The History of Mathematical Proof in Ancient Traditions

(Elle) #1

322 ian mueller


What I will try to explain is the specifi c reason why Diophantus’ attempt
to do so fails, emphasizing that, although his reasoning is mathematically
much more elaborate than Nicomachus’, his handling of generalization is
essentially the same, namely the presentation of examples which make the
general truth “obvious.”
Before turning to that material I want to signal the very fi rst statement in
On Polygonal Numbers , which concerns the fi rst (actual) polygonal number
of each kind:
[450,1] Each of the numbers starting from three which increase by one is a fi rst
polygonal aft er the unit. And it has as many angles as the multitude of units in it.
Its side is the number aft er the unit, i.e., 2. 3 is triangular, 4 square, 5 pentagonal,
and so on.
Th is is, I think, the only application of Def geo/arith that Diophantus takes for
granted, i.e., he takes for granted that:
the fi rst j +2-agonal (aft er 1) has side 2 and is j + 2.
Aft er making a remark about the ordinary conception of square
numbers, 17 Diophantus gives (450,11 and 16) the announcement of what
he is going to prove, which I have already quoted, and proves his four arith-
metical propositions. It is at this point that he fi rst reintroduces the notion
of a polygonal number in his announcement of what he intends to prove
next, which is tantamount to Def geo/arith :
[468,14] Th ese things being the case, we say that if there are numbers starting from
the unit in any multitude and in any excess, the whole is polygonal. For it has as
many angles as the number which is greater than the excess by 2, and the number
of its sides is the multitude of the numbers set out with the unit.
He now invokes Dioph 4:
[470,1] For we have shown that the sum of all the numbers set out multiplied by 8
KB plus the square of NB produces the square of PK.
Here Diophantus is working with a fi gure in which the line AKNB of Figure 6
for Dioph 4 is extended to the right so that PK is a representation of
(2 n − 1) j + 2. 18 But to get a representation of j + 2 he also extends AKNB to
the left ( Figure 9.6 ):
[470,4] But also if we posit AO as another unit, we will have KO as two, and KN is
similarly two.
17 [450,9] “It is immediately clear that squares have arisen because they come to be from some
number being multiplied by itself.”
18 Th is specifi cation of PK occurs at 466,1–2.
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