3x-2/4 -2/3= 2x+3/3 -x verify
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Galileo was born in Pisa, Tuscany, on February 15, 1564, the oldest son of Vincenzo Galilei, a musician who made important contributions to the theory and practice of music and who may have performed some experiments with Galileo in 1588–89 on the relationship between pitch and the tension of strings. The family moved to Florence in the early 1570s, where the Galilei family had lived for generations. In his middle teens Galileo attended the monastery school at Vallombrosa, near Florence, and then in 1581 matriculated at the University of Pisa, where he was to study medicine. However, he became enamoured with mathematics and decided to make the mathematical subjects and philosophy his profession, against the protests of his father. Galileo then began to prepare himself to teach Aristotelian philosophy and mathematics, and several of his lectures have survived. In 1585 Galileo left the university without having obtained a degree, and for several years he gave private lessons in the mathematical subjects in Florence and Siena. During this period he designed a new form of hydrostatic balance for weighing small quantities and wrote a short treatise, La bilancetta (“The Little Balance”), that circulated in manuscript form. He also began his studies on motion, which he pursued steadily for the next two decades.
In 1588 Galileo applied for the chair of mathematics at the University of Bologna but was unsuccessful. His reputation was, however, increasing, and later that year he was asked to deliver two lectures to the Florentine Academy, a prestigious literary group, on the arrangement of the world in Dante’s Inferno. He also found some ingenious theorems on centres of gravity (again, circulated in manuscript) that brought him recognition among mathematicians and the patronage of Guidobaldo del Monte (1545–1607), a nobleman and author of several important works on mechanics. As a result, he obtained the chair of mathematics at the University of Pisa in 1589. There, according to his first biographer, Vincenzo Viviani (1622–1703), Galileo demonstrated, by dropping bodies of different weights from the top of the famous Leaning Tower, that the speed of fall of a heavy object is not proportional to its weight, as Aristotle had claimed. The manuscript tract De motu (On Motion), finished during this period, shows that Galileo was abandoning Aristotelian notions about motion and was instead taking an Archimedean approach to the problem. But his attacks on Aristotle made him unpopular with his colleagues, and in 1592 his contract was not renewed. His patrons, however, secured him the chair of mathematics at the University of Padua, where he taught from 1592 until 1610.
Although Galileo’s salary was considerably higher there, his responsibilities as the head of the family (his father had died in 1591) meant that he was chronically pressed for money. His university salary could not cover all his expenses, and he therefore took in well-to-do boarding students whom he tutored privately in such subjects as fortification. He also sold a proportional compass, or sector, of his own devising, made by an artisan whom he employed in his house. Perhaps because of these financial problems, he did not marry, but he did have an arrangement with a Venetian woman, Marina Gamba, who bore him two daughters and a son. In the midst of his busy life he continued his research on motion, and by 1609 he had determined that the distance fallen by a body is proportional to the square of the elapsed time (the law of falling bodies) and that the trajectory of a projectile is a parabola, both conclusions that contradicted Aristotelian physics