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    Re: refraction
    From: Alexandre Eremenko
    Date: 2007 Jan 5, 02:05 -0500

    Lu and Guy,
    I hope you received the paper.
    I just read it, and I have to say
    that this is a truly outstanding example
    of popular scientific writing.
    It is long time since I read something so
    good. Maybe "Lectures on navigation"
    by Lord Kelvin can be compared.
    One has to write something like this about the
    history of refraction after Kepler.
    I knew that Snell (Snellius) discovered the
    "Snell law" which the ancients tried and failed to discover.
    (The paper says that it was really first discovered
    by an Arab whose name was Ibn Sahl).
    Actually I even have a conjecture why Ptolemy and his
    contemporaries failed to discover this law despite their
    efforts. The law says that the ratio of sines of
    the two angles is constant. The problem is that the
    Greeks did not have sines! Sines were invented by the Arabs
    some time in VIII century.
    The Greeks used chords instead: chd(t)=2sin(t/2)
    as their main trigonometric function.
    Stated in terms of chords, the law of refraction is too
    complicated to be discovered empirically as Ptolemy
    And of course I did not know (and now know from the paper)
    that the "tables of refraction" were first developed
    by Tycho Brahe, which answers the original question
    of Guy. But I guessed correctly HOW he did this:-)
    The precise theoretical law of
    atmopspheric refraction is mathematically
    complicated. It could have been possibly discovered only in
    the late XVII century, not earlier.
    But there is no doubt that by that time very accurate
    empirical tables already existed (composed on the same
    principle Tycho used).
    P.S. If anyone else is interested, go to
    I will keep it there for 2 days.
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