If anyone deserves the title of “architect of modern notation”, it is Leonhard Euler. He wrote more pages of mathematics than anyone in history, and he wrote to be read: his books were used for teaching, and the conventions he adopted spread along with them. Many symbols that seem to have always existed passed through his hands.

Leonhard Euler (1707–1783), portrait by Jakob Emanuel Handmann, 1753.
Leonhard Euler (1707–1783), portrait by Jakob Emanuel Handmann, 1753.Jakob Emanuel Handmann · public domain
  1. 1706
    William Jones uses $\pi$ for the ratio of circumference to diameter. Hardly anyone notices.
  2. 1727
    In a manuscript, Euler calls $e$ the number whose logarithm is $1$.
  3. 1734
    Euler writes $f\!\left(\frac{x}{a}+c\right)$: the birth of the notation $f(x)$ for functions.
  4. 1736
    In the Mechanica, Euler adopts Jones’s $\pi$; with the Introductio (1748) the symbol becomes standard.
  5. 1755
    Euler uses $\Sigma$ to denote a sum.
  6. 1777
    Euler uses $i$ for $\sqrt{-1}$, in a text published only after his death.

What these symbols have in common is that each turns a description into a name. Before Euler, to talk about a function you had to describe the rule (“the quantity formed from $x$ by squaring and adding one”). After $f(x)$, the rule gets a short name, and a name can go into other calculations: $f(x+h)-f(x)$, $f(g(x))$, $f'(x)$. The function becomes an object to be handled the way a number was handled. For more on each symbol, see the language Euler left us.

The function, a sentence with a hole

The word function comes from the Latin functio, from the verb fungi: to perform a task, to carry out an office. A function is something that does something. The symbol took almost a hundred years to reach $f(x)$:

Year Author How the function appeared
1637 René Descartes an equation between $x$ and $y$, tied to a curve
1667 James Gregory quantitas, a quantity obtained from others
1671 Isaac Newton fluent, the quantity that flows
1673 Leibniz the word functio
1718 Johann Bernoulli $\varphi x$, without brackets
1734 Leonhard Euler $f\!\left(\frac{x}{a}+c\right)$, with brackets
1837 Peter Gustav Lejeune Dirichlet the modern definition: any rule that assigns to each $x$ a single $y$

The old texts on this site proposed a curious way of reading a function: as a sentence with a hole in it. The function “double” would be

$$f=2\cdot(\quad),$$

an instruction waiting for a number. Putting $3$ in the hole gives $f(3)=2\cdot3=6$; putting $x$ gives $f(x)=2x$. The function “subtract two” would be $(\quad)-2$, and “$(\quad)-1$ times $(\quad)-3$” describes a parabola that crosses the axis at $1$ and at $3$.

The idea is not just a teaching device. In 1891 Gottlob Frege, whom we will meet in the text on logic, defended exactly this in the essay Function and Concept: a function is an unsaturated expression, with an empty place, such as $2\cdot(\quad)^3+(\quad)$, which only becomes a number when the empty place is filled. Euler’s $x$ is the economical way to mark that hole, and the brackets of $f(x)$ literally draw the place where the input fits.

Exercise: fill the hole

The function $g=(\quad)^2-1$ takes a number, squares it and subtracts $1$. So $g(3)=3^2-1=8$, $g(-3)=8$ and $g(0)=-1$. Writing the hole with Euler’s letter: $g(x)=x^2-1$. And $g(x+1)$? Just put the whole of $x+1$ in the hole: $g(x+1)=(x+1)^2-1=x^2+2x$. People who do not think of the hole often write $x^2-1+1$, which is a different function.

Exercise: f(x+h) − f(x)

For $f(x)=x^2$, compute $f(x+h)-f(x)$. Putting $x+h$ in the hole: $(x+h)^2-x^2=2xh+h^2$. Dividing by $h$: $2x+h$, which tends to $2x$ as $h$ tends to zero. This is the derivative of $x^2$, and the calculation is only this short because Euler’s notation lets you write “the value of the function at a neighbouring point” in four signs.

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