The History of Mathematical Proof in Ancient Traditions

311

9 Generalizing about polygonal numbers

in ancient Greek mathematics

Ian Mueller

Introduction

Th e main source for our information about the Greek handling of what are
called polygonal numbers is the Introduction to Arithmetic of Nicomachus
of Gerasa ( c. 100 ce ). 1 Heath says of the Introduction that “Little or nothing
in the book is original, and, except for certain defi nitions and refi ne-
ments of classifi cation, the essence of it evidently goes back to the early
Pythagoreans.” 2 I am not interested in this historical claim, the evidence
for which is very slight; indeed I am not interested in chronology at all but
only in certain features of Nicomachus’ treatment of polygonals, which I
discuss in Section 1 , and in the general argumentative structure of a short
treatise by Diophantus called On Polygonal Numbers ,^3 which I discuss in
Section 2.

1. Nicomachus of Gerasa

In the Introduction Nicomachus makes a contrast between the standard
Greek way of writing numbers, in which, e.g., 222 is written σκβ, where σ
represents 200, κ 20, and β 2, and what he says is a more natural way:

ii .6.2 First one should recognize that each letter with which we refer to a number...
signifi es it by human convention and agreement and not in a natural way; the
natural, direct ( amethodos ), and consequently simplest way to signify numbers
would be the setting out of the units in each number in a line side by side... :

1 Greek text: Hoche 1866 ; English translation: D’Ooge 1926 ; French translation: Bertier 1978.
Th ere is material parallel to Nicomachus’ presentation in Th eon of Smyrna (Hiller 1878 ). For
dates of individuals I use Toomer’s articles in Th e Oxford Classical Dictionary (Hornblower and
Spawforth 1996 ).

(^2) Heath 1921 : i 99.
(^3) Greek text: T1893: 450,1–476,3; French translation: Ver Eecke 1926. I do not discuss the fi nal
part of the treatise (476,4–480,2), a broken-off and inconclusive attempt to show how to fi nd
how many kinds of polygonal a given number is. Th e Oxford Classical Dictionary locates
Diophantus in the interval between 150 bce and 280 ce. Heath 1921 : ii 448 says that “he
probably fl ourished A.D. 250 or not much later.”