The History of Mathematical Proof in Ancient Traditions

Polygonal numbers in ancient Greek mathematics 317

from them. And again if any triangle is joined to any square fi gure 6 it produces
a pentagon, for example if the triangular 1 is joined to the square 4, it makes the
pentagonal 5, and if the next , that is 3, is added to the next
9 it makes the pentagonal 12, and if the following 6 is added onto the
following 16, it gives the following 22, and 25 added to 10
gives 35, and so on forever.

Nicomachus states similar results for adding triangulars to pentagonals to
get hexagonals, to hexagonals to get heptagonals, and to heptagonals to get
octagonals, “and so on ad infi nitum .” He introduces a table ( Table 9.1 ) as an
aid to memory:

and describes some of the relevant sums, results which we might formulate
as:

Th e n +1th square number is the n th triangular number plus the n +1th triangular
number;
Th e n +1th pentagonal number is the n th triangular number plus the n +1th square
number,

or, generally,

Th e n +1th k +1-agonal number is the n th triangular number plus the n +1th k -agonal
number.

At this point I would like to introduce some of Heath’s remarks about
Nicomachus’ Introduction :

It is a very far cry from Euclid to Nicomachus. Numbers are represented in Euclid
by straight lines with letters attached, a system which has the advantage that, as in
algebraical notation, we can work with numbers in general without the necessity of
giving them specifi c values.... Further, there are no longer any proofs in the proper
sense of the word; when a general proposition has been enunciated, Nicomachus
regards it as suffi cient to show that it is true in particular instances; sometimes we
are left to infer the proposition by induction from particular cases which are alone
given.... probably Nicomachus, who was not really a mathematician, intended his

Table 9.1:

Triangles 1 3 6 10 15 21 28 36 45 54
Squares 1 4 9 16 25 36 49 64 81 100
Pentagons 1 5 12 22 35 51 70 92 117 145
Hexagons 1 6 15 28 45 66 91 120 153 190
Heptagons 1 7 18 34 55 81 112 148 189 235 7

(^6) Here some exaggeration, since the triangle and the square have to “fi t together.”
(^7) Apparently the octagons are missing.