What are the direct predecessors of Lagrange's theory of quadratic forms?

The motivation for the theory of quadratic forms was representation of integers by sums of squares. This business was started by Fermat and continued by Euler, Lagrange and Legendre, then Gauss. Generalization from sums of squares to arbitrary quadratic forms is natural. Two forms represent the same integers when they are equivalent. Discriminant is the natural invariant of this equivalence relation, so it gives certain classification of quadratic forms.

Ref. André Weil, Number theory. An approach through history from Hammurapi to Legendre, Birkhäuser Boston, Inc., Boston, MA, 2007.