# NavList:

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

**Re: Tables vs. Calculators**

**From:**Vic Fraenckel

**Date:**2002 Sep 20, 15:39 -0400

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{at}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"To: 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 |