## The oldest known proof

There is evidence that Pythagoras' Theorem was
discovered very early by the Chinese and the Indians
(refer to Heath's
discussion just after I.47), but exactly how early is not known.
The
earliest tangible record of Pythagoras' Theorem
comes from Babylonian tablets dating to around
1000 B.C.
In addition to tablets containing
exercises which depend on knowing at least a few specific cases,
a number of tablets have been found
with pictures which are in effect proofs of
the Theorem in the special case where the sides of
the right triangle are equal.

In this case of
course the Pythagoras' Theorem asserts that the ratio
of the diagonal of a square
to one of its sides is equal to
the square root of 2.

This picture leads rather directly to the irrationality
of the square root of 2.
The argument we give here is variant of
one from Heath's comments
on Euclid (volume I, p.400).

Let s and d be the side and diagonal of the large square
in the figure above.
To say that s and d are commensurable, or equivalently that
the ratio
d/s (the square root of 2) is a rational number,
means that there exists some small
segment e such that d and s are both multiples of e.
Let s' and d' be the side and diagonal
of the smaller (red) square in the figure above.
We claim that if
s and d are multiples of e then so are
s' and d'. But
an argument about congruent triangles
shows that

*
s' = d - s
*

*
d' = s - s'
*
which imply the claim.

This leads to a contradiction. The two squares are similar.
Suppose the smaller one is equal to the larger
one scaled by a constant c, which
from the figure is clearly smaller than !. We can apply the argument over and over
again to see that all lengths c^{n} s and c^{n} d are
multiples of e.
This cannot happen since c^{n} becomes arbitrarily small.

### Reference

Otto Neugebauer, **Keilschrift-Texte**, Part II (especially BM 15285, Tafel 3).
Springer, 1935.
Otto Neugebauer, **The exact sciences in antiquity**, published by Dover,
1969. Especially p. 35 and plate VI.a.

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