Subject: Re: [HM] Euclid and the unique factorization theorem
From: Franz Lemmermeyer (lemmerm@mpim-bonn.mpg.de)
Date: Thu Feb 10 2000 - 06:25:52 EST
On Wed, 9 Feb 2000, Samuel S. Kutler wrote:
> As you know in IX 20, Euclid does not form the product of the primes,
> he takes the least common multiple. This is equal to the product,
> but I don't know why he did not ask us to multiply them all together.
> Probably it is because he has a procedure for finding the least common
> multiple. Do you have an opinion about this?
As a matter of fact I was afraid that this is what Euclid was
doing, but I had already returned my copy to the library. The
reason for using lcm's instead of products was explained by John.
On Wed, 9 Feb 2000, John Conway wrote:
> My guess is that he did actually "know" the unique factorization
> property, but refrained from stating it because he couldn't easily do
> so in his geometrical language. What he did instead was state exactly
> the "geometric" part of it - after all, he WAS writing a book on geometry.
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.
[...]
> A very interesting point, to my mind, is Hero(n)'s famous formula
>
> DELTA = root( s(s-a)(s-b)(s-c) ),
>
> since this involves a "non-geometrical" product of four lines. Does anyone
> know of any ancient discussion about this aspect of the 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.
franz
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