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

*formulas*^{a}. (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

*formulas*^{b}or*expressions*^{c}. (See e.g. Weitzenböck 1922, p. 29.)

Question:Has anyone anywhere everdiscussedthese choices and switches in terminology?

$\ $

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).

Let $K/k$ be a field extension, and $X$ an "object" defined over $k$. We say that a object $Y$, defined over $k$, is a $K/k$-form of $X$ it $Y$ becomes isomorphic to $X$ after extending scalars to $K$) $\endgroup$ – YCor Oct 28 '17 at 14:04