I'm just done reading The Great Ptolemaic Smackdown, and it provided a good timeline of establishing heliocentrism, broad acceptation of Kepler's system, and the initial (failed) Galileo's experiment attempting measurement of parallax of distant stars (which was based on assumption of two near stars of different brightness being a long distance apart; Galileo unfortunately picked a binary star) - all the models described within assume a fixed, mostly immutable "celestial sphere" of distant stars.
When, and how was this concept disproven?