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Re: Bowditch Table 9
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---.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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