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    Re: Fw: Chichester's Calculations ( Sine-Cosine method)
    From: Gary LaPook
    Date: 2008 Dec 21, 03:45 -0800

    I don't know if anybody noticed the similarity in the two solutions 
    using Dreisonstok (post 6744) and Weems' Line of Position Book (post 
    6745.) The computation of Hc using both methods involves the same values 
    from the Log-Trig tables. While Dreisonstok tabulates a "K" value Weems 
    tabulates a "b" value. Notice these are inverses of each other ( K = 90 
    - b) and (b = 90 - K).  The intermediate values, "K~d" for Weems and "d 
    + b" for Dreisonstok are also inverses of each other. The "A" columns in 
    the first table in each volume are identical and the "B" column in the 
    second table are inverses of each other. Both of these sets of tables 
    use the same formulas for calculating  Hc only rearranged slightly. I 
    have attached this comparison.
    
    I also wrote that Weems had been my favorite but that I had switched to 
    Dreisonstok because I didn't like the Rust diagram so much. Now that I 
    have reviewed the method shown by Weems in his 1944 edition, "New Line 
    of Position Tables," to compute the azimuth with trig I am switching 
    back to his Line of Position Book as my favorite tabular method. This 
    method can also be used with his original tables since  Table B in each 
    book is identical. Table B consists of log secants across the top and 
    log cosecants across the bottom. Azimuth is computed with the formula 
    cosecant Z = cosecant LHA times secant declination divided by secant Hc. 
    Using logs this is solved as log cosecant LHA + log secant declination - 
    log secant Hc.These values are taken from Table B. This computation 
    requires four entries to Table B and one addition and one subtraction. 
    Since computing Hc with Weems takes three table entries and two 
    additions to also compute Z with Weems takes a total of seven table 
    entries and four maths. This compares with Dreisonstok that only 
    requires four table entries and four maths. Although Weems takes three 
    more table entries this is compensated by the fact that Weems is only 54 
    pages compared to Dreisonstok's 108 pages, exactly twice as thick. I can 
    also choose to use the Rust diagam in appropriate cases.
    
    I am attaching the pages of Weems to allow the computation of azimuth to 
    go along with the illustration of finding Hc done in post 6745.
    
    gl
    
    
    
    Gary J. LaPook wrote:
    > I am attaching an example of the same computation done with Weems Line 
    > Of Position Book which provides the shortest solution of all the tabular 
    > methods. This requires only three entries to the tables to find Hc and 
    > then the azimuth is found by Rust's diagram. The example requires only 
    > two pages because two of the values are found on one page. See attached 
    > files.
    >
    > gl
    >
    >
    >
    >
    > Gary J. LaPook wrote:
    >   
    >> I also decided to work the problem using H.O. 208, Dreisonstok for 
    >> comparison. This method requires four table entries taking out eight 
    >> values and four additions.
    >>
    >> See attached example and H.O. 208 excerpts.
    >>
    >> gl
    >>
    >> Gary LaPook wrote:
    >>   
    >>     
    >>> I also decided to work Chichester's example using my Bygrave Slide Rule 
    >>> since Chichester used one of these when he flew his Gypsy Moth airplane 
    >>> from New Zealand to Australia in 1931. I came up with an Hc of 46º 26', 
    >>> Az of 57º10' the same as Chichester got. I have attached this computation.
    >>>
    >>> gl
    >>>
    >>>
    >>> Gary J. LaPook wrote:
    >>>   
    >>>     
    >>>       
    >>>> I decided to work the same example using the standard Sine- Cosine 
    >>>> method to compare it to the Haversine-Cosine method. I discovered that 
    >>>> it takes fewer table entries and additions than does the Haversine 
    >>>> method so I do not know why that method became the prefered method, any 
    >>>> ideas?
    >>>>
    >>>> I am posting and example of this computation. I am also attaching a PDF 
    >>>> of it to ensure that the format doesn't become corrupted.
    >>>>
    >>>> The normal formulas for computing Hc and azimuth are:
    >>>>
    >>>> sin Hc = sin Lat  sin Dec + cos Lat cos Dec cos LHA
    >>>>
    >>>> and
    >>>>
    >>>> sin Z = (sin LHA cos Dec)/ cos Hc
    >>>>
    >>>> rearranged to the more convenient
    >>>> sin Z = sin LHA cos Dec sec Hc
    >>>>
    >>>> These can be solved using logarithms  using this format:
    >>>>
    >>>>
    >>>>       
    >>>>     
    >>>>       
    >>>>         
    >>>     
    >>>   
    >>>     
    >>>       
    >>     
    >>   
    >>     
    >
    >
    > >
    >   
    
    
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