From conway@and.Princeton.EDU Tue May 27 11:07:11 1997
Date: Tue, 27 May 1997 10:41:36 -0400 (EDT)
From: John Conway
Subject: Re: invention of e (my guess)
To: Kermit Rose
Well, I'm afraid it ain't so: the number e was discovered
well before the notion of derivative. The following fable is
a bit closer to the truth.
The logarithms to base 2 of
1,2,4,8,16,32,64,...
are respectively 0,1,2,3, 4, 5, 6, ..., but unfortunately these
are too widely spaced to be easily interpolated. If we choose a smaller
base, say 1.1, we'll get a closer-spaced sequence of powers:
1, 1.1, 1.21, 1.331, 1.4641, 1.61051, ... with logarithms
0, 1, 2, 3, 4, 5, ...
making it easier to interpolate. Let's look at the effect of
choosing ever-smaller bases on the logarithms of 2 and 3:
We have
1.001^693 = 1.999013... , 1.001^694 = 2.001012...
1.001^1099 = 2.999516... , 1.001^1100 = 3.002515...,
so that the logs to base 1.001 of 2 and 3 are about 693.5 and 1099.2.
To bases 1.000001 and 1.0000001 we find
log 2 = 693147.53 and 6931472.15
log 3 =1098612.84 and 10986123.44.
Aha! We should obviously divide these by 10^6 and 10^7 respectively,
so getting
log 2 = .69314753 and .693147215
log 3 = 1.09861284 and 1.098612344 with respect to
the bases (1.000001)^10^6 (1.0000001)^10^7 respectively.
What ARE these bases? Well, the second is about 2.718281693... .
If we take logarithms to base 1 + 1/N and divide them by N,
we're really taking logarithms to base (1 + 1/N)^N, which gets
closer and closer to the mysterious number e as N gets larger
and larger.
This, very roughly, is what Napier did when computing his first
table of logarithms in about 1609, about 60 years before the
discovery of the calculus. It really IS "very roughly" though,
because in fact his table was of logarithmic sines, and his
logarithm of 1 wasn't 0.
John Conway