318 ian mueller
Introduction to be, not a scientifi c treatise, but a popular treatment of the subject
calculated to awaken in a beginner an interest in the theory of numbers.... Its
success is diffi cult to explain except on the hypothesis that it was at fi rst read by
philosophers rather than mathematicians... , and aft erwards became generally
popular at a time when there were no mathematicians left , but only philosophers
who incidentally took an interest in mathematics. 8
Heath’s remarks here are aimed at the whole of the Introduction , but I
wish only to consider them in relation to Nicomachus’ treatment of polygo-
nal numbers. Th ere is no question that, as Heath also notes, Nicomachus’
fl owery and imprecise language is a “far cry” from Euclid’s sparse, formal
formulations. But the representation of polygonal numbers by straight lines
would obliterate their confi gurational nature. Nicomachus shows how tri-
angular confi gurations of units can be generated as the series 1, 1+2, 1+ 2 +3,
etc. But I do not see what he could do to “prove” this fact and, therefore,
how he could “prove” any fact about polygonal numbers as confi gurations.
Of course, we know how to prove things about polygonal numbers, namely
by eliminating all geometric content and transforming Nic*, which for
Nicomachus expresses an arithmetical fact about confi gurations, into an
arithmetical defi nition in which the geometrical terminology is at most a
convenience, perhaps as follows:
[Def geo/arith ]. p is the n th j + 2-agonal number with side n if and only if p = x 1 + x 2 +⋅⋅⋅ + x (^) n , where x (^) i + 1 = x (^) i + j and x 1 = 1.
I assume that Fowler had something of this kind in mind when he
advanced the hypothesis that lying behind Nicomachus’ presentation
were ancient proofs using mathematical induction. 9 I doubt this very
much, but the more important point for me is that, unless something like
Def geo/arith is used to eliminate the confi gurational aspect of polygonal
numbers, anything like a Euclidean foundation for the theory of them lies
well beyond the scope of Greek mathematics.
2. Th e argument of Diophantus’ On Polygonal Numbers
In Tannery’s edition of On Polygonal Numbers there are four propositions.
Th e propositions are purely arithmetical and in none of them is there a
mention of polygonals. 10 I quote them and give algebraic representations8 Heath 1921 : i 97–9.
9 Fowler 1994 : 258.
10 When I say that these propositions are purely arithmetical, I only mean to point out the
absence of the notion of polygonality from the formulations and proofs of the propositions.