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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Haversine- how to derive it?
From: Lars Bergman
Date: 2015 May 18, 13:10 -0700

John Brown asked: Could a mathematician or a navigation historian on this list please explain why the versine is the poor relation of its half-size parent??

The following explanation is mainly based on "Sines, Versines and Haversines in Nautical Astronomy" by Charles H.Cotter in the (British) Journal of Navigation, vol.27, no.4.

According to Cotter, in the seventh edition of Inman's "Navigation and Nautical Astronomy for the Use of British Seamen", Portsmouth 1849, Inman introduced a novel method for computing a PZX-triangle for hour angle, namely

log hav t = log sec lat + log sec dec + 1/2 log hav (z+lat-dec) + 1/2 log hav (z-(lat-dec))

In Inman's "Nautical Tables" there were correspondingly tables of log hav, log sec and half-log hav.

Inman's tables were adopted by the Royal Navy and his method of solving the PZX-triangle was the standard method used in that service right up to the beginning of the 20th century.

In 1899, the Royal Navy Instructor H.B.Goodwin proposed the use of the formula

vers z = vers(lat-dec) + cos lat·cos dec·vers t

but Goodwin pointed out that "since in Inman we have no logarithmic versines but only log-haversines, we must divide each side by two ...".

So Cotter concludes: "The artificial function haversine appears to have been adopted instead of the natural function versine merely because the popular Inman collection of nautical tables, used by Royal Navy Officers at the end of the nineteenth century, did not contain a table of logarithmic versines."

It is to be noted that the square root of [hav(z+lat-dec)·hav(z-(lat-dec))] equals hav z - hav(lat-dec).

Lars, 59N 18E (neither mathematican nor historian)

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