Ilan Vardi's insightful remarks about Archimedes as an Experimental (Rigorous!) Mathematician

Dear Doron,

I hope you are doing well. I was looking at your website again and found 
your very nice talk What is Experimental Mathematics? In the talk you 
alluded to Archimedes, but in fact, I believe he is more significant 
because he wrote the first rigorous paper on his experimental method, 
you accurately state that the ancient Greeks tried to hide their 
reasoning. But in his paper The Method he explicitly states that it is 
easier to come up with the statements of theorems through a formal 
method, and then the proof is relatively easy.

Another example from Greek mathematics is the well known Euclid proof 
that sqrt(2) is irrational. But in fact, this is more in the line of a 
verification. To discover this fact, one simply applies the Euclidean 
algorithm on 1 and sqrt(2) and see where that leads. Working 
geometrically, one soon sees that one get a recursion so that there 
cannot be a common measure (of course this comes down to the continued 
fraction expansion of sqrt(2)). For some reason, this is rarely taught, 
I can't think of an elementary reference.

Best regards,

-ilan