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    Re: Bowditch 2002
    From: Mitch B Burrill
    Date: 2002 Oct 4, 10:19 -0400

    Jared's point on accuracy is well taken, but one must use five-place trig 
    values to get full sextant ( 0.1-0.2 min ) accuracy. I've had the privilege 
    to help train many young engineers, and I've found that one of the toughest 
    things to inculcate is "enough with the 6 decimal places already !".
    I told my HP48 to leave default mode, gave it 7 decimal places, asked it for 
    5, and it disagreed with Bowditch. How many  places should I give it ? I 
    think this is interesting because when Briggs published his 14 place log 
    tables in 1624, it contained over 1000 errors, almost all of them in the last 
    decimal place (still an astonishingly low error rate).
    Anyway, to digress : anyone out there have any practical experience with 
    longitude by Jovian moons? I found a website giving all the transit/eclipse 
    data and am ready to start. Hurricane Lili is not cooperating.....
     -----Original Message-----
    From:   Herbert Prinz [mailto:hprinz@ATTGLOBAL.NET]
    Sent:   Thursday, October 03, 2002 5:20 PM
    Subject:             Re: Bowditch 2002
    You are misunderstanding the idea about rounding to the next even or odd 
    number. This applies only to the case where you have exactly the sequence 
    5000... following the digit to
    which you are rounding. The goal is to avoid systematic rounding error while 
    still minimizing individual error. What you have suggested would produce 
    individual rounding errors
    greater than 0.5 times the place value of the place to be roundet to and would 
    thus be unacceptable as a rounding method.
    You are probably thinking of the special case where you are given n decimal 
    places and must round to n-1 places. Then you would correctly round to the 
    nearest even (or odd) decimal
    number if the n-th decimal is a 5. *) But when you produce tables such as in 
    Bowditch, you always compute your values to SEVERAL places more than you 
    tabulate, and you avoid
    systematic rounding error altogether by simply picking the nearest value to the full computed result.
    *) Interesting enough, the US Power squadron does not think so. For them you 
    must round up, or else you flunk their tests. Could it be that they want to 
    be able to check your work
    with a pocket calculator?
    Interesting enough
    Mitch B Burrill wrote:
    > The only algorithms I know are :
    > 1. Always round up, like a modern electronic calculator (which I think leads 
    to serious error accumulation in hand calculation)
    > 2. Round up if the digit to the right of the 5 is odd (or even)
    > and 3. Round up to make the digit to the left of the 5 odd (or even).
    > To get the values below (which are published in Bowditch 2000 AND the 23rd 
    edition of the CRC math tables),
    > Bowditch is using Round if Right is Odd.
    > But then that algorithm fails with the published values for 32-22 and 32-33.
    > I'm stymied (for now)  !!
    >  -----Original Message-----
    > From:   Herbert Prinz [mailto:hprinz@ATTGLOBAL.NET]
    > Sent:   Thursday, October 03, 2002 3:01 PM
    > Subject:             Re: Bowditch 2002
    > Well, Mitch, which rounding algorithms are you aware of?
    > Sin(32d 07') = 0.53164497... and Sin(32d 43') = 0.54048508...
    > So, rounded to 5 places after the decimal point, we would expect to see 
    0.53164 and 0.54049, using ANY rounding algorithm that I can possibly think 
    > What does Bowditch have?
    > Herbert Prinz
    > Mitch B Burrill wrote:
    > > Yes, examine the sine of  32-07, 32-22, 32-31, 32-33, 32-43, and 32-59. 
    The values given for 32-07 and 32-43 seem mutually exclusive using the 
    rounding algorithms I'm aware of.

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