Historical basis of e
Matthew P Wiener
>I sometimes heard that the historical ground is such:
For a detailed account, see Eli Maor E: THE STORY OF A NUMBER (Princeton,
1994).
>You define a^x and log_a for any positive a. Trying to evaluate the
>derivative of these functions, you immediately notice, that you need
>only the derivative of log_a at 1. This is (if it exists, which all
>historical men had no doubt about) lim_{n to infty} n log_a(1+1/n) =
>lim log_a (1+1/n)^n. You are automaticall led to the limit of
>(1+1/n)^n.
This is true as mathematicis, but revisionist as history.
Napier, who invented logarithms, more or less worked out a table of
logarithms to base 1/e, 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
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 1/100
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, 1/e 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!=-1, but not for y=1/x. 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.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)