Pythagoras and the Pythagoreans
18
Now use the gnomon. Begin by placing the gnomon around
n
2
.
The next number is
2
n
+ 1
, which we suppose to be a square.
2
n
+ 1 =
m
2
,
which implies
n
=
1
2
(
m
2
−
1)
,
and therefore
n
+ 1 =
1
2
(
m
2
+ 1)
.
It follows that
m
2
+
m
4
4
−
m
2
2
+
1
4
=
m
4
4
+
m
2
2
+
1
4
This idea evolved over the years and took other forms. The essential fact
is that the Pythagoreans were clearly aware of the Pythagorean theorem
Did Pythagoras or the Pythagoreans actually prove the Pythagorean the-
orem? (See the statement below.) Later writers that attribute the proof
to him add the tale that he sacrificed an ox to celebrate the discovery.
Yet, it may have been Pythagoras’s religious mysticism may have pre-
vented such an act. What is certain is that Pythagorean triples were
known a millennium before Pythagoras lived, and it is likely that the
Egyptian, Babylonian, Chinese, and India cultures all had some “proto-
proof”, i.e. justification, for its truth. The proof question remains.
No doubt, the earliest “proofs” were arguments that would not
satisfy the level of rigor of later times. Proofs were refined and retuned
repeatedly until the current form was achieved. Mathematics is full of
arguments of various theorems that satisfied the rigor of the day and
were later replaced by more and more rigorous versions.
19
However,
probably the Pythagoreans attempted to give a proof which was up
to the rigor of the time. Since the Pythagoreans valued the idea of
proportion, it is plausible that the Pythagoreans gave a proof based on
proportion similar to Euclid’s proof of Theorem 31 in Book VI of
The
Elements
. The late Pythagoreans (
e
400 BCE) however probably did
supply a rigorous proof of this most famous of theorems.
19
One of the most striking examples of this is the Fundamental Theorem of Algebra, which
asserts the existence of at least one root to any polynomial. Many proofs, even one by Euler,
passed the test of rigor at the time, but it was Carl Friedrich Gauss (1775 - 1855) that gave
us the
Þ
rst proof that measures up to modern standards of rigor.