Some more examples can be found in Schrodingers book "Nature and the Greeks".

One that stuck with me is the "discovery" of atoms. Some philosopher (don't remember the name) in Greece was thinking about the volume of a pyramid. And he imagined what would happen if we dissected the pyramid through the middle and looked at two squares formed on the bases. He reasoned that the squares should have equal areas, however if that was the case for every such dissection the pyramid would never form and instead we would have a cube.

So his reasoning was that in order to explain this paradox objects in the real world should be composed of small elements, so that after dissection the squares are slightly different.