*
80
PYTHAGOREAN ARITHMETIC
numbers 3, 4, 5 is right angled. This fact could not but add
strength to his conviction that all things were numbers, for it
established a connexion between numbers and the angles of
geometrical figures. It would also inevitably lead to an
attempt to find other square numbers besides 5 2 which are
the sum of two squares, or, in other words, to find other sets
of three integral numbers which can be made the sides of
right-angled triangles; and herein we have the beginning of
the indeterminate analysis which reached so high a stage of
development in Diophantus. In view of the fact that the
sum of any number of successive terms of the series of odd
numbers 1, 3, 5, 7 . .. beginning from 1 is a square, it was
only necessary to pick out of this series the odd numbers
which are themselves squares; for if we take one of these,
say 9, the addition of this square to the square which is the sum
of all the preceding odd numbers makes the square number
which is the sum of the odd numbers up to the number (9) that
we have taken. But it would be natural to seek a formula
which should enable all the three numbers of a set to be imme
diately written down, and such a formula is actually attributed
to Pythagoras. 1 This formula amounts to the statement that,
if m be any odd number,
m 2
•m 2 — 1 % 2 ,m 2 + 1
2 ' ~ C 2
)•
Pythagoras would presumably arrive at this method of forma
tion in the following way. Observing that the gnomon put
round n 2 is 2-11+1, he would only have to make 2n+l a
square. *
If we suppose that 2 n + 1 = m 2 ,
we obtain n — \ (m 2 — 1),
and therefore n + 1 = \ (m 2 +1).
It follows that
m 2
m 2 -I-1
~ 2
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1 Proclus on Eucl. I, p. 487. 7-21,
1523