Polygonal numbers in ancient Greek mathematics 321

As I have said, the material described thus far in this section is purely arith-
metical. However, if one accepts Def geo/arith , what Diophantus has shown is
that:

[Dioph 4 geo/arith ] if p is the n th j + 2-agonal number, then
p 8 j + (j − 2)^2 = ((2n − 1)j + 2)^2.
It is clear from Diophantus’ initial less specifi c statement of what he will
show that he does think that he can establish this :

[450,11] Here it is established ( edokimasthê ) that if any polygonal is multiplied by a
certain number (which is a function ( kata tên analogian ) of the multitude of angles
in the polygonal) and a certain square number (again a function of the multitude of
angles in it) is added, the result is a square.

if p is a j + 2-agonal number, there are functions f and c such that
f( j + 2)p + c( j + 2) is a square number.

Aft er announcing this result Diophantus states the goal of the treatise:

[450,16] We will establish this and indicate how one can fi nd a prescribed polygonal
with a given side and how the side of a given polygonal can be taken.

Th at is,

how to fi nd (1) the j + 2-agonal p with side n and (2) the side of a j + 2-agonal p.

Th is last subject is the concern of the fi nal part of the treatise (472,21–
476,3). Nic* allows one to solve these in a slightly cumbersome mechanical
way, but what Diophantus proves enables him to give what amounts to
formulae for the solutions:

(1) p=

((2 1) 2)

2

(2)

2

8

nj j

j

−+ −−

,

(2) n
pj
j

=^1

2 

8j ( 2)^22

(^1) 
+−–
+.
Th is last material is quite mundane, and I shall not discuss it. My major
concern will be with the material immediately following the presentation
of the arithmetical results Dioph 1 to 4. For those four propositions are
purely arithmetical; they do not say anything about polygonal numbers and
certainly do not establish anything about spatial confi gurations of units.
It is in the remainder of the treatise that Diophantus tries to establish a
general truth corresponding to Nic*, but as I have indicated, I believe that it
is impossible to prove this truth within the confi nes of Greek mathematics.