324 ian mueller
Th ere is no more basis for this generalization than for the generalizations
we have seen in Nicomachus; indeed, in a sense there is even less since
Diophantus has considered only fi rst polygonals and shown that they
satisfy Dioph 4. Like Nicomachus, he clearly could show the same thing for
any particular example, but that hardly proves his claim or the next one:
[470,27] And what is said by Hypsicles in a defi nition 20 has been demonstrated,
namely:
If there are numbers in equal excess in any multitude starting from the unit,
then, when the excess is one the whole is triangular, when it is two, square,
three, pentagonal. Th e number of angles is said to be greater than the excess
by two, and its sides are the multitude of numbers set out with the unit.
It is not clear exactly what the defi nition of Hypsicles was. 21 In Diophantus’
representation he said something about the fi rst three polygonals, but it
seems reasonable to suppose that he at least intended a generalization and
so can be credited with Def geo/arith. 22 But we have no information about how
he used it – if he did. In any case Diophantus would have been on fi rmer
footing had he made the defi nition the basis of his treatise rather than pur-
porting to do the impossible, namely demonstrate it. Had he done this he
would not have had to worry about Dioph 1 and the special case of it which
he invokes to deal with fi rst j + 2-agonal numbers.
Diophantus now applies Hypsicles’ defi nition and his own results to tri-
angular numbers.
[472,5] Hence, since triangulars result when the excess is one and their sides are the
greatest of the numbers set out, the product of the greatest of the numbers set out
and the number which is greater by one than it is double the triangular indicated.
If p = x 1 + x 2 +... + xn with xi+ 1 = xi + 1 and x 1 = 1, then p is a triangular
with side xn and xn(xn + 1) = 2 p. 23
Diophantus returns again to fi rst polygonal numbers. He recalls the appli-
cation of Dioph 1 at 470,6.
[472,9] And since OB has as many angles as there are units in it, if it is multiplied
by 8 multiplied by what is less than it by two (that is by the excess; that will be
20 D’Ooge 1926 : 246 endorses Gow’s ( 1884 : 87) suggestion that en horôi might mean “in a book
called Defi nition .” In itself this suggestion seems to me unlikely, but the recurrences of the
word horos in 472,14 and especially 472,20 seem to me to rule it out completely.
21 S t a n d a r d fl oruit : c. 150 bce.
22 Contrast Nesselmann 1842 : 463.
23 Proof: It follows from Hypsicles’ defi nition that x 1 + x 2 +... + xn is a triangular number p with
side n. But by Dioph 2 xn = (n − 1). 1 + 1 = n. And by Dioph 3 2(x 1 + x 2 +... + xn) =
n(xn + x 1 ) = xn(xn + 1).