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    Converting a Lunar Distance to GMT
    From: Bruce Stark
    Date: 2003 May 5, 22:39 EDT

    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
    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
    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

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