Most sources attribute the first use of "function" in the context of mathematics to Leibniz. But D'Alembert, Lacroix and Dini claim the following:

D'Alembert in Encyclopédie 1757:

les anciens géometres, ou plûtôt les anciens analystes ont appellé fonctions d'une quantité quelconque $x$ les différentes puissances de cette quantité.

Lacroix in Traité du calcul différentiel et du calcul intégral 1797, p.1:

Les anciens Analystes comprenoient en général sous la dénomination de fonctions d'une quantité, toutes les puissances de cette quantité.

Dini in Fondamenti per la teorica della funzioni di variabili reali 1878, p.35

Gli antichi usarono dapprima la parola funzione per esprimere le varie potenze du un stessa quantità, e solo da Leibnitz, dai Bernoulli e più specialmente poi da Eulero fu esteso il concetto di funzione [...]

None of them explain who they mean by ancients, but Dini makes it quite clear that he doesn't mean Leibniz. So who were these ancients? The greeks?

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    $\begingroup$ This may come from Leibniz himself in Historia et origo calculi differentialis (written 1714-16 in reply to the famous Commercium epistolicum; p. 2; transl. Child): "before [me] no other functions were admissible but $x$, $xx$, $x^3$ , $\surd x$, etc., that is to say, powers and roots." Note 1) He doesn't claim those precursors (Kepler, Cavalieri, Fermat, Huygens, Wallis, Vieta, Descartes) said "function". $\endgroup$ Jul 26, 2017 at 1:43
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    $\begingroup$ And 2) This remained a manuscript until 1846, so it's not clear that D'Alembert could have seen it. (Lagrange also repeats D'Alembert's version in Théorie des fonctions analytiques.) $\endgroup$ Jul 26, 2017 at 1:48

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The Encyklopädie article "Principles of the theory of functions" (1899, p. 3) relays the assertion, quoting Lagrange and Lacroix, but notes it's unconfirmed. In an "Encyclopedia Chat Room", Archiv der Math. u. Phys. 3 (1902) 317-319, G. Eneström comments:

Lagrange's assertion is no doubt literally false, presumably all he meant is that the first analysts generally dealt with no functions other than powers.

The much-expanded translation in Encyclopédie (1912, p. 3) traces the quotes of D'Alembert and Leibniz we have above, considers the possibility that D'Alembert used "function" to translate the latin "dignitas" once used for powers (Tartaglia, Bombelli), but in the end, sides with Eneström:

This assertion of J. d'Alembert is no doubt inaccurate (...) one can affirm that J. d'Alembert likely didn't bother to find out whether early analysts actually used the word "function", or not.


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