Polygonal numbers in ancient Greek mathematics 313
Nicomachus also mentions hexagonal, heptagonal, and octagonal
numbers, and there is no question that he has the idea of an n -agonal
number, for any n , but he only expresses this with words like “and so on
forever in the direction of increase” ( aei kata parauxêsin houtôs ; ii .11.4). It
is clear that Nicomachus intends to make some kind of generalization, but
it is not at all clear what, if any, theoretical or mathematical ideas underlie
it. Any connection between what he says and the natural representation
of numbers is at best indirect. Nicomachus is relying on the idea that the
numbers go on forever, but much more central to his account of polygonal
numbers is the geometric fact that an n -agon is determined by the n points
which are its vertices. If induction lies behind the reasoning, it is not made
at all explicit.
I turn now to some further features of what Nicomachus says. Th e fi rst
sentence of his description of triangular numbers is quite opaque, but it is
clearly intended to bring out their confi gurational aspect. I quote it in the
translation of d’Ooge:
II.8.1 A triangular number is one which, when it is analyzed into units, shapes into
triangular form the equilateral placement of its parts in a plane. Examples are 3, 6,
10, 15, 21, 28, and so on in order. For their graphic representations ( skhêmatograph-
iai ) will be well-ordered and equilateral triangles....
Here again we have the thought of continuing indefi nitely. Nicomachus
now indicates the arithmetical procedure for generating these triangular
numbers, again insisting on the distinction between the unit and a number
even though leaving it aside would simplify his description.
And, proceeding as far as you wish, you will fi nd triangularization of this kind,
making the thing which consists of a unit fi rst of all most elementary, so that
the unit may also appear as potentially a triangular number, with 3 being actually
the fi rst.
ii .8.2 Th e sides
of the potentially fi rst being one, that of the actually fi rst (i.e., 3) two, that of the
actually second (i.e., 6) three, that of the third four, of the fourth fi ve, of the fi ft h six,
and so on forever.
Figure 9.1 Geometric representation of polygonal numbers.
αααα α
ααα
α α
α α