Polygonal numbers in ancient Greek mathematics 325
8 × KB)
duces a square.
j + 2 is a j + 2-agonal and (j + 2)8j + (j − 2)^2 is a square.
Th is, too, is immediately generalized with no justifi cation.
[472,14] And this will be a defi nition ( horos ) of polygonals:
Every polygonal multiplied by 8 multiplied by what is less by two than the
multitude of its angles plus the square of what is less than the multitude of
angles by 4 makes a square.If p is j + 2-agonal, p 8 j + ( j − 2)^2 is a square.
[472,20] In this way we have demonstrated simultaneously this defi nition of polyg-
onals and that of Hypsicles.
In this case the truth which Diophantus purports to establish as a defi ni-
tion is not a defi nition in the standard sense at all, since n 8 j + ( j − 2)^2 can be
a square even when n is not a j + 2-agonal; 2·8·3 + (3 − 2)^2 = 7 2 , but 2 is not
pentagonal. 24 And his claim to have demonstrated it is just as weak as his
claim to have established Def geo/arith.
Conclusion
It is certainly not surprising that Diophantus’ treatise on polygonal numbers
shows great mathematical skill. And it is perhaps also not surprising that its
sense of logical rigor is at times not superior to that of Nicomachus. Within
the limits of Greek mathematics there can be no mathematical demonstra-
tion of an arithmetical characterization of confi gurationally conceived
polygonal numbers. Within those limits Aristotle ( Posterior Analytics 1.6
(Ross)) was correct to insist that the generic diff erence between arithmetic
a n d g e o m e t r y c a n n o t b e b r e a c h e d.
24 Th is shortcoming is already pointed out in the editio princeps of the Greek text of Diophantus
(Bachet 1621 : 21 of the edition of On Polygonal Numbers ). What Diophantus says at 472,14
could serve as a defi nition for triangulars and squares. For, ignoring complications that would
arise if one tried to avoid “numbers” less than 1, it is easy to prove that:
p = 1 + 2 +... + n (i.e., p is a triangular) if and only if p 8·1 + (3 – 4)^2 is a square (i.e., if and only
if 8 p + 1 is a square);
p = 1 + 3 +... + 2 n − 1 (i.e., p is a square) if and only if p 8·2 + (4 – 4)^2 is a square (i.e., if and
only if 16 p is a square, i.e., if and only if p is a square).
It is tempting to think, although it cannot be proved, that Diophantus was misled by the
truth of these biconditionals to the false notion that Dioph 4 was the basis of a defi nition of
polygonality in general.