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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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