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From: Courtney Thomas (no email)
Date: Mon May 30 2005 - 11:09:46 EDT
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?
> >>>>>
> >
> >
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