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Gary LaPook writes:

Thanks George, I needed that.

gl

George Huxtable wrote:

>Alexander Walster asked-
>
>| I have a book called "Practical Navigation for Second Mates" and it
>| details the procedure for sight reduction using haversines.
>|
>| I am from the scientific calculator age and I was wondering if someone
>| had a simple explanation on how haversines can be used?
>
>=====================
>
>Response from George.
>
>Today, versines  and haversines have little purpose. But in their day, the
>period when logarithms were used for all precise calculation, they had
>significant advantages over sines and cosines.
>
>First, what is a versine? Versine of an angle A is abbreviated as vers A,
>and
>vers A = (1- cos A). It's as simple as that. And a haversine, or hav A, is
>just half that value.
>
>It's also true, as Henry Halboth has explained, that
>hav A = (sin A/2) squared, which has its uses.
>
>The main difficulty with calculating by logs is that the log of a negative
>number is meaningless. There are fiddles to get around the problem, as you
>will find used in Chauvenet. Usually, complex rules would be introduced into
>a calculation that used logs, to ensure that all quantities remained
>positive. The trouble in using sines and cosines with logs is that for
>angles less than zero, sines become negative, and for angles greater than
>90, cosines become negative.
>
>This led to recasting formulae that had used sines or cosines into using
>versines, which can never go negative. Just like a sine, versine of angle 0
>degrees is 0, and for 90 degrees, it's 1, though it's shape differs greatly
>from that of a sine curve between those values. However, above 90, the
>versine keeps on rising, until at 180 degrees, it's exactly 2; then it
>starts to fall again.
>
>Why was it called a versine? That has puzzled me. I have seen it "explained"
>in terrms of using a shortened form of "reversed sine", which doesn't make
>sense to me, because a versine isn't any sort of reversed sine.
>
>Anyway the awkwardness of a table of versines was that it varied from 0 up
>to a maximum of 2, whereas sines and cosines never exceeded 1. So you had to
>keep an eye on the digit to the left of the decimal point, which could
>always be omitted in tables of sines and cosines. The next step was simply
>to use halved versines instead, and these became known as haversines. You
>then had tables of hav A and also tables of log hav A.
>
>When pocket calculators and computers came in, tnen all the complications
>involved in twisting the trig to make it suitable for logs became
>unnecessary, and we could go back to using the simple basic formulae,
>breathing a sigh of relief. There's never any need for versines and
>haversines today, and they are never available as a separate function on
>calculators. But you can always get a haversine from a calculator using (1 -
>cos A) / 2, if you really need to.
>
>Today, haversines appear only within special tables that use logs, in a
>disguised form, such as in the sight reduction tables of Bennett's
>"Celestial Navigator". In that table, the middle column is actually a
>tabulation of
>-13,030 log hav LHA, and the right column is
>200,000 hav (lat ~ dec). [For completeness, the left column is
>-13,030 log cos (lat or dec)].
>
>My apologies, if all this is more than Alexander really needed to know.
>
>George.
>
>contact George Huxtable at george@huxtable.u-net.com
>or at +44 1865 820222 (from UK, 01865 820222)
>or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
>
>
>
>>
>
>
>


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