Galileo Galilei began teaching mathematics at Pisa in 1589 and moved to Padua in 1592. He studied motion, mechanics, and mathematical instruments, combining theoretical argument with experiments and practical work. His investigations challenged Aristotelian explanations of falling bodies and helped develop descriptions of accelerated motion. The story that he proved his ideas by dropping objects from Pisa's leaning tower is not securely documented and should not substitute for his surviving research.

In 1609 Galileo improved a telescope based on reports of a Dutch invention and applied it to astronomy. Observations of lunar mountains, numerous stars, and four satellites of Jupiter appeared in The Starry Messenger in 1610. The discoveries brought fame and appointment to the Medici court in Florence. Observations of the phases of Venus undermined the traditional Ptolemaic arrangement, though they did not alone prove Earth's motion.

Galileo defended Copernican astronomy and argued that biblical interpretation should accommodate demonstrated knowledge of nature. In 1616 church authorities prohibited treating heliocentrism as established truth, and he was instructed not to defend it. His Dialogue Concerning the Two Chief World Systems, published in 1632, compared competing cosmologies through a literary conversation. Its treatment of arguments and the earlier restrictions provoked proceedings before the Roman Inquisition. In 1633 he was judged vehemently suspected of heresy, required to recant, and placed under house arrest. His tidal explanation for Earth's motion was also scientifically mistaken, an example of the limits within his otherwise influential work.

During confinement near Florence, Galileo continued research and correspondence. Two New Sciences, published in the Netherlands in 1638, brought together his studies of the strength of materials and motion. It provided important foundations for later mechanics, including treatment of projectile trajectories and uniformly accelerated movement. Blindness restricted his final years, and he died in 1642. His career changed the practice and authority of mathematical natural inquiry through instruments, experiments, and persuasive writing. The conflict with the church was historically complex, involving theology, patronage, institutional authority, and his own claims about proof; it should not be reduced to an uncomplicated contest between an infallible scientist and a single undifferentiated opponent.