Images from Wikimedia Commons, all in the public domain (or CC0): Portrait of Euler: Jakob Emanuel Handmann, 1753. Basel at 1615: Matthäus Merian. Johann Bernoulli: Rijksmuseum (CC0). Saint Petersburg Academy of Sciences: G. Kachalov, after M. Makhaev, 1753. Frederick II: Antoine Pesne's studio, 1745. Introduction to analysin infinityum, 1748. Lettres à une princesse d'Allemagne, 1768. Catherine the Great: Alexander Roslin. Euler at 1778: Joseph Friedrich August Darbes. Königsberg at 1652: Merian-Erben. Figure of bridges: Leonhard Euler, 1736. The images in the sections on the square root, the ages of mathematics, the number $e$ and the spring are from the collection of Euler’s Eyes. The interactive figures were made for this website.
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.
John Napier was so important for applications in finance that he motivated Leonhard Euler to venture into this area. One of the irrational numbers most important with application in finance, is due to these two mathematicians. Compound Interest The history of compound interest and the number $e$ was thanks to Jacques Bernoulli between (1654-1705), who became interested in discovering the maximum value of interest on loans using the compound interest technique.