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