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    Re: Sight Reduction by the Cosine Haversine Method
    From: Herbert Prinz
    Date: 2004 Oct 7, 16:53 -0400

    George Huxtable wrote:
    > 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.
    But take note of a subtlety. When the geometrical arguments lead to a dead end,
    Ptolemy is prepared to drop the whole apparatus of pure geometry and resorts to
    a numerical procedure. Starting out with 72d and 60d from the pentagon and
    hexagon, he obtains the chords of angles 12d, 6d,...,..3/4d by continued
    bisection. But how to get the chord of 1/2d? He knows that the trisection of the
    angle is impossible with rule and dividers. With the help of some proportions,
    he estimates an upper and lower limit and observes that they agree to the
    required number of decimals. That's good enough for his purposes. He takes
    astronomy out of the realm of philosophy and treats it like an engineer.
    Trying hard to give the appearance that what we are discussing here has anything
    to do with navigation, I shall mention in passing the name of Cotter. (I know,
    George, that you maintain this list...)
    In A History etc., p.16 he writes that "Ptolemy had the theorem
        "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
    "This theorem is usually known as Ptolemean Theorem."
    Now that you have seen the relevant chapter in the Almagest, would you agree
    with this?
    Herbert Prinz

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