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Galileo studied with the Camaldolese Order of monks at Vallombrosa and foud that he really enjoyed the quiet, yet disciplined, life of the monks that he learned from. When he was 15 years old, he announced that he wanted to be a monk and even prepared to join them, but his father had words for Galileo over this plan; thus, he was quickly pulled from the monastery where he had been studying for the past four years and he returned to the family in Florence (St. Andrews University).
In Florence, Galileo continued his educational studies from another group of Camaldolese monks. In 1581, Galileo, who was now 17 years old, moved back to Pisa to live with his uncle again. In order to appease his father, Galileo started to learn about medicine in the University of Pisa, although his heart really was not into it. However, he was extremely interested in learning about math and philosophy and took several classes on these subjects. In 1582, Galileo took a course about Euclid's Elements and was intrigued by this subject (St. Andrews University).
While learning math, Galileo had a professor named Filippo Fantoni, whom he invited to his home in Florence one summer day; Galileo wanted him to talk his father about allowing Galileo to study more about math. At first, Vincenzo was strongly against the idea but eventually relented, thus allowing Galileo to learn more about Euclid and Archimedes. Two years later, he gave up his offical course of study in medicine and left the university in order to study math full time (St. Andrews University).
At first Galileo taught math to private students until he landed a position teaching at his alma mater, Vallombrosa. Here, he wrote his first book called "La Balancitta" (The Little Balance) (St. Andrews University). This book describes "an accurate balance for weighing things in air and water" (Van Helden). Because of this book, Galileo's theories and how he describes them in theoretical and practical terms is what gave him the edge over his comtempories (Van Helden).
Galileo continued to make advancements in the world of mathematics as he made progress in his experiments with gravity. In 1588, Galileo was invited to speak at the prestigious Florence Academy about the size and the location of hell in Dante's "Inferno". Because of this and glowing recommendations from Christopher Clavius, a renowned Jesuit mathematician and astronomer, Galileo was offered Fantoni's position at the University of Pisa. The salary was not much but he could continue to work on his theories. He wrote a set of essays, which were not published, called "De Motu" regarding his theory of motion, which included one of his most important finding--that one could slow down the rate of descent by using an inclined plane (St. Andrews University).