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    Re: Converting a Lunar Distance to GMT
    From: Arthur Pearson
    Date: 2003 May 11, 18:07 -0400

    For fans of Bruce's tables, I am providing links to these supplemental
    explanations (this being the second) on the Nav-L section of
    www.LD-DEADLINK-com.  As a user of the tables and an enthusiastic
    follower of how they were developed and how they work, I hope Bruce will
    continue this series of postings about his tables.
    
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of Bruce Stark
    Sent: Monday, May 05, 2003 10:39 PM
    To: NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM
    Subject: Converting a Lunar Distance to GMT
    
    Dan Allen, George Huxtable and others pointed out some time ago that the
    cookbook explanations provided with my Tables for Clearing didn't
    satisfy
    everyone. Some people like to know more than just WHAT to do. It's a
    good
    point, and I finally began, in an April 28th posting, to deal with it.
    This
    is a continuation of that posting.
    
    The next step has to do with converting a cleared distance to GMT. That,
    in
    turn, will lead to a discussion of the whys and hows of the Tables for
    Clearing the Lunar Distance.
    
    Until about ninety years ago the Nautical Almanac gave pre-calculated
    comparing distances every third hour. Suppose you'd measured and cleared
    the
    distance between the moon and Regulus. You'd find, in the Almanac, the
    two
    tabulated distances of Regulus from the moon that your observed distance
    fit
    between. Then, proportioning change in time to agree with change in
    distance,
    you'd find what your watch would have read, had it been keeping
    Greenwich
    time, at the moment you measured your distance.
    
    The Almanac doesn't give distances now, but it does give the GHAs and
    declinations of the bodies for every hour. With an electronic calculator
    and
    the law of cosines for spherical triangles you can work out comparing
    distances yourself.
    
    But if you don't like to depend on electronics you'll need a more
    refined
    formula than the law of cosines. The cosine-haversine is ideal. You can
    use
    it with a set of nautical tables, such as Norie's, or those in the WW II
    era
    Bowditch.
    
    Or, if you like, you can do the job with my Tables for Clearing. They
    include
    a form for entering the Almanac data and the functions you'll need from
    the
    Tables. Then, after you've calculated comparing distances, you can use
    tables
    7 and 8 to proportion for GMT. There are advantages to using the Tables:
    
    1) You don't have to know anything about logarithmic calculation.
    
    2) You don't have to interpolate, or do any other mental arithmetic.
    
    3) The reliable precision will be slightly better, since nothing is lost
    in
    interpolation or in the conversion of logarithms to natural values.
    
    The formula I use to calculate comparing distances is simply the
    cosine-haversine. Fred has pointed that out. But the formula that clears
    the
    distance, and that led to the development of the tables, is more
    complicated.
    I had combined the old time sight formula with the cosine-haversine, and
    was
    trying to work it into an all-haversine equation. Three quarters of the
    way
    through a sheet of notebook paper the term (cos M cos S)/(cos m cos s)
    appeared in the equation. Everything else was in haversines, and that
    ratio
    of cosines obviously had a narrow range of values. Might its logarithm
    fit
    into a table? Later I realized it already was in a table, the
    "logarithmic
    difference" table used with Dunthorne's, Borda's, and similar methods of
    clearing.
    
    Here is the equation. Since I don't know how else to indicate it in this
    e-mail program, the phrase "sq. root of" will have to stand in for the
    radical sign.
    
    hav D = sq. root of {hav [d - (m ~ s)] * hav [d + (m ~ s)]} * [(cos M *
    cos S
    )/(cos m * cos s)] + hav (M ~ S)
    
    I've already pointed out that you don't have to understand logarithms to
    use
    the Tables for Clearing. You don't have to know you're using logs. In
    case
    anyone is interested, here's a brief explanation of why:
    
    The log of a number greater than one is positive. The log of a number
    less
    than one is negative. Nautical tables were designed to handle
    calculations
    that included a mix of positive and negative logarithms. Some
    calculations
    called for summing from three to six logs at once. Not handy if some
    were
    positive and some negative.  So +10 was applied to everything that went
    into
    the trig-log table. That way all the logs could be treated the same. But
    the
    navigator had to discard and borrow tens to suit his calculation.
    
    In the equation above only the logarithmic difference (the log of that
    ratio
    of cosines) and log haversines are needed. Both are always negative. So
    is
    the log cosine used to calculate comparing distances from the Almanac.
    All
    three are left negative. The Gaussian log, used to get past the + sign
    in
    front of "hav (M ~ S)," is always positive, so is subtracted. This
    simplifies
    matters, and saves figures.
    
    I'll try before long to post something about the individual tables in
    the
    set.
    
    Bruce
    
    
    

       
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