We don't know what scientists and mathematicians of Alexandria really knew. Some modern discoveries are stunning.

1. In a recent book of Lucio Rossi `La Rivoluzione Dimenticata' (Forgotten Revolution), published in Italian in 1996 (I know it only by a review in Notices of the American Mathematical Society, May 1998, p. 601) its author Lucio Russo makes a strong case that Alexandrian astronomers knew the correct law of gravity, and understood that it is gravity which holds the planets rotating around the Sun, and Moon around the Earth. Apparently this discovery was credited to Hipparchus of Nicea. None of his writings survived. Judging by references to his work in the books of other writers of that time, he was a scientist of the same caliber as Archimedes.

2. In several recent issues of American Mathematical Monthly, (104 (1997) 344-350 and May 1998, 446) two mysterious numbers, mentioned in Plutarch are discussed. The first of them, 103,049 was recently identified as solutions of a rather complicated combinatorial problem, which previously thought to have been solved first only in the end of XIX century. This is the so-called 10-th Schröder number. About the second number, 310,952 has not been reliably identified so far. Plutarch attributes the discovery of these numbers to Hipparchus. No one knows how could Hipparchus possibly solve such combinatorial problems.

Returning to the Law of Refraction, Feynman cites in one of his Lectures on Physics (strongly recommended reading!) a table given by Claude Ptolemy (second century A. D.) with experimental data on refraction. The data fit the real law of refraction very well. But it seems that nobody could guess the correct law before 1621. One possible explanation is that Ptolemy lived in the time of decline of science in Alexandria (the peak of it was about 4 centuries before, this is more the distance between Kepler and our time!) so many earlier discoveries could have been forgotten by the time of Ptolemy. There are several such examples. Or maybe the law of refraction was indeed discovered in 1621 for the first time.

By the way, ancient Greeks, and Alexandrian scientists did not use the sine as their main trigonometric function. They used the chord instead, chd t=2 sin(t/2). So they had tables of chords instead of our tables of sines. The sine was introduced only in the fifth century A. D. This could make a discovery of the correct law of refraction much harder, because as you understand chd(t) is not proportional to the sin(t). (This is my own conjecture).