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I read in a not always reliable source (David Foster Wallace's Everything and More, p.104), that Leibniz introduced the terms constant, variable, and function, the latter as an alternative to Newton's fluent. (Perhaps Leibniz used the German words Konstante, Veränderliche, Funktion?) Can anyone verify or undermine this attribution of fundamental terms of our current mathematical language?

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    $\begingroup$ It is worth pointing out that while Leibniz introduced the names (along with "coordinate" and "parameter"), and boosted development of analysis with systematic terminology, he did not introduce the concepts. Variable first appears in Diophantus's Arithmetica c. 250 AD, this is what first $x$ looked like $\varsigma$. Diophantus does not use letters for constant terms, so he is forced to assign numerical value to them instead of working generally. Nemorarius writes constant terms c.1225. $\endgroup$ – Conifold May 27 '15 at 4:27
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    $\begingroup$ By "function" Leibniz meant a segment in a diagram that "serves a function" there. Even "function" of 18th century, as analytic expression, was only introduced by Johann Bernoulli in 1698, prompting Leibniz to write "I am pleased that you use the term function in my sense",which he really wasn't. Euler came up with the $f(x)$ notation. Modern notion of function does not appear until Fourier in 1822, and even he did not appreciate full generality of his own definition. hsm.stackexchange.com/questions/2035/… $\endgroup$ – Conifold May 27 '15 at 4:40
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Wallace may have read Kline's Mathematical thought from ancient to modern times, which on p. 340 makes this attribution to Leibniz in De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu, Acta Eruditorum (1692) 168-171. This short text, of which a French translation is available here, has indeed functiones, constans, variabilis (and more) italicized by Leibniz himself.

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  • $\begingroup$ Thanks, Francois, you are surely right in your conjecture of DFW's source. (And thanks for correcting my Deutsch!) $\endgroup$ – Joseph O'Rourke May 26 '15 at 21:10
  • $\begingroup$ Not German, it was Latin... :) $\endgroup$ – Frunobulax Jan 20 '16 at 9:23
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You can see into :

Historia et Origo Calculi Differentialis (1714-1716).

The original text is in Latin and you can find there occurrences of :

"considerando differentias $dx$ [...] funciones quasdam ipsarum $x$" ["considering the differences $dx$ [...] as functions of the $x$'s"] (page 2)

and :

"ita jam quantitates [consideratae] ut variantes, habere novas functiones nempe differentias." ["also such quantities [considered] as were variable had new functions, namely, differences."] (page 17).

Newton instead, into his Introductio ad Quadraturam Curvarum (published 1704), writes :

Quantitates Mathematicas non ut ex partibus quam minimis constantes, sed ut motu continuo descriptas hic considero.

He consider curves as described by continuous motion, and define his :

Method of determining the Quantities from the Velocities of their Motions or Increments, by which they are generated; and by calling the Velocities of the Motions, or of the Augments, by the Name of Fluxions, and the generated Quantities Fluents [...].


But I think that the main difference between the two approach was not regarding "terminology", but "symbolism".

The "algebraic" symbols invented by Leibniz was one of the strong point supporting the success of his new calculus.

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  • $\begingroup$ Oh, yes, of course he wrote in Latin! $\endgroup$ – Joseph O'Rourke May 25 '15 at 11:01

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