Euler and functions: giving a rule a name
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.Jakob Emanuel Handmann · public domain 1706William Jones uses $\pi$ for the ratio of circumference to diameter. Hardly anyone notices. 1727In a manuscript, Euler calls $e$ the number whose logarithm is $1$. 1734Euler writes $f\!\left(\frac{x}{a}+c\right)$: the birth of the notation $f(x)$ for functions. 1736In the Mechanica, Euler adopts Jones’s $\pi$; with the Introductio (1748) the symbol becomes standard. 1755Euler uses $\Sigma$ to denote a sum. 1777Euler 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.
