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    Re: Martelli's Navigational Tables
    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?
    > >>>>>
    > >
    > >
    
    
    

       
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