We begin the account of the events leading to the development of the theory of shock waves (reviewed in Section 1.2) around the time of the creation of the École Polytechnique of Paris, just prior to the end of the eighteenth century. A better understanding of those events requires a review of the changes in mathematical thinking that took place during the preceding half century. Of particular importance is the emergence of the concept of a function and the introduction of partial differential equations as models of physical phenomena. The latter was a paradigm shift in which equations replaced axioms for the description of physical phenomena. The key players in these events were the mathematical savants Jean d’Alembert, Leonhard Euler, Luigi de la Grange (Lagrange), and the experimentalist Daniel Bernoulli.