316 ian mueller
Nicomachus proceeds through the octagonal numbers without fi gures,
making clear that:[Nic*]. Th e sum of the fi rst n numbers x 1 , x 2 ,... , x (^) n which are such that x (^) i + 1 = x (^) i + j
is the n th j +2-agonal number and its side is n.
He then turns to showing that his presentation of polygonal numbers is in
harmony with geometry (<hê> grammikê < didaskalia >), something which
he says is clear both from the graphic representation and from the following
considerations:
ii .12.1 Every square fi gure divided diagonally is resolved into two triangles and
every square number is resolved into two consecutive triangulars and therefore is
composed of two consecutive triangulars. For example, the triangulars are:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc.,
and the squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
If you add any two consecutive triangulars whatsoever you will always produce
a square, so that in resolving any square you will be able to make two triangulars
Figure 9.4 Th e graphic representation of the fourth pentagonal number.
α
αα
ααα
αααα
αααα
αααα
αααα
1 + 4 + 7 + 10