Kepler's eight-arcminute refusal to fudge
Using Tycho Brahe's precise naked-eye observations of Mars, Kepler built a circular-orbit model that matched the data to within 8 minutes of arc, an error most astronomers of his time would have accepted. Because he trusted Tycho's measurements more than his own cherished belief in perfect circles, he discarded years of work and eventually found that planets move in ellipses with the Sun at one focus. A small, stubborn mismatch between theory and careful data was what overturned two millennia of circular astronomy.
