Galileo's was a mathematican, physicist, and astronomer. So, he had many different inventions and findings in all three subjects. 

Law of the Pendulum

At age twenty, Galileo noticed a lamp swinging overhead while he was in a cathedral. Curious to find out how long it took the lamp to swing back and forth, he used his pulse to time large and small swings. Galileo discovered something that no one else had ever realized: the period of each swing was exactly the same. The law of the pendulum, which would eventually be used to regulate clocks, made Galileo Galilei instantly famous.
Mathematics

To earn a living, Galileo Galilei started tutoring students in mathematics. He did some experimenting with floating objects, developing a balance that could tell him that a piece of, say, gold was 19.3 times heavier than the same volume of water. He also started campaigning for his life's ambition: a position on the mathematics faculty at a major university. Although Galileo was clearly brilliant, he had offended many people in the field, who would choose other candidates for vacancies.

The Leaning Tower of Pisa

At the time that Galileo arrived at the University, some debate had started up on one of Aristotle's "laws" of nature, that heavier objects fell faster than lighter objects. Aristotle's word had been accepted as gospel truth, and there had been few attempts to actually test Aristotle's conclusions by actually conducting an experiment!

According to legend, Galileo decided to try. He needed to be able to drop the objects from a great height. The perfect building was right at hand--the Tower of Pisa, 54 meters tall. Galileo climbed up to the top of the building carrying a variety of balls of varying size and weight, and dumped them off of the top. They all landed at the base of the building at the same time  Aristotle was wrong.

The Telescope

The telescope was introduced to astronomy in 1609 by the great Italian scientist Galileo Galilei, who became the first man to see the craters of the moon, and who went on to discover sunspots, the four large moons of Jupiter, and the rings of Saturn. Galileo's telescope was similar to a pair of opera glasses in that it used an arrangement of glass lenses to magnify objects. This arrangement provided limited magnification--up to 30 times for Galileo--and a narrow field of view; Galileo could see no more than a quarter of the moon's face without repositioning his telescope.

This is a letter Galileo wrote to the Doge to try and sell his telescope:

"Galileo Galilei, a most humble servant of Your Serene Highness, being diligently attentive, with all his spirit, not only to discharging the duties pertaining to the lecturing of mathematics at the University of Padua, but also to bringing extraordinary benefit to Your Serene Highness with some useful and remarkable invention, now appear before You with a new contrivance of glasses, drawn from the most recondite speculations of perspective, which render visible objects so close to the eye and represent them so distinctly that those that are distant, for example, nine miles appear as though they were only one mile distant.  This is a thing of inestimable benefit for all transactions and undertakings, maritime or terrestrial, allowing us at sea to discover at a much greater distance than usual the hulls and sails of the enemy, so that for two hours or more we can detect him before he detects us... 

....(the telescope is) one of the fruits of the science which he has professed for the past 17 years at the University of Padua, with the hope of carrying on his work in order to present You greater ones, if it shall please the Good Lord and Your Serene Highness that he, according to his desire, will pass the rest of his life in Your service."
                                                                         -Galileo Galilei

The Rudimentary Thermometer

What Galileo needed was to come up with some sort of device that could make him a tidy profit. A rudimentary thermometer (which, for the first time, allowed temperature variations to be measured) and an ingenious device to raise water from aquifers found no market.

The Rudimentary Thermometer, thanks to Galileo Galilei.

Pendulum Clock, thanks to Galileo Galilei.

Tower of Pisa experiment, thanks to Galileo Galilei.

The great invention of the telescope, which helped the world discover what was beyond, thanks to Galileo Galilei.


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