Pythagoras and the Pythagoreans
31
that
a
+
c >
2
√
ac
, or
b
a
≥
b
g
. Next, we note that
b
h
= 2
ac
a
+
b
or
b
2
g
=
b
h
b
a
. Thus
b
a
≥
b
g
≥
b
h
.
What is not quite as well known is that the fourth mean, some-
times called the
subcontrary to the harmonic
mean is larger than all
the others except the seventh and the ninth, where there is no greater
than or less than comparison over the full range of
a
and
c
. The proof
that this mean is greater than
b
a
is again straight forward. We easily
see that
b
=
a
2
+
c
2
a
+
c
=
(
a
+
c
)
2
−
2
ac
a
+
c
= 2
b
a
−
b
2
g
b
a
≥
b
a
by (1). The other proofs are omitted.
Notice that the first six of the proportions above are all of a
specific generic type, namely having the form
a
−
b
b
−
c
=
· · ·
. It turns
out that each of the means (the solution for
b
) are comparable. The
case with the remaining five proportions is very much different. Few
comparisons are evident, and none of the proportions are much in use
today. The chart of comparison of all the means below shows a plus
(minus) if the mean corresponding to the left column is greater (less)
than that of the top row. If there is no comparison in the greater or less
than sense, the word “No” is inserted.