# Grassmann's “forms”

In Grassmann's famous article Ausdehnungslehre from 1844 (the one where he introduces what has come to be famous as Grassmann algebra) he uses the termionology "form" in place of, as he explains in his introduction, "quantity" or really "magnitude" (namely German: Größe). He says that he does so because he finds "quantity"/"magnitude" to be too restrictive to capture what he has in mind.

Indeed, from a modern perspective what he has in mind and what he discusses at length in the bulk of the article is what today we call differential forms on Euclidean space. Maybe what he considers are only the constant differential forms on Euclidean spaces, but anyway.

Certainly one sees from this why he found "magnitude" to be too restrictive. But besides this it seems striking that where he says "form" we still say "form". Or is it again instead of still?

That's my question: is there a continuity of usage from Grassmann's terminology "form" to modern terminology? Or is it a coincidence that when we say "differential form" today, deriving from "linear form", that this happens to use the same word for the same concept as Grassmann did, who seems to have been motivated from more general ideas about "forms of thought".

• I have not read Grassmann because I cannot read in German, but the word "form" is a traditional name for what is also called "homogeneous polynomial", and I di not think this meaning changed since Grassmann times. – Alexandre Eremenko Mar 6 '15 at 19:45
• I think that the stress is on "formal", meaning abstract (from magnitude, either geometrical or arithmetical), like in "formal logic". See Ausdehnungslehre, §3 : "reine Mathematik Formenlehre". – Mauro ALLEGRANZA Jan 22 '17 at 20:26
• Compare with Alfred North Whitehead. A Treatise on Universal Algebra (1898), where the debt to Grassmann is explicit from the Preface. "Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning. The reasoning is formal in the sense that the meaning of propositions forms no part of the investigation." – Mauro ALLEGRANZA Jan 22 '17 at 20:34
• And page 4 : "When once the rules for the manipulation of the signs of a calculus are known, the art of their practical anipulation can be studied apart from any attention to the meaning to be assigned to the signs." And page 11 : "Algebra does not depend on Arithmetic for the validity of its laws of transformation." – Mauro ALLEGRANZA Jan 22 '17 at 20:35
• Of course, Whitehead's concept of "uninetrpreted calculus" is due also to Boole with multi-interpreted algebra. – Mauro ALLEGRANZA Jan 22 '17 at 20:41

The site Earliest Known Uses of Some of the Words of Mathematics cites Katz, who in The History of Stokes' Theorem (1979) attributes the first use of "exterior differential form" to Elie Cartan in 1922.

Katz writes p.154:

The mathematician chiefly responsible for clarifying the idea of a differential form was Elie Cartan. In his fundamental paper of 1899 [Sur certaines expressions diffrentielles et sur le probleme de Pfaff], he first defines an "expression differentielle" as a symbolic expression given by a finite number of sums and products of the $n$ differentials $dx_1,dx_2,\ldots,dx_n$, and certain coefficient functions of the variables $x_1,x_2,\ldots,x_n$. A differential expression of the first degree, $A_1 dx_1 + A_2 dx_2 + \ldots + A_n > dx_n$, he calls an "expression de Pfaff".

[...]

By 1922, Cartan had extended his work on differential expressions in Leçons sur les invariants intégraux. It is here that he first uses the current terminology of "exterior differential form" and "exterior derivative."

I'm not sure if "differential form" (without the exterior) was used prior to that. I randomly picked a work of Cartan from 1901 L'intégration des systèmes d'équations aux différentielles totales and a search for "forme" didn't yield any "forme differentielle". But I found the following in a footnote p.255:

Par example, si les équations de $E_p$ sont $$P_1 =P_2 = \ldots = P_{r-p}=0,$$ où les $P$ sont des formes linéaires en $dx_1,\ldots,dx_r$, les equations de $E_{p+1}$ sont [...]

This suggests, as you did in the question, that if there is a continuity between Grassmanns "form" and "differential form", it might be through the use of "linear form". But I don't know when the use of the latter startet.

• I think it much predates Cartan, although the word form is so generic it's sometimes hard to tell. E.g. Darboux’ Sur le problème de Pfaff (1882; 1882) talks of differential expressions “of this form”, “of that form”, but also (p. 835) “Consider a form $\theta_d=X_1dx_1+\dots+X_ndx_n \dots$”, (pp. 23, 25, 58) “Denote by $\smash{\Theta^n_d}$ a differential form...” – Francois Ziegler Aug 16 '17 at 2:58
• Earlier Frobenius in Ueber das Pfaffsche Problem (1877) has plenty of linear and bilinear forms, and points to earlier ones in Stickelberger (1874), Kronecker (1874, 1868),... – Francois Ziegler Aug 16 '17 at 2:58
• ...Christoffel (1869): “eine in den Differentialen $\underset1\partial\, x, \underset2\partial\, x,\dots\underset\mu\partial\, x$ $\mu$-fach lineare form”, Lipschitz (1869), Weierstrass (1868),... (Time to start a community-wiki errata for the “earliest known uses” site?) – Francois Ziegler Aug 16 '17 at 3:11
• The errata sounds like a good idea. – Michael Bächtold Aug 16 '17 at 6:05

“Differential forms” seem to have also evolved from “differential formulas” — a terminology used by e.g. Cousin (1777):

(p. 175): la quadrature et la rectification des lignes courbes (...) se réduiront visiblement à intégrer une différentielle de cette forme $X\,dx$, par $X$ j'entens une fonction de $x$ et de constantes: réciproquement, on pourra toujours faire dépendre l’intégrale d’une formule différentielle telle que $X\,dx$, de la quadrature ou de la rectification de quelque courbe (...)

(pp. 287–289): On a la formule différentielle homogène $(hx+iy)dx-(kx+ly)dy$ qu’il faut rendre exacte (...) Soit $\mu$ un facteur propre à rendre intégrable la formule différentielle quelconque du premier ordre $\alpha\,dx + ϐ\,dy$ (...)

(p. 403): Chapitre VII: De l'intégration des formules différentielles qui ne renferment qu’une seule variable. Il va être question de l’intégration de la formule différentielle $X\,dx$ (...)

Other examples: Agnesi (1775), du Bourguet (1810), Garnier (1812), Abel (1826, 1829), Liouville (1835, 1851, 1856). Initially some authors1 only used “differential formula” for exact forms, calling the general $\sum A_i\,dx_i$ another name such as “differential”2, “formula”3, or “differential expression”4.

1 Bernoulli (1712), Euler (1767, 1793, 1793), Lagrange (1786).
2 Le Seur–Jacquier (1768), Sauri (1774).
3 Euler (1770), Lexell (1772), Bossut (1798), Poisson (1811).
4 Gauss (1815), Jacobi (1845), Sturm (1877), Frobenius (1877, 1879).

As mentioned in comments to M. Bächtold’s answer, the switch to “differential form” (and merger with the multilinear terminology of Grassmann(?), Kronecker, Weierstrass) seems to have occurred in papers of Christoffel (1869), Lipschitz (1869), Darboux (1882, 1882).

• Thanks for this very detailed collection of pointers to the literature! – Urs Schreiber Oct 25 '17 at 14:52
• @UrsSchreiber: You’re welcome; I added some more to try and cover 19th century evolution. – Francois Ziegler Oct 26 '17 at 0:51