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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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