[FOM] Archimedes's "Method"

Alasdair Urquhart urquhart at cs.toronto.edu
Mon Nov 19 13:09:03 EST 2007

I've read the book by Netz and Noel, "The Archimedes Codex."
It's an entertaining piece of popular history/science,
but it is full of tasteless and foolish exaggeration,
as well as silly bragging.

The manuscript in question, most of which is devoted to
Archimedes's "Method," was first discovered by Heiberg
in Constantinople around 1907.  Most of the palimpsest
was deciphered by Heiberg and published by him.
Netz and Noel make a great song and dance about how
they have improved on Heiberg's work, but I take
this with a grain of salt.  There is an English translation
of the text by Heath published as part of the Dover
reprint of Archimedes's works (Netz and Noel turn up
their noses at Heath, but that is only to be expected).

The codex subsequently disappeared, and suffered from
all kinds of horrendous damage, including the insertion
of a couple of forged paintings.  It recently came
to light, and was sold at auction for $2,200,000.
The owner is a reclusive and very wealthy man (not named
in the book), who allowed full access to the codex.

Anyway, as part of their popular book, Netz and Noel make a claim
that Archimedes had anticipated Georg Cantor.  How so?
Well, "The Method" is given over to a physically inspired
method of computing volumes of solids.  The idea is to
imagine two solids balanced at the ends of a bar,
and then compare vertical cross-sections of the corresponding
solids.  Then if you can show that there is some
proportion between the cross-sections, and you know the
volume of one solid, you can compute the volume of the
other.  Archimedes emphasizes strongly that this is a purely
heuristic method, and the final proof should be done using
the classical method of exhaustion.  All of this has been
known since 1907, and Netz and Noel have added nothing to all
of this.

But now they make the claim that Archimedes anticipated Cantor,
and this is their "great discovery."  Well, at some point,
Archimedes observes that there is a one-to-one correspondence
between the cross-sections of one solid, and the cross-sections
of the other, as his method requires.  This is the basis
for their ridiculous claim that Archimedes anticipated Cantor.
I quote:

 	He apparently was stating that this infinite set
 	was equal to that infinite set, because there was
 	a one-to-one relationship between the two sets (p. 198).

To this I can only say:  "Gimme a break!"

Anyway, I enjoyed reading the book, and recommend it strongly
to FOM subscribers, in spite of my criticisms above.

Alasdair Urquhart

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