• FIGURED NUMBERS 
77 
ot‘ side n. The particular triangle which has 4 for its side is 
mentioned in a story of Pythagoras by Lucian. Pythagoras 
told some one to count. He said 1, 2, 3, 4, whereon Pytha 
goras interrupted, ‘Do you see 1 What you take for 4 is 10, 
a perfect triangle and our oath V This connects the know 
ledge of triangular numbers with true Pythagorean ideas. 
(ß) Square numbers and gnomons. 
We come now to square numbers. It is easy to see that, if 
we have a number of dots forming and filling 
up a square as in the accompanying figure repre- —-—14-' 
senting 16, the square of 4, the next higher 
square, the square of 5, can be formed by adding 
a row of dots round two sides of the original 
square, as shown; the number of these dots is 
2.4 + 1, or 9. This process of forming successive squares can 
be applied throughout, beginning from the first square 
number 1. The successive additions'are shown in the annexed 
figure between the successive pairs of straight 
lines forming right angles; and the succes- _J 
sive numbers added to the 1 are , J 
3, 5, 7 ... (2 n + 1), ~~~ .' 
that is to say, the successive odd numbers. ——-—'—1—11 * 
This method of formation shows that the . .—;—;—;— 
sum of any number of successive terms 
of the series of odd numbers 1, 3, 5, 7 . . . starting from 
1 is a square number, that, if n 2 is any square number, the 
addition of the odd number 2 n + 1 makes it into the next 
square, (n+ l) 2 , and that the sum of the series of odd num 
bers 1 + 3 + 5 + 7 + . . . + {2n+ 1) = (n + l) 2 , while 
1 + 3 + 5 + 7 +... + (2n— 1) = n 2 . 
All this was known to Pythagoras. The odd numbers succes 
sively added were called gnomons ; this is clear from Aristotle’s 
allusion to gnomons pläced round 1 which now produce different 
figures every time (oblong figures, each dissimilar to the pre 
ceding one), now preserve one and the same figure (squares) 2 ; 
the latter is the case with the gnomons now in question. 
2 Arist. Phys. iii. 4, 203 a 13-15. 
Lucian, Bieav TT[nnTis, 4.