il A 
THE TREATISE ON POLYGONAL NUMBERS 517 
regression, the 
seeding column. 
mtions another 
as. Every tri- 
increased by 1 
The last proposition, which breaks off in the middle, is : 
Given a number, to find in how many ways it can he 
polygonal. 
The proposition begins in a way which suggests that 
Diophantus first proved geometrically that, if 
8 P(a — 2) + {a - 4) 2 = {2 + (2 n — 1) {a — 2) } 2 , 
On Polygonal 
different from 
roof is strictly 
e, of being long 
try propositions 
ves that, if a be 
leal progression 
3, 2s = n{l+a). 
-{n—l)h be an 
then 2P = n {2+ (n—l){a — 2)}. 
Wertheim (in his edition of Diophantus) has suggested a 
restoration of the complete proof of this proposition, and 
I have shown (in my edition) how the proof can be made 
shorter. Wertheim adds an investigation of the main pro 
blem, but no doubt opinions will continue to differ as to' 
whether Diophantus actually solved it. 
& 
, as we have it 
+ 1 = (2 n + l) 2 . 
finition and also 
les. 
a-gonal number