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    Re: Tables vs. Calculators
    From: Arthur Pearson
    Date: 2002 Sep 20, 18:12 -0400

    Very well stated argument, I agree. Even for those of us who can't
    program, replicating table results with a calculator or a spreadsheet,
    or even solving problems with two different tables or methods and
    reconciling results, leads to a much better understanding of what is
    under the covers. I would argue that robust navigational practice should
    always be comparing the different sources and methods and applying
    judgment in the face of what are often inconsistent or conflicting data
    ("my DR says X, my fathometer says Y, my distance off that mountain
    suggests Z, I believe I am...").  The same applies to sight reduction in
    that comparing methods and their differences leads to a greater
    understanding what variables have the impact the accuracy of the
    results. My only real objection to any black box (from 229 to GPS) is
    when complete faith is placed in one and only one method of obtaining
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of Chuck Taylor
    Sent: Friday, September 20, 2002 7:32 AM
    Subject: Tables vs. Calculators
    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
    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.
    and dividing 5-digit sines and cosines can be a bit tedious, however.
    traditional solution was to use more tables, specifically tables of
    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
    and numerous others produced by various hydrographic offices around the
    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
    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
    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
    the main tables by computer, but I have been stumped at trying to
    engineer the the interpolation tables (difference and
    tables).  I even asked the folks at NIMA who publish the tables, and
    couldn't give me a satisfactory answer.  If I can't program it, I don't
    I would be very grateful if one of you could provide me with a set of
    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
    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
    but it still has many of the other characteristics of a black box.  (It
    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
    myself with going through a Taylor series expansion.  Because I can
    independently reproduce  what a calculator does, I trust it.  I don't
    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
    than the use of sight reduction tables such as Pub 229.  In that sense I
    argue that the use of calculators (programmable or otherwise) is fully
    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
    methods as tables of logarithms.  Why should we be shy about using more
    labor-saving devices?
    Chuck Taylor
    Everett, WA, USA

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