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rebellionkid comments on Archimedes's Chronophone - Less Wrong
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I would speak highly of sharing scientific research with China and India - in the hopes that it would come out as a paean on the benefits of sharing research with other cultures - hopefully leading to a sharing of information with the great Islamic scientists. Most of which are after Archimedes' time... but having a culture-of-sharing in place would be a good way to spark things off.
Archimedes died in 212 BC. Justinian closed the pagan schools and killed of the classical research project in 529 AD. The first book of the Quran was delivered in 610 AD. So sharing with the Islamic scientists (who had all read Aristotle anyway) doesn't make sense.
But an expansion of the communication between places like Alexandria, Athens, Cos etc could be a really good plan.