Pythagoras and the Pythagoreans
19
There are numerous proofs, more than 300 by one count, in the
literature today, and some of them are easy to follow. We present three
of them. The first is a simple appearing proof that establishes the
theorem by visual diagram. To “rigorize” this theorem takes more than
just the picture. It requires knowledge about the similarity of figures,
and the Pythagoreans had only a limited theory of similarity.
(
a
+
b
)
2
=
c
2
+ 4(
1
2
ab
)
a
2
+ 2
ab
+
b
2
=
c
2
+ 2
ab
a
2
+
b
2
=
c
2
b
a
b
a
b
a
b
a
c
c
c
c
This proof is based upon Books I and
II of Euclid’s
Elements
, and is sup-
posed to come from the figure to the
right. Euclid allows the decomposi-
tion of the square into the two boxes
and two rectangles. The rectangles
are cut into the four triangles shown
in the figure.
b
a
b
a
b
a
b
a
Then the triangle are reassembled into the first figure.
The next proof is based on similarity and proportion and is a
special case of Theorem 31 in Book VI of
The Elements
. Consider the
figure below.