NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Martelli's Navigational Tables
From: Courtney Thomas
Date: 2005 May 30, 10:09 -0500
From: Courtney Thomas
Date: 2005 May 30, 10:09 -0500
Lu, Thank you for a very clear and concise reply to this matter. I only wish that 5% of all the math/physics textbook authors could write half as well as do you :-) Cordially, Courtney On Sun, 2005-05-29 at 21:56, Lu Abel wrote: > Courtney: > > Sometimes it's easy to forget how different sight reduction was from the > earliest days of celestial navigation until about 25 years ago when PCs > and pocket calculators arrived. Navigators had to rely on sight > reduction tables and/or longhand calculations of the celestial triangle > formulae. Given that accurate navigation requires 4~5 digit accuracy > in answers (dd mm.m), and a rule of thumb is that calculations should be > carried out with at least one more digit of accuracy than desired in the > final answer, longhand paper calculations must have been daunting indeed! > > One way to make things easier is to use logarithms for multiplication > instead of actually trying to multiply a pair of six-digit numbers. But > there's a problem: Logarithms are defined only for positive numbers and > sines and cosines can be negative as well as positive. Enter the > versine: Versine (x) = 1 - cos(x). As you can see, this simply > inverts the cosine curve and adds 1 to it, making it range between 0 and > 2. It's a bit more convenient to have a function that runs between 0 > and 1, so it's divided in half, giving the half versine or haversine: > hav(x) = (1 - cos(x))/2. > > The celestial triangle formulae involving sines and cosines can be > restated in terms of haversines. By using a trig function that is > always positive, it can be solved with the aid of a table of logarithms. > > In a quick search I can't find the celestial formulae exactly, but > here's a link to the formula for a great circle. Hc is simply 90 > degrees minus the great circle distance to the GP of the body. > http://www.mathdaily.com/lessons/Haversine_formula > > By the way, your GPS likely calculates great circle distances using this > formula rather than the traditional spherical triangle formula. That's > because calculating short distances using the traditional formula > requires taking the difference between two large numbers that are fairly > close to one another using the traditional formula. Tiny differences > due to rounding and a limited number of significant digits can result in > significant errors. (Interestingly, errors can creep into the haversine > formula with very long distances, but I suspect a one mile error in > calculating the distance between New York and Beijing isn't as > significant as a one-mile error in a local distance.) Calculations > aren't actually made using haversines, the haversine formulae can be > re-expressed in terms of ordinary sines and cosines and that's what's used. > > Lu Abel > > Courtney Thomas wrote: > > Being unfamiliar with the haversine cosine formula, can this be > > programmed into a calculator and subsequently submit the variables that > > immediately pertain, hence getting the Martelli result without carrying > > around tables ? > > > > If yes, where can this modified formula be found, please ? > > > > What is gained by the tables via-a-vis currnent methods, if anything, or > > is it be more appropriately deemed, an historical step in celnav's > > evolution ? > > > > Thank you again, > > > > Courtney > > > > > > On Sun, 2005-05-29 at 01:37, Victor Garand wrote: > > > >>Courtney, > >> > >>"The tables are based on a modified form of the haversine cosine formula. > >>They provide a rapid solution of spherical triangles of the celestial or > >>terrestrial sphere." > >> > >> > >>----- Original Message ----- > >>From: "Courtney Thomas"> >>To: > >>Sent: Saturday, May 28, 2005 2:38 PM > >>Subject: Re: Martelli's Navigational Tables > >> > >> > >> > >>>Please excuse my ignorance, but what is the value of Martelli's tables ? > >>> > >>>Thank you, > >>>C. Thomas > >>> > >>> > >>>On Sat, 2005-05-28 at 12:36, Victor Garand wrote: > >>> > >>>>Henry, > >>>>The 1952 edition (new GHA edition with additional examples and quick > >>>>reference charts (59 pages) ...) includes the following: > >>>>-Position Line (sun or star), longitude, latitude and intercept (St. > >>>>Hilaire > >>>>or calculated altitude) methods. > >>>>-Position Line (circumpolar star), longitude, latitude and intercept (St. > >>>>Hilaire or calculated altitude) methods. > >>>>-Deviation of magnetic compass. > >>>>-High-altitude ex-meridian. > >>>>-Amplitudes. > >>>>-Identification of stars. > >>>>-Great Circle distance and initial course. > >>>>-Calculation of points on Great Circle. > >>>> > >>>>Googling, I found that some used book dealers have a copy of these tables > >>>>but I couldn't ascertain the vintage. > >>>> > >>>> > >>>>----- Original Message ----- > >>>>From: "Henry C. Halboth" > >>>>To: > >>>>Sent: Friday, May 27, 2005 9:22 PM > >>>>Subject: Re: Martelli's Navigational Tables > >>>> > >>>> > >>>> > >>>>>I have used the 1914 edition and still do for the time sight solution. I > >>>>>really did not know that these tables had continued in print as late as > >>>>>1952 and must assume them to have been modernized to allow for an > >>>>>intercept and azimuth solution. > >>>>> > >>>>>On Wed, 25 May 2005 10:17:27 -0600 Victor Garand > >>>>>writes: > >>>>> > >>>>>>Is there anyone on the list who still uses these? My edition is a > >>>>>>1952 edition, is there a later edition? > >>>>> > > > >