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Re: Sight Reduction by the Cosine Haversine Method
From: Herbert Prinz
Date: 2004 Oct 6, 15:50 -0400
From: Herbert Prinz
Date: 2004 Oct 6, 15:50 -0400
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.