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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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

```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-----
[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
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
simply does
what you could do with more time and effort.  There is nothing
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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