Pythagoras and the Pythagoreans
32
i/j
1
2
3
4
5
6
7
8
9
10 11
1
+
+
-
-
-
No No No No +
2
-
+
-
-
-
No No
-
No No
3
-
-
-
-
-
-
No
-
No No
4
+
+
+
+
+ No + No +
+
5
+
+
+
-
+ No + No +
+
6
+
+
+
-
-
No No No No +
7
No No + No No No
No
-
No No
8
No No No
-
-
No No
No +
+
9
No +
+ No No No + No
No No
10
No No No
-
-
No No
-
No
No
11
-
No No
-
-
-
No
-
No No
Comparing Pythagorean Proportions
Linking qualitative or subjective terms with mathematical propor-
tions, the Pythagoreans called the proportion
b
a
:
b
g
=
b
g
:
b
h
the
perfect
proportion. The proportion
a
:
b
a
=
b
h
:
c
was called the
musical
proportion.
8
The Discovery of Incommensurables
Irrationals have variously been attributed to Pythagoras or to the Pythagore-
ans as has their study. Here, again, the record is poor, with much of
it in the account by Proclus in the
4
th
century CE. The discovery is
sometimes given to
Hippasus of Metapontum
(
5
th
cent BCE). One
account gives that the Pythagoreans were at sea at the time and when
Hippasus produced (or made public) an element which denied virtually
all of Pythagorean doctrine, he was thrown overboard. However, later
evidence indicates that Theaetetus
23
of Athens (c. 415 - c. 369 BCE)
23
the teacher of Plato