# Tracing the word “form”

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

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

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

3. 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).

• On an $n$-dimensional vector space, a linear form $\varphi$ is the same thing as a homogeneous polynomial in $n$ variables (indeterminates) of degree 1 once you express $\varphi$ in terms of a basis. Thus your second and third mathematical objects are closely related. – KConrad Oct 28 '17 at 4:34
• You can also think of forms of of an algebraic group (groups defined over the same field and isomorphic to it over some extension), or to (automorphic) form as particular elements in an automorphic representation. – Desiderius Severus Oct 28 '17 at 6:40
• @DesideriusSeverus it's not specific to algebraic groups, but to plenty of structures defined over a base ring or field (scheme, variety, algebra, vector space with a quadratic form, etc). Serre's book "Galois cohomology", Chap 3, starts with a paragraph "forms" in this general sense (albeit with no general definition). (quoth: 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 $K$) – YCor Oct 28 '17 at 14:04
• Not to be confused with the question "Forming the word 'trace'." – R. van Dobben de Bruyn Oct 29 '17 at 22:12
• Formally, "formula" is the diminutive form of "forma". – Pietro Majer Oct 29 '17 at 22:14