* 
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