From the many contributions Galileo gave to science and mathematics, several were improved upon and used as a base for future development. The telescope being the most direct and
obvious improvement. As tools and designs improved, the telescopes would have to improve as well. I am not saying that his changes were revolutionary, but it pushed what was thought to be the best beyond that. With improvements to the telescope, we have been able to (without any doubt) prove that the solar system is heliocentric.
Some of Galileo's ideas of motion include; concept of motion in terms of velocity, idea of force as a cause for motion, the natural state of an object (rest) has uniform motion, and objects resist changes in motion. Through the years these are improved by Newton and have become normal in schools across the world. Newton's concepts are along the lines of; a change in velocity is acceleration which is caused by an outside force, inertia is proportional to the mass of the object, momentum is the quantity of motion energy with the equation E=mc^2, and total momentum during an interaction is conserved.
These ideas are possible because Galileo challenged what had been developed and understood before him. He wanted to be sure of the world around him, bringing facts and truth to the forfront. It caused him trouble over the years, but I'm sure it was all worth it in the end. And if he were able to see where math and science was today, I bet he would be astonished that he started the movement that caused it all.

