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summing distances in the geometrical progression corresponds to multiplying ele-
ments of the arithmetical progression. Napier realized that numbers such as 1
01
100
,
.
n
with large
n
were useful for re-
alizing his intuition. All these numbers are greater than 2 and smaller than 3, and
for increasing values of
n
they approximate a real number
e
, later formally defined
by Euler. We will present, in Sect. 5.6.5, an argument showing the importance of
e
in many developmental processes. Here, we report an analysis of Euler which links
this number to the imaginary unit and to Archimedes' constant
001
1000
, that is, having the form
or 1
.
(
1
+
1
/
n
)
. The first impor-
tant result of Euler, about
e
, was its representation as an infinite sum. The starting
point of this result is Newton's formula:
π
n
k
a
k
b
n
−
k
n
k
=
0
n
(
a
+
b
)
=
where
k
denotes the
binomial coefficient
, that is, the number of different
k
-subsets
of
n
elements, which is given by the following formula, where
n
! is the factorial of
n
, that is the product 1
×
2
×...×
(
n
−
1
)
×
n
:
n
k
n
!
=
!
.
k
!
(
n
−
k
)
The argument of Euler is informal, but extremely fascinating. He expresses
e
by
the following formula, where
ω
stands for an infinite number (
numerus infinite
magnus
):
/
ω
)
ω
e
=(
1
+
1
but, being, for any number
x
,also
ω
/
x
an infinite number, we can put:
/
ω
)
ω
/
x
e
=(
1
+
x
therefore
e
x
/
ω
)
ω
/
x
x
/
ω
)
ω
=((
1
+
x
)
=(
1
+
x
and by Newton's formula, dealing with
ω
as if it were a natural number, this can be
written as:
k
x
k
ω
k
=
0
e
x
/
ω
)
ω
=
k
=(
1
+
x
/
ω
but, by definition of binomial coefficients and factorials, we get:
k
x
k
ω
k
=
0
e
x
k
)
x
2
2
)
x
3
3
=
/
ω
=
1
+
ω
x
/
1!
ω
+
ω
(
ω
−
1
/
2!
ω
+
ω
(
ω
−
1
)(
ω
−
2
/
3!
ω
...
and for the infinity of
ω
we can assume
ω
=(
ω
−
1
)=(
ω
−
2
)
...
therefore we can
conclude that:
e
x
x
2
x
3
=
1
+
x
/
1!
+
/
2!
+
/
3!
...