It soon became clear that medicine was not Galileo's first love. Rather, the young scholar became intrigued by mathematics, and found inspiration in the form of Ostilio Ricci, a mathematician in the court of the Tuscan Grand Duke. Ricci, impressed by Galileo's curiosity, agreed to tutor him privately. From 1581 to 1585, Galileo continued to formally pursue a degree in medicine, while Ricci educated him in geometry and applied mathematics. These years were a formative period for the young man, and in 1583 he made his first famous discovery: that each swing of a pendulum, regardless of width, takes an equal amount of time to swing between the extremes of its arc. This suggested that pendulums could be used to mark small intervals of time, and professors at the university quickly pounced on the notion, creating a device called a "pulsilogia," to keep track of a medical patient's heartbeat.
Galileo's discovery temporarily restored him to the good graces of his professors, but he showed a patent lack of interest in medicine, and skipped so many of the required lectures that he was soon in danger of failing out of the university. Meanwhile, the Galilei family was running short of funds, and Galileo's father suggested that he apply for a scholarship offered by the Duke of Tuscany. Galileo did so, but his poor record as a medical student doomed his application, and in 1585 he was forced to leave the University of Pisa without a degree. For the next four years, he earned his keep in Florence by working as a private tutor in mathematics, and continued to make experiments on his own. He invented a device to measure the relative weights of alloys in a metal, improving on the work of the ancient mathematician Archimedes, and distinguished himself in a city-wide contest by most convincingly determining the location, shape, and dimensions of Hell as portrayed in Dante's famous Inferno. (This was no small issue, since Florence was Dante's birthplace, and the great Italian poet's 16th-century fans treated him as a kind of secular saint.) During this time Galileo also earned the praise of the noted Jesuit mathematician, Christopher Clavius, the man responsible for the great calendar reform that the Church had put into effect in January 1582. But Galileo wanted not only praise from Clavius, but patronage: he aspired to become a university professor, but had been meeting with repeated failure. By 1589, he had been rejected as under-qualified for positions all across northern Italy. Despairing, he and a friend considered leaving for Constantinople and the Turkish East in search of employment. But in the summer of 1589, a position teaching mathematics opened at the University of Pisa, in the city of his birth- -and this time, Galileo's application was accepted.
At the University, Galileo quickly developed a rivalry with the older, more established professors, particularly a conservative philosopher named Girolamo Borro. Philosophy, in the 16th century, incorporated a wider range of investigations than it does today, including questions of natural science and physics, and Borro had written extensively on the ocean's tides, as well as the properties of motion. He was an Aristotelian, like most scholars of his day, meaning that he based his work on the ancient Greek philosopher Aristotle. But Borro did not accept Aristotle's precepts on faith alone, and advocated a science based on experimentation. Galileo much doubted many Aristotelian claims, and resolved to test one of the more famous ones, namely, that "the downward motion of a mass of gold or lead, or of any other body endowed with weight, is quicker in proportion to its size." In other words, the heavier an object, the faster it falls. According to legend, Galileo ascended to the top of the famous Leaning Tower of Pisa, and dropped balls of different weights and sizes to the ground. Each pair, regardless of composition or weight, hit the ground at the same time, laying Aristotle's clever, but mistaken theory to rest.
If this event really took place, it would have been a brilliant act of scientific theater. And it does seem to fit with Galileo's character during his three years in Pisa: he was a gadfly, a sharp-tongued teacher who earned the esteem of students but the enmity of older professors like Borro for his theatrics and disrespect for their authority. Possibly because of the enemies he had made, or possibly by his own request, Galileo's contract at the university was not renewed, and by summer 1592 he was out of a job. By now, however, he had established a reputation as one of the bright lights of the mathematical field, and he carried this reputation with him out of the Duchy of Florence and into the territory governed by that great maritime power, the Serene Republic of Venice. There in autumn of 1592, the Venetian Senate chose him as the chair in mathematics at the University of Padua.