 13, 20:36 +0100

    In reply to Martin Gardner-
    I am pleased to hear, from Herbert Prinz, that in a later edition of
    Bowditch these errors have been corrected.
    Here is the expression I used, on a Casio FX730P pocket calculator.
    I haven't bothered to implement any minutes-to-degrees conversion, so you
    have to enter angle A in decimal degrees.
    If I put in A=1(degree) and H=1000(feet) this gives me D=8.8821322(miles),
    which compares well with the 8.9 miles given in table 9. It agrees,
    everywhere I've bothered to check, with the result in table 9.
    To break it down further, with those given inputs,
    TANA=.017455, TANA/.0002419=72.158, (TANA/.0002419)^2=5206.8
    5206.8+1360.7=6567.5, SQR(6567.5)=81.040
    If Martin is still having trouble, perhaps he can quote the exact
    expression he has used, and the machine he has used it on, give us a sample
    result, and we may be able to get to the bottom of it. But the first step,
    in a situation such as this, should always be to do a rough check
    calculation by hand rather than with a calculator/computer.
    A likely source of trouble occurs where a machine insists on working its
    trig functions in radians rather than degrees. The Casio calculator
    provides a choice (or grads, even!).
    George Huxtable.
    >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@huxtable.u-net.com
    >> George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    >> Tel. 01865 820222 or (int.) +44 1865 820222.
    >> ------------------------------
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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