Irrationals

As it happens, I have been reading Neugebauer "The Exact Sciences
in Antiquity" and Heath. Archimedes stated that 3 1/7 < pi < 3 10/71,
but Heron says that actually Archimedes found that 211872/67441 <
pi < 195882/62351, but that these numbers were impractical for
computation (the numbers in Heron's text are somewhat corrupt as
211875 and 197888, but Tannery restored the values above) (Heath
vol 2, 329). Neugebauer claims that Greek computational methods
developed independently from a basis of Babylonian mathematics
and were more sophisticate, if less rigorous, than usually
thought. However, the Babylonians often used just 3 for pi.

--John


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