René Descartes’ La Géométrie marked a pivotal transformation in mathematics during the Scientific Revolution, converting geometry into a method of scientific analysis. To understand this transformation, we must examine what preceded it—the geometry of Apollonius of Perga (c. 262–190 BC). This chapter explores Apollonius’s Conics, which seventeenth-century mathematicians studied extensively, along with crucial Arabic additions from the tenth and eleventh centuries. Using classical geometry concepts, including similarity and the Pythagorean theorem, we analyze how conic sections emerge as intersections of planes with cones. The chapter demonstrates Apollonius’s use of geometric means and proportions to establish properties of parabolas, ellipses, and hyperbolas, showing how these static geometric relationships prefigured coordinate geometry. The authors examine Arabic contributions, including Ibn Sina’s construction methods for drawing conics, and discuss pedagogical approaches that combine physical constructions with modern geometry software. By tracing the evolution from three-dimensional cone sections to two-dimensional analytical representations, we establish the mathematical foundations necessary for understanding Descartes’ dynamic geometry and his critique of classical mathematics. This historical perspective reveals how the transition from Apollonian geometry to Cartesian analysis involved not merely a shift from geometry to algebra, but a fundamental reconceptualization from static to dynamic geometry involving motion and time.