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    Re: Tables vs. Calculators
    From: Robert Eno
    Date: 2002 Sep 21, 11:32 -0400

    Interesting argument. I have to agree. I reduce my sights using a simple
    calculator and the spherical trigonometric formulas. It is fast and simple.
    When I am engaged practicing my sight-taking at home, I always reduce a few
    of them long-hand using the formulas in conjunction with tables of trig and
    log functions.
    Some will argue that a calculator is in itself, a black box, however, my
    argument is that it is simply an anti-drudgery device. It is no different
    than washing your clothes with a automatic washer, vs. a washboard and pail.
    What I find most useful about using the formulas and calculator, as well as
    trig and log tables, is that after a while, one develops a "feel" for the
    answers. In otherwords, one can sense when the answer "doesn't look right".
    This advantage is totally lost with most black box methods.
    ----- Original Message -----
    From: Arthur Pearson 
    Sent: Friday, September 20, 2002 6:12 PM
    Subject: Re: Tables vs. Calculators
    > Chuck,
    > 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
    > position.
    > Arthur
    > -----Original Message-----
    > From: Navigation Mailing List
    > [mailto:NAVIGATION-L@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
    > performing
    > 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.
    > 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
    > subtraction.
    > 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
    > log-haversines.
    > 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
    > it.
    > 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
    > 229.
    > 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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