The evolution of the concept of a *form* from arithmetic to algebra is discussed on page 20 and following of <A HREF="https://arxiv.org/abs/0712.2566">arXiv:0712.2566 </A> (F. Brechenmacher, 2007).

> 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 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.