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    Re: Bowditch Table 15
    From: Jim Thompson
    Date: 2005 Jan 25, 10:20 -0400

    > -----Original Message-----
    > From: George Huxtable
    > I transcribed into emailese, writing A instead of Alpha, the expression in
    > my edition (1981 vol 2) of Bowditch as-
    > d = sqrt[(tan A/.0002419 )^2 + ((H-h)/.7349) - (tan A/.002419)]
    > Is the expression and the text the same in later editions?
    Jim: 2002 Edition:
    d = sqrt[(tan A/.0002419)^2] + ((H-h)/.7349) - (tan A/.0002419)
    > I wrote- To me, it seems a bit suspicious that that text refers to "The
    > constants .0002419 and 0.7349", although a third, and different, constant
    > of .002419 also appears to be used in the expression. Is that a
    > misprint, I wonder?
    > But it seems me there's more wrong with the expression I quoted above than
    > a simple matter of the number of decimal places in a constant term. When
    > the angle =0, then the expression gives the same results as does
    > the table,
    > but the two seem quite inconsistent in the way they vary with the angle.
    Jim: That's one of the things bugging me.  The angles also seem to vary the
    opposite when I draw solutions graphically.
    > I've no reason to believe that the numbers in the table are wrong (on the
    > other hand, no reason to be certain that they are correct). But
    > the numbers
    > in Table 9 and the expression in the text for Table 9 are quite
    > inconsistent.
    > Just to check that my Table 9 and the later Table 15 are the
    > same, here are
    > two spot values to compare-
    > angle = 0',  H-h = 100 ft., distance 11.7 miles
    > angle = 1 deg 50', H-h = 450 ft., distance = 2.3 miles
    > Does Table 15 give those same values?
    Jim: 2002 Bowditch: Yes, they are the same.
    Jim: In the past Nav-L posting
    Martin Gardner wrote this:
    "A couple days ago, I was fooling around putting some formulas on my Palm.
    One of them was the formula to compute table 9 in Bowditch. 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. More serious: as I
    read the formula, we are taking the sqrt of three terms of the form
    A**2 + B - A
    Which will get larger with increasing A (if A is > 1 - which is true for
    angles over 1 minute)
    So the formula, as I understand it, is going the wrong way - producing
    larger distances for increasing angles....
    Please help me find my error or -- gasp -- an error in Bowditch."
    Jim: George Huxtable wrote this in reply on Nav-L
    "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. "
    Jim: But I cannot see a bracket to close the sqrt opening bracket.
    Presumably George meant to put the closing bracket after the second term,
    Distance = sqrt[((tan A)/0.0002419)**2 + (H-h)/0.7349)] - (tan A)/0.0002419
    Jim: In any event, I think that Table 15 (was 9) and its formula remain a
    very dark secret, at least to me.
    Jim Thompson

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