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Re: Sight Reduction by the Cosine Haversine Method
From: George Huxtable
Date: 2004 Oct 7, 00:21 +0100
From: George Huxtable
Date: 2004 Oct 7, 00:21 +0100
I wrote- >> 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). Clearly, this had several historical errors. I'm glad I wrote it, however, because it stimulated Herbert Prinz to put me right with a fascinating account of early trig. I had never even heard of Aryabhata, al Battani, or even Peuerbach, ashamed to admit. Herbert explained the advanage of hav A as being equivalent to sin^2(X/2), and then asked >Where the haversine is concerned, I do not see the advantage of the reduced >image range. Could George elaborate on this? Well, my thinking was very simple. Versine tabulations range between 0 and 2, so the table requires an initial digit, with value 0 or 1, before the decimal point. Haversines are always between 0 and 1, so that initial digit, always 0, could be omitted, which would save a bit of space in the tabulation. Similarly, log versines switch, when versine = 1, between a negative log with a prefix of 9, to a positive log with a prefix of zero. This somewhat-awkward transition is avoided in a table of log haversines. I was thinking of nothing more sophisticated than ease of number-handling. ================== On thr related matter of early trig, now that it's been raised by Herbert, I have recently been dipping into an English translation by Taliaferro of Ptolemy's "Almagest" , which is collected, with Copernicus and Kepler, in one of the Great Books series from the Encyclopaedia Britannica (1952.) And what amazed me was this, in the first few pages (my dipping hasn't got very far yet). Ptolemy writes down what is effectively a table of sines (actually, chords of a circle). At half-degree intervals of angle A Ptolemy computes the lengths of the chord of a circle which subtends angle A, effectively 2 sin (A/2), from A = 0 to 180 deg, to remarkably high accuracy, entirely from geometrical arguments. So this is effectively a table of sines, at quarter-degree intervals, between 0 and 90 degrees. No doubt further revelations will result from delving deeper. George. ================================================================ contact George Huxtable by email at george@huxtable.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================