Pythagoras and the Pythagoreans
34
it was absolutely impossible to observe (in this case, the incommensu-
rability of a square’s diagonal with its side).
The discovery of incommensurability was attended by the intro-
duction of indirect proof and, apparently in this connection, by the
development of the definitional system of mathematics.
26
In general,
the proof of irrationality promoted a stricter approach to geometry, for it
showed that the evident and the trustworthy do not necessarily coincide.
9
Other Pythagorean Contributions.
The Pythagoreans made many contributions that cannot be described in
detail here. We note a few of them without commentary.
First of all, connecting the concepts of proportionality and relative
prime numbers, the theorem of Archytas of Tarantum (c. 428 - c. 327
BCE) is not entirely obvious. It states that there is no mean proportional
between successive integers. Stated this way, the result is less familiar
than using modern terms.
Theorem.
(Archytas) For any integer
n
, there are no integral solutions
a
to
A
a
=
a
B
where
A
and
B
are in the ratio
n
:
n
+ 1
.
Proof.
The proof in Euclid is a little cumbersome, but in modern
notation it translates into this: Let
C
and
D
be the smallest numbers
in the same ratio as
A
and
B
. That is
C
and
D
are relatively prime.
Let
D
=
C
+
E
Then
C
D
=
C
C
+
E
=
n
n
+ 1
which implies that
Cn
+
C
=
Cn
+
En
. Canceling the terms
Cn
, we
see that
E
divides
C
. Therefore
C
and
D
are not relatively prime, a
contradiction.
26
A. Szabo ”Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden?”, Acta
Antiqua, 4 (1956), p. 130.