The evolution of the concept of a form from arithmetic to algebra is discussed on page 20 and following of F. Brechenmacher (arXiv:0712.2566; revised version published in 2016).
Whereas such terms as forms and transformations had been given an explicit mathematical definition in the arithmetic of quadratic forms in relation to the notion of equivalence relation that had been introduced by Gauss’ 1801 Disquitiones arithmeticae, they pointed to various and mostly implicit meanings within the algebraic framework of the discussion.
Kronecker implicitly referred to the legacy of the works of Gauss and Hermite on the arithmetic of quadratic forms in 1866 – when he preferred to make use of the term form to name what others would designate as a function [Weierstrass, 1858] or as a polynom [Jordan, 1873].
Kronecker blamed algebraic methods, notably by Jordan, for their tendency to think in term of the “general” case with little attention given to the arithmetic difficulties that might be caused by assigning specific values to the symbols. He appealed to the tradition of Gauss on behalf of his claim that the theory of forms should be considered as belonging to arithmetic and should consequently focus on the characterisation of equivalence classes in establishing arithmetical invariants thanks to some effective procedures such as g.c.d computations.
