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](https://archive.org/stream/werkecarlf01gausrich#page/n129)), renaming what others had called ***formulas***a. (See e.g. Bachmann [1922](https://www.emis.de/cgi-bin/JFM-item?48.1108.03), [p. 17](https://archive.org/stream/s2p1werkehrsgvon10gausuoft#page/n22).)
2. A scalar-valued linear or multilinear map. Apparently started by Kronecker ([1866](https://books.google.com/books?id=mxFLAQAAMAAJ&pg=PA597)) / Weierstrass ([1868](https://books.google.com/books?id=qClJAAAAcAAJ&pg=PA310)), rather out of the blue.
3. A field of forms in the sense 1 or 2. Apparently started by Christoffel ([1869](https://books.google.com/books?id=9D9bAAAAcAAJ&pg=PA56&q=%22lineare+Form%22#v=onepage)) / Lipschitz ([1869](https://books.google.com/books?id=9D9bAAAAcAAJ&pg=PA74&q=%22Formen+von%22#v=onepage)), renaming what others called differential ***formulas***b or ***expressions***c. (See e.g. Weitzenböck [1922](https://www.emis.de/cgi-bin/JFM-item?48.0517.01), [p. 29](https://archive.org/stream/encyklomath303encyrich#page/n654).)



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> **Question:** Has anyone anywhere ever *discussed* these choices and switches in terminology?
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References: e.g.  

a)
Euler ([1770](https://books.google.com/books?id=wB1aAAAAcAAJ&pg=PA379&dq=Formel), [1774](https://archive.org/stream/elemensdalgbre02eule#page/195), [1827](https://archive.org/stream/elementsofalgebr00eule#page/387)), Lagrange ([1773](https://archive.org/stream/uvresdelagrange03lagr#page/695), [1774](https://archive.org/stream/elemensdalgbre02eule#page/636)), Liouville ([1852](https://books.google.com/books?id=y5tbAAAAQAAJ&pg=PA478)).  

b)
Bernoulli ([1712](https://books.google.com/books?id=Db9eAAAAcAAJ&pg=PP739&dq=formule)),
Euler ([1755](https://archive.org/stream/bub_gb_sYE_AAAAcAAJ#page/n229), [1768](https://archive.org/stream/institutionescal020326mbp#page/n297)),
Agnesi ([1775](https://books.google.com/books?id=XQ1bAAAAQAAJ&q=%22formule+diff%C3%A9rentielle%22)), Cousin ([1777](https://books.google.com/books?id=obF1teMTx6oC&pg=PA289&dq=%22formule+diff%C3%A9rentielle%22)),
Lagrange ([1786](https://archive.org/stream/mmoiresdelacadm04turigoog#page/n229)), Bossut ([1798](https://books.google.com/books?id=pro_AAAAcAAJ&pg=PA127&dq=formule)), Poisson ([1811](https://books.google.com/books?id=zt2wKlNovI8C&pg=PA291&dq=formule)), Abel ([1826](https://books.google.com/books?id=ZppGAAAAcAAJ&pg=PA185&dq=Differential+Formel)), Liouville ([1852](https://books.google.com/books?id=y5tbAAAAQAAJ&pg=PA478), [1856](https://books.google.com/books?id=mMVDfrvRyb0C&pg=PA445&dq=%22formule+diff%C3%A9rentielle%22)).  

c)
Gauss ([1815](https://archive.org/stream/werkecarlf03gausrich#page/n245)), Jacobi ([1845](https://books.google.com/books?id=EgVCAAAAcAAJ&pg=PA240&dq=expressio+differentialis)), Riemann ([1867](https://books.google.com/books?id=P8qTb9q1T9QC&pg=PA8&dq=Differentialausdruck)), Sturm ([1877](https://books.google.com/books?id=N-I3AAAAMAAJ&pg=PA38&dq=expression+diff%C3%A9rentielle)), Frobenius ([1879](https://books.google.com/books?id=85HxAAAAMAAJ&pg=PA1&dq=Differentialausdruck)), Darboux ([1882](http://www.numdam.org/item?id=BSMA_1882_2_6_1_14_1)), Cartan ([1899](https://doi.org/10.24033/asens.467)).



Tracing the word “form”
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](https://arxiv.org/abs/0712.2566); revised version published in [2016](http://www.ams.org/mathscinet-getitem?mr=3617469)):



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> 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. (...)
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> Kronecker had been implicitly referring to the legacy of the works of Gauss and Hermite on the arithmetic of quadratic forms in [1866](https://books.google.com/books?id=mxFLAQAAMAAJ&pg=PA597) — [when] he preferred to make use of the term “form” to name what others would designate as a function ([Weierstrass, [1858](https://books.google.com/books?id=V-Q6AQAAMAAJ&pg=PA207)]) or as a “polynom” ([Jordan, [1873](http://gallica.bnf.fr/ark:/12148/bpt6k3034n/f1487)]) (...)
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> 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.
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