This chapter examines the mathematical foundations available to 17th-century scholars at the dawn of the Scientific Revolution. The authors trace how Euclidean geometry and Apollonius's work on conic sections, preserved through Roman and Islamic civilizations, formed the cornerstone of 1600s mathematical knowledge. The chapter explores Euclid's axiomatic approach and ruler-and-compass constructions, demonstrating how the printing press enabled reliable reproduction of figures and catalyzed new mathematical thinking. We show how mathematicians like Descartes began bridging geometric and algebraic reasoning by constructing arithmetic operations geometrically. Through detailed analyses of the Pythagorean Theorem and similarity, the authors illustrate the evolution from Euclidean synthetic geometry through Cartesian coordinate geometry to modern transformational geometry. This historical context reveals how different mathematical representations emerged as complementary "windows" for modeling measurement and physical phenomena, establishing the groundwork for calculus and modern mathematics. Understanding these historical intentions and transformations is essential for effective mathematics education, illuminating why certain concepts remain central to today's curriculum.