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An easy example: How do you know how far away the sun is?
Terry Tao has a really cool presentation on that topic: The Cosmic Distance Ladder.
Parallax effects are a surprisingly good reason to reject heliocentrism. Wrong, of course, but it does seem to fail the sniff test -- and about all the Greeks had to work with were sniff tests of varying sophistication.
Although now I kind of wonder how Aristarchus' critics explained his observations.