My question is about linear forms, quadratic forms, n-linear forms, differential forms and so on. The first term of these names seem clear to me, but I cannot make a link between these mathematical objects and the usual meanings of the word "form". I was unable to find an answer on the Internet. Knowing the first occurence of this use of "form" in mathematics would probably help.

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    $\begingroup$ The terminology goes back to Gauss's Disquisitiones Arithmeticae, where he writes (in Latin):"In this section we shall treat in particular functions in two unknowns $x, y$ of the form $axx + 2bxy + cyy$ where $a, b, c$ are given integers. We will call these functions forms of the second degree or simply forms." For this and similar questions see Jeff Miller's website Earliest Known Uses of Some of the Words of Mathematics. $\endgroup$ – Conifold Jan 27 at 20:36
  • $\begingroup$ From Latin with the meaning of shape, pattern. Today, we would say expressions. $\endgroup$ – Mauro ALLEGRANZA Jan 28 at 13:03

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