By looking into space with the telescope.
Galileo's discoveries showed faults with the old Ptolemaic theory, which had the Sun at the centre. So he was right to publicise these but not to proclaim that the heliocentric theory of Copernicus was correct. Tycho produced a geocentric theory that explained the phenomena discovered by Galileo, which were (mainly) the full range of phases dislplayed by Venus. Eventually Johannes Kepler brought out a new heliocentric theory with novel elliptical orbits for the planets. It was later backed up by Newton's theoretical discoveries and by later measurements, and it is the model used and accepted today.
Galileo's early telescope observations provided crucial evidence for the heliocentric model by revealing celestial phenomena inconsistent with the geocentric view. He discovered the phases of Venus, which showed that it orbits the Sun, not the Earth, as the geocentric model suggested. Additionally, Galileo observed Jupiter's moons, demonstrating that not all celestial bodies revolve around the Earth. These findings supported the idea that the Sun is at the center of the solar system, with planets, including Earth, orbiting around it.
Galileo's observations with his early telescope provided crucial evidence for the heliocentric model by revealing celestial phenomena that contradicted the geocentric view. He discovered that Jupiter had moons orbiting around it, demonstrating that not all celestial bodies revolve around the Earth. Additionally, his observations of the phases of Venus showed that it could not be explained by the geocentric model, as Venus would not exhibit a full set of phases if it orbited the Earth. These findings supported the idea that the planets, including Earth, orbit the Sun.
None of it but Galileo's discoveries with the telescope were very important in raising questions about the old Ptolemaic theory, which was geocentric. However Galileo's discovery of Venus's phases was not a proof of the heliocentric principle because Tycho produced a geocentric model that explained Venus's phases.
i think it's Kepler
This shift in understanding was largely due to the work of astronomers like Nicolaus Copernicus and Johannes Kepler, who proposed heliocentric models of the universe in the 16th and 17th centuries. Their theories were later confirmed by Galileo Galilei's observations using a telescope, which provided empirical evidence supporting the heliocentric model and eventually led to the acceptance of the idea that the planets orbit the sun.
Apex: conclusion
deductive
If a hypothesis is supported by new data gathered over a period of time, it may become a theory. A theory is a well-supported explanation that has been extensively tested and confirmed through observation and experimentation.
In telescopes that use them, mirrors are the principal mechanism of magnification. The mirror is what gathers the light, and the more light gathered the better. The eyepiece is not the principal magnifier; the eyepiece serves to focus the gathered light so that it can make a clear image on the retina. The larger the mirror, the more light is gathered, and the better the telescope is.
Bruno did a pretty good job of it, and was burned at the stake for his troubles. Galileo's observations of the solar system were very thorough, and he had a decent telescope with which to record his findings. The evidence Galileo gathered was quite compelling.