In an attempt to develop a new way of teaching mathematics, we can use the history to find the important ideas that articulated the now classical mathematical theories. My goal is to illustrate through an example, the Partial Differential Equations of the first order, how the different disciplines interact within a theory. This contribution aims indeed to show how the PDEs of first order constitute a point of convergence between geometry, algebra and analysis in the 19th century. Differential calculus of several variables has its origin in the study of the geometric properties of curves and some mechanical problems. Resolutions of PDEs use first analytical methods until Lagrange and Monge who provide a geometrical interpretation. The research about generalization of PDEs of firstorder innvariables oblige the mathematicians to use uniquely analytical methods in the first half of 19th century (Pfaff, Cauchy, Jacobi). The development of projective geometry, the birth of theory of groups, the geometricalvision of algebraic theory of invariants give the conditions for reinterpreting the general PDEs with geometrical methods (Lie, Klein). A few observations upon the teaching of Lie’s Theory will follow by way of conclusion.