Subject: Re: [HM] Euclid and the unique factorization theorem
From: John Conway (conway@math.Princeton.edu)
Date: Fri Feb 11 2000 - 12:55:27 EST
On Thu, 10 Feb 2000, Franz Lemmermeyer quoted and wrote:
> > As you know in IX 20, Euclid does not form the product of the primes,
> > [...] Do you have an opinion about this?
>
> On Wed, 9 Feb 2000, John Conway wrote:
>
> > My guess is that he did actually "know" the unique factorization
> > [but only stated] the "geometric" part of it [...]
> He surely was - does that mean we are distorting his views when
> we translate his geometry into our arithmetic? It seems we're
> back to the old question whether he was thinking arithmetic
> and writing geometric.
[... do we] know of any ancient [talk about Heron's] formula?
>
> That was a long time after Euclid. Diophantus, who lived at most
> 200 years after Heron, had no qualms about using powers beyond
> the third.
I agree, except that so little is known about Diophantus's life that
it's not even possible to say which century he lived in - it might well have
been >200 years after Heron. But Diophantus was writing about Numbers
rather than about geometry. Other people used such products of numbers
even before Euclid's time - in fact that was precisely my point - that
it's silly to "infer" from Euclid's book that he didn't know about products
of many numbers, when a more conservative position is that in his geometry
book he only stated the geometrically expressible part of number theory.
I doubt very much that Heron stated the formula in a way that involved
the product of 4 terms. I hope very much that some knowledgeable person
will be able to tell us about some ancient discussion of the theorem.
John Conway
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