*Banca, trapeza, mensa*: the table of the first bankers, between 1450 and 1650.

1. Customer: initial capital

The person in front of the counter represents someone bringing money for a transaction. In this article’s example, we start with 100 units of money: this is the initial capital.

2. Service: checking the accounts

The person behind the counter represents service and checking amounts. To follow an investment in our model, we need the capital, the rate and the number of periods.

3. Coins: interest on interest

In the mathematical example at 100% a year, one annual payment turns 100 into 200. With two half-year periods at 50%, 100 becomes 150, then 150 becomes 225: the second payment also earns interest on the earlier interest.

4. Cash box: saving is not compounding

Stored money does not grow by itself. The interest in this example follows a mathematical payment rule. Explore the chart below to compare annual, monthly and increasingly frequent payments.

Try compound interest ↓ Enlarge the engraving
Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650.Illustrative reconstruction with AI based on an engraving from the Euler’s Eyes collection

Before reaching Euler, the number $e$ passed through the banks. The vocabulary of money comes from Latin: depositum (what was put, from ponere), creditum (what was trusted, from credere, to believe) and juro (from jus, juris, the right).

Imagine R$ 100 applied to **100% per year**. If the bank pays you just once, at the end of the year, you receive R$ 200. But what if he divides the year into periods and pays interest on interest?

Periods in the year Rate per period Account At the end of the year
1 (annual) 100% $(1 + \frac{1}{1})^1 = 2$ R$ 200,00
2 (six-monthly) 50% $(1 + \frac{1}{2})^2 = 2{,}25$ R$ 225,00
4 (quarterly) 25% $(1 + \frac{1}{4})^4 \approx 2{,}4414$ R$ 244,14
12 (monthly) ≈ 8,33% $(1 + \frac{1}{12})^{12} \approx 2{,}6130$ R$ 261,30
365 (daily) ≈ 0,27% $(1 + \frac{1}{365})^{365} \approx 2{,}7146$ R$ 271,46

With $n$ periods, the capital $C_0$ becomes

$$C_n = C_0\left(1 + \frac{1}{n}\right)^n.$$
The dashed curve represents continuous capitalization; the steps, the interest paid in each period. Use the control to compare the balance at the end of the year.

The final value grows, but not without limit: it approaches a number that is neither an integer nor a fraction.

$$\begin{aligned} \left(1 + \tfrac{1}{10}\right)^{10} &= 2{,}5937\ldots \\ \left(1 + \tfrac{1}{100}\right)^{100} &= 2{,}7048\ldots \\ \left(1 + \tfrac{1}{1000}\right)^{1000} &= 2{,}7169\ldots \\ \left(1 + \tfrac{1}{10\,000}\right)^{10\,000} &= 2{,}7181\ldots \\ \left(1 + \tfrac{1}{100\,000}\right)^{100\,000} &= 2{,}7182\ldots \end{aligned}$$
$$\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e = 2{,}718281828\ldots$$
When n grows without limit, (1 + 1/n)ⁿ approaches e. The expression is a limit, not “1 divided by infinity”.
When n grows without limit, (1 + 1/n)ⁿ approaches e. The expression is a limit, not “1 divided by infinity”.Original illustration Euler’s Eyes
John Napier (1550–1617), the inventor of logarithms.
John Napier (1550–1617), the inventor of logarithms.Colorized portrait provided for this article

Jakob Bernoulli had already studied this limit in 1683, without giving it a name. Euler named it $e$ and showed that it is also the sum of a series:

$$e = 1 + \frac{1}{1} + \frac{1}{2 \cdot 1} + \frac{1}{3 \cdot 2 \cdot 1} + \frac{1}{4 \cdot 3 \cdot 2 \cdot 1} + \cdots = \sum_{k=0}^{\infty}\frac{1}{k!}$$

In Introductio (1748), Euler calculated the 23 first decimal places of $e$.

Scribatur pro numero whose logarithmus is unitas, e, qui est 2,7182817… (Write e for the number whose logarithm is the unit.)

— Leonhard Euler, manuscript of 1727–1728

The path to the letter $e$:

  1. 1614
    John Napier publishes Mirifici Logarithmorum Canonis Descriptio, the first table of logarithms.
  2. 1618
    An appendix to the English translation of Napier’s work, made by Edward Wright, contains logarithms in the base that we today call $e$.
  3. 1647
    Grégoire de Saint-Vincent shows that the area under the hyperbola grows as a logarithm.
  4. 1668
    Nicolau Mercator gives the name logarithmus naturalis (natural logarithm) in Logarithmotechnia.
  5. 1683
    Jakob Bernoulli studies the limit of compound interest.
  6. 1727
    Euler uses the letter $e$ in a manuscript, published only in 1862.
  7. 1748
    In Introductio, Euler writes $e$ as a series and calculates 23 decimal places.
The logarithm to the base e is called the natural logarithm: logₑ(x) = ln(x).
The logarithm to the base e is called the natural logarithm: logₑ(x) = ln(x).Original illustration Euler’s Eyes

Why “natural”? Because it is the logarithm with the simplest derivative. Using the limit that defines $e$ (with $n = x/\Delta x$, which goes to infinity when $\Delta x$ goes to zero):

$$\begin{aligned} \frac{d}{dx}\ln x &= \lim_{\Delta x \to 0}\frac{\ln(x + \Delta x) - \ln x}{\Delta x} = \lim_{\Delta x \to 0}\frac{1}{\Delta x}\ln\left(1 + \frac{\Delta x}{x}\right) \\ &= \lim_{\Delta x \to 0}\frac{1}{x}\cdot\frac{x}{\Delta x}\ln\left(1 + \frac{\Delta x}{x}\right) = \frac{1}{x}\lim_{n \to \infty}\ln\left(1 + \frac{1}{n}\right)^{n} \\ &= \frac{1}{x}\ln e = \frac{1}{x}. \end{aligned}$$

And then comes the most famous property of $e^x$. As $\ln(e^x) = x$, deriving both sides:

$$\frac{1}{e^x}\cdot\left(e^x\right)' = 1 \quad\Longrightarrow\quad \left(e^x\right)' = e^x.$$

$e^x$ is the function that is equal to the derivative itself. This property is the key to the discovery 4, about the differential equations.

All topics by Leonhard Euler · ← The Basel problem · Euler’s formula and the “most beautiful equation” →

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