TITLE: How was $e$ first calculated?
QUESTION [6 upvotes]: I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed.  What mathematical principles are behind the value of $e$?

REPLY [6 votes]: I am quoting here from an ancient USENET post of Matthew P. Wiener, who points out that explanations involving $\sum\frac1{n!}$ and $\left(1+\frac 1n\right)^n$ were not actually the first appearances or calculations of $e$:

Napier, who invented logarithms, more or less worked out a table of
  logarithms to base $\frac1e$, as follows:
0 1 2 3  4  5  6   7   8   9   10 ...
1 2 4 8 16 32 64 128 256 512 1024 ...

The arithmetic progression in the first row is matched by a geometric
  progression in the second row. If, by any luck, you happen to wish to
  multiply 16 by 32, that just happen to be in the bottom row, you can
  look up their "logs" in the first row and add 4+5 to get 9 and then
  conclude 16·32=512.
For most practical purposes, this is useless. Napier realized that what
  one needs to multiply in general is $1+\epsilon$ for a base—the intermediate
  values will be much more extensive. For example, with base 1.01, we get:
  0 1.00   1 1.01   2 1.02   3 1.03   4 1.04   5 1.05
  6 1.06   7 1.07   8 1.08   9 1.09  10 1.10  11 1.12
 12 1.13  13 1.14  14 1.15  15 1.16  16 1.17  17 1.18
 18 1.20  19 1.21  20 1.22  21 1.23  22 1.24  23 1.26
 24 1.27  25 1.28  26 1.30  27 1.31  28 1.32  29 1.33
 30 1.35  31 1.36  32 1.37  33 1.39  34 1.40  35 1.42
[...]
 50 1.64  51 1.66  52 1.68  53 1.69  54 1.71  55 1.73
[...]
 94 2.55  95 2.57  96 2.60  97 2.63  98 2.65  99 2.68
100 2.70 101 2.73 102 2.76 103 2.79 104 2.81 105 2.84
[...]

So if you need to multiply 1.27 by 1.33, say, just look up their logs,
  in this case, 24 and 29, add them, and get 53, so 1.27·1.33=1.69. For
  two/three digit arithmetic, the table only needs entries up to 9.99.
Note that $e$ is almost there, as the antilogarithm of 100. The natural
  logarithm of a number can be read off from the above table, as just $\frac1{100}$ the corresponding exponent.
What Napier actually did was work with base .9999999. He spent 20 years
  computing powers of .9999999 by hand, producing a grand version of the
  above. That's it. No deep understanding of anything, no calculus, and
  $e$ pops up anyway—in Napier's case, $\frac1e$ was the 10 millionth entry. (To
  be pedantic, Napier did not actually use decimal points, that being a new
  fangled notion at the time.)
Later, in his historic meeting with Briggs, two changes were made. A
  switch to a base $> 1$ was made, so that logarithms would scale in the
  same direction as the numbers, and the spacing on the logarithm sides
  was chosen so that $\log(10)=1$. These two changes were, in effect, just
  division by $-\log_e(10)$.
In other words, $e$ made its first appearance rather implicitly.
The calculus connection came later. Fermat had successfully solved the
  quadrature problem for $y=x^n$ for $n\ne -1$, but not for $y=\frac1x$. Fermat's
  method was to use geometrically spaced intervals on the $x$ axis, and to
  add the resulting areas. It took a bit of time for a contemporary to
  notice that this method produced arithmetically spaced areas under the
  hyperbola—ie, that there's a logarithm going on.