Galileo believed that properly chosen mathematical proofs can be used to explore any quantitative problem. Galileo put forward the first set of mechanical problems for himself, which involved the scale effect. When Galileo investigated the scale effect, he studied the quantity of matter later called mass, and later explored the dynamic problems involving time measurement and speed measurement in the same way. The central problem of Galileo's research is the problem of falling objects under the action of gravity, which overturns Aristotle's argument that objects with different weights fall at different speeds.
According to Aristotle's physics, it is the persistent action of force that keeps an object moving at a constant speed. But Galileo's experimental results prove that the object does not move at a uniform speed under the lasting influence of gravity, but the speed increases after a certain time. An object continues to maintain its speed at any point and is aggravated by gravity. If there is no gravity, the object will continue to move at the speed obtained at that point. This is the principle of inertia. This principle makes it clear that an object will keep its original static state or uniform motion state as long as it is not affected by external forces.
Galileo developed the flight trajectory theory of projectiles from the principle of inertia, thus showing the scientific value of mathematical proof. He observed the movement of a ball rolling across the table at a uniform speed and then falling to the floor from the curved track at the edge of the table. At any point on this falling trajectory, the ball has two speeds: one is the speed along the horizontal plane, which is always constant according to the principle of inertia, and the other is the speed in the vertical direction, which accelerates with the passage of time under the influence of gravity. In the horizontal direction, the ball travels the same distance in the same time, but in the vertical direction, the distance traveled by the ball is proportional to the square of time. This relationship determines the trajectory form of the ball, that is, semi-parabola. So when an object is thrown at an angle of 45 degrees, the distance will be the farthest.
Galileo developed some instruments. He made the first thermometer to measure temperature and used a pendulum to measure time. Galileo also improved the refracting telescope and used it for astronomical observation.
1609, Galileo heard that an optician in middelburg, the Netherlands, built a "telescope" that can magnify distant things, so Galileo studied the optical characteristics of synthetic lenses and built several improved telescopes for his own use. He made astronomical observations with a new telescope, found sunspots on the sun and potholes on the surface of the moon, and calculated their heights according to the length of the edge shadows. He also found that the Milky Way is made up of many stars. In addition, Galileo also discovered the phase of Venus, that is, Venus and the moon have the same phase change, which will gradually change from the new moon to the full moon; He also discovered four moons of Jupiter. All these findings supported Copernicus' Heliocentrism and seriously challenged Ptolemy's ancient Greek astronomical view and geocentric theory recognized by the Roman church at that time. He compiled these findings into a dialogue about Ptolemy and Copernicus, with the aim of calming down opposition and avoiding church sanctions. Galileo was summoned by the Roman inquisition, where he was forced to make a statement admitting his mistake.