It's not clear to me the swindle described has anything to do with n parties.

Archimedes lived two centuries after Pythagoras and knew more math in spite of being known as an engineer. http://en.wikipedia.org/wiki/Archimedes#Mathematics

However, calculus is obvious now, so that won't work.

But the task is now easy -- I'd talk to Archimedes about plant rights (see http://www.overcomingbias.com/2007/03/morality_of_the.html).

Tom, the Greeks were colonizers, e.g., Syracuse is not in the homeland. http://en.wikipedia.org/wiki/Colonies_in_antiquity#Greek_colonies_.28.22apoikiai.22.29

The obvious non-meta version of my suggestion is to teach Archimedes calculus.

The sort of thing I have in mind is mathematical proofs and engineering designs that go just beyond what he was able to manage without my help, not imparting the scientific method or liberal morality. Given what the chronophone does more ambitious communication seems really risky.

I would attempt to send useful facts that the chronophone should not distort -- facts that classical Greek culture was ignorant of, but not biased against -- discoveries that would have been accepted by Greek culture, had they been made. I'd have to think about what those would be.