The general theory of quadratic forms was initiated by Lagrange in 1775 in his "Recherches d'Arithmétique". Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms." He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since". Lagrange showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic class number. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of infrastructure. In 1798, Legendre published "Essai sur la théorie des nombres", which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.