Subject: Re: [HM] About Euclid
From: Ivo Schneider (Ivo.Schneider@UniBw-Muenchen.de)
Date: Thu Jan 27 2000 - 12:25:25 EST
Dear Laura Levi,
let me first come to your father's suggestion "that Euclid could be even
somewhat older that it results from Proclos information". Since I do not
know about the reasons which led Beppo Levi to this impression except for
the "Platonic (or Socratic) way of thinking" he found in Euclid I can only
speculate about them and perhaps harmonize them with my own. If your
father based his idea on books of the elements which usually are ascribed
to Theaetetus and Eudoxus, two mathematicians who were close to Plato,
such an impression might appear reasonable. However, if Euclid's editing
of the work of his predecessors did not change the Platonic flavour
contained in it this does not necessarily mean that Euclid himself was
personally influenced by Plato or his immediate followers.
My reasons to propose as a possible solution for a problem (not to
conclude) that Euclid lived after Archimedes presuppose two points.
1. The observation based on statements of later Greek mathematicians like
Heron or Pappus that some of the Archimedean tracts were not preserved in
their original form.
2. The good political and economical relationships between Syracuse and
Alexandria at Archimedes' times and his personal contacts to Dositheos or
Eratosthenes, who are (believed to be) connected with the school of
Alexandria, make it at least probable that Archimedes was (perhaps well)
informed about the state of the mathematical art in Alexandria.
If the or part of the Archimedean tracts do not refer to the Euclidean
Elements it seems plausible to assume that Archimedes was not acquainted
with them at least when he wrote these tracts. Of course, such a
non-acquaintance can be explained in different ways. One of them would be
that he was not acquainted because the Euclidean Elements did not exist yet
at this time.
Archimedes introduced in the letter to Dositheos preceding the quadrature
of the parabola a "lemma" later called the Archimedean axiom which he
applied in his proofs in the same way as predecessors who proved four
theorems which he names.
These four theorems are contained in Elements XII,2,7,10, and 18. The
proofs of these theorems are based on theorem Elements X,1 which again is
proved on the basis of Definition 4 of Elements V. I argued that if
Archimedes would have been familiar with Elements XII in this form there
would have been no reason to introduce and justify his axiom as he did.
The late Wilbur Knorr dealt with the same problem in much more detailed
form in an article which was not known to me when I finished the manuscript
of my book in 1977. In this article Wilbur R. Knorr, Archimedes and the
pre-Euclidean proportion theory, Archives internationales d'histoire des
sciences, vol. 28, 1978, p. 183-244 contends (convincingly for me) that
(p.221) "referring to a treatment of the Eudoxean 'exhaustion' method
different from Elements XII, Archimedes sought in his axiom the basis of
a proof of the bisection-principle, which figured in his sources not as
a theorem, but as an assumption." Knorr who assumes that Archimedes'
father Phidias was "of Euclid's generation" and that Phidias was
responsible for the first mathematical education of Archimedes finds a
pre-Euclidean orientation of Archimedes plausible. Knorr is convinced as
many others including myself that the only explicit reference to Euclid's
Elements in Sphere and Cylinder I, 2 is an interpolation. The same holds
for two other references just to Elements in Sphere and Cylinder I, 6 and
On method II.
In the light of all this a possible explanation for the lack of real (not
interpolated) references to the Euclidean Elements in Archimedes' works
would be to give up an order relation in time, popular for centuries,
according to which Euclid is older than Archimedes.
Ivo Schneider
> Alexander Jones and Ivo Schneider have recently noted that very little
> is known about the time of Euclid's life. Ivo Schneider suggests that he
> possibly lived after Archimedes, as he discussed in his book published
> in 1979.
> I am not a specialist on this subject, but recently I had to read
> something about Euclid, in the occasion that the book "Leyendo Euclides"
> ("Reading Euclid") was being prepared for a new edition. The first edition
> was published in 1947 by my father, the mathematician Beppo Levi. In his
> book, Beppo Levi is especially interested in some correspondence he finds
> between the Euclid and Platonic (or Socratic) way of thinking, and
> consequently suggests that Euclid could be even somewhat older that it
> results from Proclos information. I would be very interested to know the
> reasons considered by Ivo Schneider, to conclude that, on the contrary,
> Euclid could have been younger than Archimedes, but I have no access here
> to the book he makes reference to. Could it be possible to have from Ivo
> Schneider some more detailed information? I would be very thankful.
> Laura Levi
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