This chapter explores the influential empirical mathematical methods of John Wallis (1616–1703), focusing on his landmark 1655 work, Arithmetica Infinitorum. Wallis's contribution centered on championing the use of fractional exponents, which, though not his invention, he rigorously validated across multiple representations (numerical tables, algebra, and geometry). Wallis's work was a foundational text that inspired Isaac Newton (1642–1722) to formulate the general binomial series, which, in turn, served as the basis for Leonhard Euler's (1707–1783) explorations of continuous functions, including natural logarithms and exponentials. Notably, all three mathematicians relied on empirical methods—using extensive conjectures, comparisons, and confirmations—to establish mathematical truth before the advent of the formal proofs that would arise in the nineteenth century. To fully grasp this intellectual context, the chapter first examines Blaise Pascal's philosophy of science. By advocating for an empirical and investigatory approach over a purely formal logical analysis, Wallis aimed to develop a more powerful theory of knowledge, which he demonstrated through successful interpolation and coordination across different mathematical representations to solve complex problems, such as deriving the infinite product for π.