This chapter examines René Descartes's revolutionary approach to mathematics in the 17th century, where he rejected classical Euclidean methods in favor of a practical, mechanistic framework. Descartes developed linkage machines and curve-drawing devices that established fundamental connections between geometry, algebra, and physical motion. His work introduced algebraic degree as a measure of curve complexity and demonstrated how mechanical devices could generate equations, rather than equations graphing curves (as taught in the 21st century curricula). Key contributions explored include his hyperbola construction device, his linkage machine for finding geometric means, and an innovative geometric construction of logarithmic curves. Descartes viewed the universe as a mechanical system governed by mathematical principles, enabling him to unify physical measurements, numerical tables, and algebraic representations into a coherent mathematical language. This integration of geometry, algebra, and mechanics laid essential groundwork for calculus and transformed how mathematics could be used to solve real-world problems in engineering and natural philosophy.