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    Tables vs. Calculators
    From: Chuck Taylor
    Date: 2002 Sep 20, 11:32 US/PACIFIC

    Sight reduction tables have long been widely used by celestial navigators.  Why?
    The formulas for sight reduction by the law of cosines have long been known. The
    answer is pretty straightforward:  Tables are used to save labor in performing
    One can perform sight reduction by the law of cosines with with a set of
    trigonometric tables (sines, cosines, etc.) and a pencil and paper.  Multiplying
    and dividing 5-digit sines and cosines can be a bit tedious, however.  The
    traditional solution was to use more tables, specifically tables of logarithms,
    so that multiplication could be converted to addition, and division to
    The next logical step was to combine trigonometric an logarithm tables, so that
    one could look up, for example, the log-sine of an angle (the logarithm of the
    sine).  Then came variations on the same theme, such as tables of haversines and
    Next came various other sets of tables intended to speed up the process of sight
    reduction by combining various steps, relieving the navigator of still more of
    the labor of computation.  Examples include HO 214, Pub 229, Ageton's Tables,
    and numerous others produced by various hydrographic offices around the world.
    Many of us object to the exclusive use of "black boxes" such as GPS units on the
    grounds that it takes all the sport out of navigating if all you have to do is
    turn on the black box and observe your position (either the lat/lon or a mark on
    a chartplotter). We call it a "black box" because most of us don't fully
    understand how it operates, and we certainly can't duplicate its results by
    other means such as pencil and paper.
    We also believe that it is important to use the traditional methods in order to
    maintain our skills.  Who knows, the black box may fail some day.
    I would argue that tables such as Pub 229 are an early form of "black box". At
    least many of us treat it as such.  We open to the appropriate page and extract
    numbers, trusting on faith that they are correct.  How many of us have tried to
    verify that those numbers are correct?  I have.  I can successfully reproduce
    the main tables by computer, but I have been stumped at trying to reverse
    engineer the the interpolation tables (difference and double-second-difference
    tables).  I even asked the folks at NIMA who publish the tables, and they
    couldn't give me a satisfactory answer.  If I can't program it, I don't trust
    I would be very grateful if one of you could provide me with a set of algorithms
    to reproduce the various difference and double-second-difference tables in Pub
    How can we logically dismiss the use of the "GPS black box" while simultaneously
    embracing the "Pub 229 black box"?  I'll grant you that the Pub 229 black box is
    less susceptible to failure due to causes beyond the control of the navigator,
    but it still has many of the other characteristics of a black box.  (It is
    certainly easier to carry a spare GPS than a spare set of the various volumes of
    Pub 229.)
    To me a calculator is less of a black box than a set of tables.  I can
    reproduce the calculator's results using pencil and paper and a bit of time and
    effort. I could even reproduce the sines and cosines if I wanted to trouble
    myself with going through a Taylor series expansion.  Because I can
    independently reproduce  what a calculator does, I trust it.  I don't trust
    tables that I can't reproduce.  (I do trust the Ageton tables, because they are
    more easily reproduceable).
    In this sense, the use of a calculator is arguably less of a black-box operation
    than the use of sight reduction tables such as Pub 229.  In that sense I would
    argue that the use of calculators (programmable or otherwise) is fully in
    keeping with the spirit of traditional navigation.  The calculator simply does
    what you could do with more time and effort.  There is nothing mysterious about
    it.  Those who came before us weren't a bit shy about using such labor-saving
    methods as tables of logarithms.  Why should we be shy about using more modern
    labor-saving devices?
    Chuck Taylor
    Everett, WA, USA

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