34 karine chemla
fi rst terms of the sequences. However, this is how Nicomachus argues in
favour of the general statement, whereby his establishment of this general
statement diff ers from modern standards. In the second step, Nicomachus
further brings to light patterns in the modes of generation of the fi rst
sequences, thereby indicating the general structure of the set of sequences
of polygonal numbers and pointing to further relationships between these
sequences. Again, Nicomachus indicates the general pattern and argues for
it by highlighting the pattern for the fi rst sequences. And again this is where
his approach falls short of modern standards. Th e most general statement
by which Nicomachus summarizes his procedure of proof consists of a table
of numbers. It collects in its rows the sequences introduced and more. Since
it displays the pattern of relationship between the lines, the table allows
Nicomachus to generate subsequent lines by deploying the same pattern
further and thereby determining the value of any polygonal number.
Th e textual elements brought into play (diagrams, sequences and table
of numbers) and the ways of using and articulating them by modes of rea-
soning contrast sharply with how Diophantus approaches the same topic.
Th e core of Diophantus’ treatise On Polygonal Numbers consists of purely
arithmetical and general propositions. Th ese propositions state arith-
metical properties in the form of relationships holding between numbers.
Diophantus proves these relationships through representations of numbers
as lines, using the representations in a way that is specifi c to this branch
of inquiry. Diophantus attempts to formulate the value of the n th j -agonal
number as a proposition of this kind. Th e diagrams used and the proposi-
tions stated thus exhibit a style completely diff erent from Nicomachus’.
However, their kind of generality is precisely what constitutes the problem
for concluding the proof. It is in Diophantus’ attempt to connect these
general statements to polygonal numbers with full generality that Mueller
identifi es the gap in Diophantus’ proof. Th e tools Diophantus uses here are
too general to allow him to recapture the details of the general objects that
polygonal numbers represent. He manages to establish the link only for the
fi rst n -agonal numbers.
Th ese two texts devoted to the same topic illustrate quite vividly the
plurality of practices in Greek mathematics, the study of which Lloyd advo-
cated. Mueller highlights diff erences in the ways of making diagrams and
relating them to the mathematical objects being studied. He shows the dis-
tinct ways in which diagrams are employed in the arguments being devel-
oped, thereby bringing to light two distinct kinds of arithmetical methods.
Additional interest in this case study derives from what is revealed when the
proofs are considered from the viewpoint of generality. Clearly, both texts