314 ian mueller
If we ignore the distinction between a unit and a number, 5 we may express
Nicomachus’ claim here as:
Th e side of the n th (actual or potential) triangular number is n.
Nicomachus now turns to deal more explicitly with the question of the
relationship between the sequence of triangular numbers and the “natural”
numbers:
ii .8.3 Triangular numbers are generated when natural number is set out in
sequence ( stoikhêdon ) and successive ones are always added one at a time
starting from the beginning, since the well-ordered triangular numbers are
brought to completion with each addition and combination. For example,
from this natural sequence
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
if I take the very fi rst item I get the potentially fi rst triangular number, 1:
α
then, if I add to it the next term, I get the actually fi rst triangular number,
since 3 is 2 and 1, and in its graphic representation it is put together as
follows: two units are placed side by side under one unit and the number is
made a triangle:
α
α α
And then, following this, if the next number, 3, is combined with this and
spread out into units and added, it gives and also graphically represents 6,
which is the actually second triangular number:
α
α α
α α α
Nicomachus continues in this vein for the fi rst seven (potential and actual)
triangular numbers, essentially showing that:
Th e n th triangular number is the sum of the fi rst n “natural” numbers.
5 As I shall sometimes do, without – I hope – introducing any confusion or uncertainty.