Tracing the word “form” Today the word form can refer to (at least) three different kinds of mathematical object:


*

*A homogeneous polynomial. This was apparently started by Gauss (1801), renaming what others had called formulasa. (See e.g. Bachmann 1922, p. 17.)

*A scalar-valued linear or multilinear map. Apparently started by Kronecker (1866) / Weierstrass (1868), rather out of the blue.

*A field of forms in the sense 1 or 2. Apparently started by Christoffel (1869) / Lipschitz (1869), renaming what others called differential formulasb or expressionsc. (See e.g. Weitzenböck 1922, p. 29.) 

Question: Has anyone anywhere ever discussed these choices and switches in terminology?

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References: e.g.
a)
Euler (1770, 1774, 1827), Lagrange (1773, 1774), Liouville (1852).
b)
Bernoulli (1712),
Euler (1755, 1768),
Agnesi (1775), Cousin (1777),
Lagrange (1786), Bossut (1798), Poisson (1811), Abel (1826), Liouville (1852, 1856).
c)
Gauss (1815), Jacobi (1845), Riemann (1867), Sturm (1877), Frobenius (1879), Darboux (1882), Cartan (1899).
 A: The evolution of the concept of a form from arithmetic to algebra is discussed on pp. 20, 21, 27 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 Disquisitiones arithmeticae, they pointed to various and mostly implicit meanings within the algebraic framework of the discussion. (...)
Kronecker had been implicitly referring 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.s computations.

