The theory was vastly extended and refined by Gauss in Section V of "Disquisitiones Arithmeticae". Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a group under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. (Gauss and many subsequent authors wrote 2"b" in place of "b"; the modern convention allowing the coefficient of "xy" to be odd is due to Eisenstein).