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    Re: Tables vs. Calculators
    From: Vic Fraenckel
    Date: 2002 Sep 20, 15:39 -0400

    Are you acquainted with the works of Jean Meeus, specifically his
    "Astronomical Algorithms"? He has a chapter devoted to Interpolation and
    covers the subject somewhat by discussing 3 and 5 variable interpolation,
    interpolation with LaGranges method, extremum and zero valued interpolation.
    I also draw your attention to "Fundametal Ephemeris Calculations" by Paul
    J. Heafner.
    Victor Fraenckel - The Windman                 vfraenc1@nycap.rr.com
    KC2GUI                                                      www.windsway.com
          Home of the WindReader Electronic Theodolite
                                   Read the WIND
    "Victory at all costs, victory in spite of all terror, victory however long
    and hard the road may be; for without victory there is no survival."
    - Winston [Leonard Spencer] Churchill (1874 - 1965)
    Dost thou not know, my son, with how little wisdom the world is governed?
    -Count Oxenstierna (ca 1620)
    ----- Original Message -----
    From: "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.
    | 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
    | calculations.
    | 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.  The
    | traditional solution was to use more tables, specifically tables of
    | so that multiplication could be converted to addition, and division to
    | subtraction.
    | The next logical step was to combine trigonometric an logarithm tables, so
    | one could look up, for example, the log-sine of an angle (the logarithm of
    | sine).  Then came variations on the same theme, such as tables of
    haversines and
    | log-haversines.
    | Next came various other sets of tables intended to speed up the process of
    | 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 reverse
    | engineer the the interpolation tables (difference and
    | 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
    | it.
    | 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
    | 229.
    | 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 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
    | 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 in
    | keeping with the spirit of traditional navigation.  The calculator simply
    | what you could do with more time and effort.  There is nothing mysterious
    | 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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