This chapter delves into the philosophy and methodology of Gottfried W. Leibniz, whose work laid the foundation for the widely accepted symbolic language of calculus that we use today. Leibniz believed that truth resided in the logical structure and symbolism of mathematics itself, viewing mathematical logic as the essence of the universe. His goal was to create a universal language capable of modeling all of geometry, essentially establishing a scientific grammar and syntax for tables and calculations. A critical test case for his evolving system was the cycloid, a curve whose area, tangents, and arc length properties were already known through pure geometry (like the experiments of Galileo and the geometric methods of Roberval and Wren) before a concise equation for it was written. Leibniz demonstrated the consistency and universality of his calculus through a geometric technique known as the transmutation of curves. This technique showed how an area problem (integration) could be systematically converted into a tangent problem (differentiation), forming the conceptual basis for the Fundamental Theorem of Calculus. By applying this method to the circle, Leibniz derived the infinite series for π/4. This result served as a powerful confirmation of his new symbolic language against long-established geometric knowledge. His work, perpetuated by followers like Leonhard Euler, secured the foundation for modern mathematics, emphasizing that abstract notation could successfully and consistently model the physical world.