The word comes from the Latin functus, participle of the verb fungor: fulfill a task, exercise an activity. The function is what “does the work” of taking one number to another. Already defunctus is someone who no longer performs any function, and hence the word defunct.

A Leibniz notation for functions, from letters to Johann Bernoulli.
A Leibniz notation for functions, from letters to Johann Bernoulli.Euler’s Eyes Collection

The name and symbol took more than a century to establish itself:

  1. 1637
    Descartes, in Geometry, describes curves by equations between $x$ and $y$.
  2. 1667
    James Gregory calls quantitas a quantity obtained from others by operations (Vera Circuli et Hyperbolae Quadratura).
  3. 1671
    Newton calls quantities that vary fluents (Methodus Fluxionum, published only in 1736).
  4. 1673
    Leibniz uses the word functio for the first time, in a manuscript.
  5. 1698
    In the letters between Leibniz and Johann Bernoulli, the term takes on today’s mathematical meaning.
  6. 1718
    Johann Bernoulli gives the first definition: a quantity composed in any way of a variable and constants.
  7. 1734
    Euler creates the notation $f(x)$: “f(x) indicates any function of x”.
  8. 1748
    Euler, in Introductio, puts the function at the center of the calculation.
  9. 1797
    Lagrange uses $f'(x)$ for the derivative (Théorie des fonctions analytiques).
  10. 1837
    Dirichlet defines a function as any rule that associates each $x$ with a single $y$.

All topics by René Descartes · ← Descartes and the problem of cannons · Physis: deciphering the traits of nature →

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