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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
intended:-)

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).

Alex

P.S. If anyone else is interested, go to
http://www.math.purdue.edu/~eremenko/Refraction1.pdf
I will keep it there for 2 days.


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