Pythagoras and the Pythagoreans
30
2
a
−
b
b
−
c
=
a
b
geometric
ac
=
b
2
3
a
−
b
b
−
c
=
a
c
harmonic
1
a
+
1
c
=
2
b
The most basic fact about these proportions or means is that if
a > c
,
then
a > b > c
. In fact, Pythagoras or more probably the Pythagore-
ans added seven more proportions. Here is the complete list from the
combined efforts of Pappus and Nicomachus.
Formula
Equivalent
4
a
−
b
b
−
c
=
c
a
a
2
+
c
2
a
+
c
=
b
5
a
−
b
b
−
c
=
c
b
a
=
b
+
c
−
c
2
b
6
a
−
b
b
−
c
=
b
a
c
=
a
+
b
−
a
2
b
7
a
−
c
b
−
c
=
a
c
c
2
= 2
ac
−
ab
8
a
−
c
a
−
b
=
a
c
a
2
+
c
2
=
a
(
b
+
c
)
9
a
−
c
b
−
c
=
b
c
b
2
+
c
2
=
c
(
a
+
b
)
10
a
−
c
a
−
b
=
b
c
ac
−
c
2
=
ab
−
b
2
11
a
−
c
a
−
b
=
a
b
a
2
= 2
ab
−
bc
The most basic fact about these proportions or means is that if
a > c
,
then
a > b > c
. (The exception is 10, where b must be selected
depending on the relative magnitudes of
a
and
c
, and in one of the
cases
b
=
c
.) What is very well known is the following relationship
between the first three means. Denote by
b
a
, b
g
,
and
b
h
the arithmetic,
geometric, and harmonic means respectively. Then
b
a
> b
g
> b
h
(1)
The proofs are basic. In all of the statements below equality occurs if
and only if
a
=
c
. First we know that since
(
α
−
γ
)
2
≥
0
, it follows
that
α
2
+
γ
2
≥
2
αγ
. Apply this to
α
=
√
a
and
β
=
√
b
to conclude