The correct options that describe how Nicolaus Copernicus, Tycho Brahe, Johannes Kepler, and Galileo Galilei were able to convince other scientists of their time include:
C. They used telescopes to make observations, such as the moons of Jupiter, that directly contradicted the geocentric model.
D. They provided detailed mathematical models and precise observational data that accurately predicted planetary motions.
Options A and B do not accurately reflect the methods of these scientists, as they did engage with evidence and used empirical observation to support their conclusions rather than ignoring conflicting evidence or relying solely on personal beliefs.