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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!
...
 
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