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    Re: Sight Reduction by the Cosine Haversine Method
    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---.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.
    ================================================================
    
    
    

       
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