# 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 $e$.

• Euler, the Master of Us All shows how, I think. It's available online for free. – Akiva Weinberger Oct 13 '15 at 16:43
• You might like to look at `Higher Trigonometry, Hyperreal Numbers and Euler's Analysis of Infinities' by Tuckey and McKinzie where they rehabilitate Euler's derivation of the series for the exponential function using nonstandard analysis. It's pretty readable. – IIM Aug 29 at 10:57

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.

• Rigth but I don't think he actually derives the infinite series in it. – Roland Oct 13 '15 at 14:19