Galileo did not actually invent the telescope himself. Rather he heard about the invention, designed his own, and was the first to turn it towards the heavens. He is therefore widely given credit for inventing the astronomical telescope. His first telescope was a very poor telescope by modern standards. A modern child's toy telescope will produce better images. It however still produced images of the heavens vastly superior to anything else available in Galileo's time. He was therefore able to make many new discoveries with his telescope.
Prior to Galileo's time scientists and philosophers thought that the Moon must be a perfect sphere because Aristotelian physics, taught by the ancient Greeks, decreed that any objects in the heavens must be unchanging perfect unblemished spheres. Galileo's telescope for the first time revealed that the Moon was blemished and imperfect. Galileo discovered a surface pockmarked with craters. He also discovered relatively smooth areas that he called maria (singular: mare), which derives from the Latin word for seas. Galileo thought the surface of the Moon had seas just like Earth's surface.
Galileo also observed the sun with his telescope. (Aside: NEVER NEVER EVER look at the Sun through a telescope that is not properly filtered. It will blind you instantly!) He discovered small dark spots on the surface of the Sun, which we now call sunspots. Like the Moon the Sun's surface is also blemished.
When Galileo observed Jupiter he discovered its four largest Moons: Io, Europa, Ganymede, and Callisto. These moons are still called the Galilean moons of Jupiter. Galileo is known for his troubles and disagreements with the Roman Catholic Church of his time, but he apparently shared observations of Jupiter's moons with the Jesuits of the day.
Galileo also almost discovered the rings of Saturn. He observed that Saturn seemed to have bumps on either side, but could not make out the details. If his telescope had been just a little better, he would have resolved those bumps into Saturn's rings.
When he observed Venus, Galileo discovered that it has phases similar to the Moon's. When Venus is full, it appears smaller in size, and when it is a thin crescent it appears larger in size. This effect is just what is predicted if Earth and the other planets orbit the Sun as had recently been suggested by Copernicus. These observations can not be explained in the context of the Ptolemaic model which had the Sun and planets orbiting Earth.
When Galileo turned his telescope towards the Milky Way, he observed that it was composed of a vast number of stars to faint to be seen individually with the naked eye. We now know that it is the plane of our galaxy. Prior to this observation, ancient Greeks believed that there were a small number of fixed stars.
Galileo's observations supported the Copernican idea that Earth and the planets orbit the Sun. Direct support comes from his observations of the phases of Venus, which are just as predicted by the Copernican model. Indirect support for Copernicus comes from Galileo's observations of blemished imperfect heavenly bodies. The Ptolemaic idea that all heavenly bodies orbit Earth was based on Aristotelian physics, which is now known to be incorrect. Aristotle taught that all heavenly bodies must be perfect unchanging spheres. Discovering imperfections in the Sun and Moon was like pulling bricks out of the foundation of the edifice of ancient Greek astronomy. Without the foundation, it soon tumbled down.
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