Hum... We can't present Archimedes with the scientific method, but we could present him with simple experiments we did to confirm and infirm some of our simpler beliefs. That might be enough to give him a hint of the method...

Thanks Eliezer, for this example! Even though it's been making my head spin - there seem to be ways of cheating the chronophone (such as presenting the scientific method through specific experiments), but they don't seem "fair" to your project - deciding what is best for us today, if we don't know the destination.

Since what we'd most want to do is to get Archimedes to doubt some of his assumptions, maybe what we'd need most today is doubt?

But the one thing I'd most like to tell him is "watch out for that roman soldier!"