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Subject: Re: Tables vs. Calculators
From: Vic Fraenckel (vfraenc1@XXX.XXX)
Date: Fri Sep 20 2002 - 15:39:33 EDT
Chuck,
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.
HTH
Vic
________________________________________________________
Victor Fraenckel - The Windman vfraenc1@XXX.XXX
KC2GUI www.windsway.com
Home of the WindReader Electronic Theodolite
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-Count Oxenstierna (ca 1620)
----- Original Message -----
From: "Chuck Taylor" <ctaylor@XXX.XXX>
To: <NAVIGATION-L@XXX.XXX>
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
|
|