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From: Herbert Prinz (no email)
Date: Wed Oct 06 2004 - 15:50:59 EDT
I would like to correct the notion that the versine (sinus versus) was newly
created for the purpose of facilitating logarithmic computation. The versine,
like the sine, is a Hindu invention. Aryabhata tabulates both functions side by
side. The cosine appears later under various names, but throughout the Arabic
period and the Renaissance it never gained the same acceptance as the versine.
Around 900 A.D., al Battani solved the time sight problem by means of versines.
Around 1450, Peuerbach used versines for proofing (again) an old recipe for
finding the altitude of the sun from LHA. When Regiomontanus came up with what
we now call the cosine theorem, he formulated it entirely in terms of sines and
versines. Logarithms of versines appear for the first time in Cavalieri's
"Directorium generale uranometricum", 1632, together with those of the sine,
tangent and cotangent (but not the cosine). This choice cannot be explained
with the negative values of the cosine in the second quadrant, because he
tabulates only the first quadrant anyway. He just continued a tradition.
The cosine gained importance from a theoretical point of view only after Euler
and others began the analytical treatment of the trigonometric functions. But
the versine always had its place in applied mathematics.
Where the haversine is concerned, I do not see the advantage of the reduced
image range. Could George elaborate on this?
The real benefit of this function seems to be found in the equality
1/2 * versine(x) = sin^2(x/2)
Its tabulation permits thus the direct solution for A in the frequently
occurring formula
sin^2(A/2) = (sin(s-b) * sin (s-c)) / (sin b * sin c)
and its many variants. Another use is in distance computations. For very small
distances, the haversine formula shows a better numerical behaviour than the
cosine formula.
From an analytical point of view, it is hardly appropriate to grant the
haversine the status of a trigonometric function in its own right. But for
practical purposes it is useful to distinguish it from the versine. It evolved
gradually from the latter. It has been tabulated under various names before the
nautical community settled on "haversine". In Mendoza's Tables, 1805, it simply
appears as versine while the user is alerted to the fact that the entries
actually correspond to half the value of that function. Clearly, the difference
between the two logarithms is just the constant 0.30103.
Can someone tell us who coined the name "haversine"?
Herbert Prinz
George Huxtable wrote:
> That's all correct, but Chuck has omitted the important reason WHY it was
> necessary to avoid negative values. The reason was the USE OF LOGS.
>
> [...]
> First, a new function, the "versine", (1 - cos), was introduced, in
> addition to the usual sin and cos; clearly, this was always positive,
> varying between 0 and +2. Later, it became clear that a function which
> never exceeded 1 would be more useful still, so the versine was simply
> halved, becoming the haversine (hence its name). Then the altitude formula
> was reconstructed to use the haversine and avoid negative quantities.
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