This chapter examines Galileo Galilei's revolutionary contributions to scientific methodology by integrating mathematical analysis with experimental observation. The authors explore how Galileo transformed "experience" into "experiment," using geometry to describe natural phenomena and challenging Aristotelian philosophy. Key topics include Galileo's inclined plane experiments demonstrating laws of motion, his use of the geometric mean for analyzing falling bodies, and his invention of practical calculating instruments. The chapter also covers historical methods for computing square roots and John Napier's development of logarithms, which revolutionized seventeenth-century mathematics. Through detailed examples and historical context, we demonstrate how Galileo established mathematical modeling as fundamental to physics, marking a critical moment in the Scientific Revolution. The chapter concludes with implications for mathematics education, emphasizing the importance of understanding how abstract mathematics came to describe physical reality and how computational tools evolved.