Polygonal numbers in ancient Greek mathematics 323
But now Diophantus is only interested in OB ( j + 2 = 1 + (1 + j )), KB ( j ), and
BN ( j − 2), and, in his only application of Dioph 1, he says:
[470,6] Th erefore OB, BK, BN will exceed one another by an equal amount.
Th erefore, 8 times the product of the greatest OB and the middle BK plus the square
of the least BN makes a square the side of which is the sum of the greatest OB and
2 of the middle BK. Th erefore OB multiplied by 8 KB plus the square of NB is equal
to the square of OB and 2KB together.
(j + 2)8j + (j − 2)^2 = (j + 2 + 2 j)^2.
Th is is, of course, just the special case of Dioph 4 in which n = 2. To make
this point clear Diophantus argues that j + 2 + 2 j = (2·2 − 1) j + 2:
[470,13] And the side minus two (OK) leaves 3 KB, which is KB multiplied by three.
But three plus one is 2 multiplied by 2.
Diophantus underlines the analogy with Dioph 4 and then points out that
OB ( j + 2) is the fi rst j + 2-agonal number:
[470,17]... the sum of the numbers set out with the unit produces ( poiei ) the same
problem as OB, but OB is a chance number and is the fi rst polygon {of its kind}
aft er the unit (since AO is a unit and the second number is AB), and {OB} has two
as side.
So, in addition to proving Dioph 4, Diophantus has proved a special case
of it in which n = 2, a case for which he has asserted that p is the fi rst
j + 2-agonal number. Th ese two propositions by themselves do not imply
that whenever the conditions of Dioph 4 hold, p is a j + 2-agonal number
with side n. But this is precisely what Diophantus asserts: 19
[470,21] Th erefore also the sum of all the numbers set out is a polygon with as many
angles as OB and having as many angles as it is greater by 2 (i.e., by OK) than the
excess, KB; and it has as side GH, which is the number of the numbers set out with
the unit.
19 Commentators have standardly approved this “reasoning,” or at least not raised any doubts
about it. See Poselger 1810 : 34–5; Schulz 1822 : 618; Nesselmann 1842 : 475; Heath 1885 : 252;
Wertheim 1890 : 309; Massoutié 1911 : 26; and Ver Eecke 1926 : 288.
Figure 9.6 Diophantus’ diagram, Polygonal Numbers.
OAK N B P(2n − 1)j + 2 + 2
11 2 j − 2
OAK N B j + 2
KN B j