Euler's derivation of e? Does anyone know where I can read Euler's original derivation of the infinite series used to define $e$?
I mean the series as defined in the wikipedia page about $e$.
 A: It seems that Euler used the letter $e$ to represent the number $2.71828...$ in one of his earliest works, a manuscript entitled "Meditation
upon Experiments made recently on the firing of Cannon," dated 1727 (published in 1862).
The first "published" $e$ by Euler seems to be in :

Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68 :
Erit enim $\frac{dc}{c} = \frac{dy ds}{rdx}$ seu $c = e^{\int\frac{dy ds}{rdx}}$ ubi $e$ denotat numerum, cuius logarithmus hyperbolicus est $1$. (So it [i.e., $c$, the speed] will be $\frac{dc}{c} = \frac{dy ds}{rdx}$ or $c = e^{\int\frac{dy ds}{rdx}}$, where $e$ denotes the number whose hyperbolic [i.e., natural] logarithm is $1$.)


Jakob Bernoulli was the first to point out the connection between $\lim_{n \to \infty} (1 + 1/n)^n$ and the problem of continuous compound interest. By expanding the expression $(1 + 1/n)^n$ according to the binomial theorem (see
this post), he showed that the limit must be between 2 and 3.

For $e^z = \lim_{n \to \infty} (1 + z/n)^n$, see Leonhard Euler, Introductio in analysin infinitorum : Tomus I (1748, new ed.1797), page 90.
