Euclid vs Eratosthenes Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes sieve.
Thus I'd like to ask mathematical historians and everybody else: is there any clear proof that Euclid and Eratosthenes are two different people?

I wonder if Euclid's Elements were considered by the top geometers of Archimedes times (before and afterward for awhile) as teaching materials (or even philosophical too) rather than mathematical research. Especially that Elements were not mentioned much in those years, they became popular quite a bit later.

For a longer time, even after Archimedes, nobody talked about Euclid of Alexandria. Instead, they talked only  about "the author of Elements". Possibly, the geometric part of Elements was a cumulation of the work by several mathematicians, of which Eudoxes was a big part. But the number theoretical part of Elements and the Eratosthenes Sieve were perhaps by the same author, namely Erathostenes.

The Elements had no name nor any pseudonym assigned to it. That's why nobody was talking explicitly about the author.
Bourbaki used to have meetings together in a single room (perhaps a different room at the different occasions). They have established common notation and conventions, etc. Thus their textbooks had a fairly consistent style. They used to write in that style also monographs by single Bourbakist authors. In the case of the ancient authores such consistency was impossible due to the time and space span. This is why I conjecture that Eratosthenes combined the past results by several authors (especially by Eudoxes) and Eratosthenes rewritten these results in his own hand. This would explain why there was no author mentioned but the style of Elements was consistent.
 A: Eratosthenes lived around -200 (more precisely -276 to -195/194) and was a correspondent of Archimedes.  Archimedes communicated his Method to Eratosthenes in the form of a letter. Euclid lived around -300.  He would have had to enjoy extraordinary longevity to coincide with Eratosthenes.
A: C.K. Raju goes to some length to argue that Euclid did not exist at all, in Good-Bye Euclid!
He starts from the established fact that, while Euclid was first mentioned by the 5th century philosopher Proclus, our sources for Proclus are a translation from Arabic made in Toledo around the year 1000. Raju has a political agenda for wishing to debunk Euclid, to which I do not subscribe, but some of his arguments are intriguing:

Possibly, the name "Euclid" was inspired by a translation error made
  at Toledo regarding the term uclides which has been rendered by some
  Arabic authors as ucli (key) + des (direction, space). So,
  uclides, meaning “the key to geometry”, was possibly misinterpreted as a Greek name Euclides.

.

In attributing the Elements to an early Greek called Euclid, we are
  supposing that there was a fixed text which was repeatedly copied out
  without any significant change by subsequent scribes. But the
  available papyri on geometry from Alexandria do not correspond to the
  received text, and do not show any such evidence of the existence of a
  fixed, early text.


Mikhail Katz pointed me, via a MSE thread, to a review by José Ferreirós on Raju's book, since it's behind a paywall I quote the relevant paragraph:

In his interest to revise traditional historiography and oppose
  proof-centred mathematics, Raju devotes a lot of effort to questioning
  the existence of Euclid and insisting that the text of the Elements
  originates at the earliest in 370 CE (with Theon) or perhaps even in
  the tenth century. In my opinion, this is useless and does not help
  advance the author’s main theses. For historical purposes, what is
  relevant is that Elements represents a systematisation of a large
  portion of geometrical knowledge in the Greek-speaking world before
  the common era. (‘Euclid’ is simply the name of its otherwise unknown
  author, whose dates—it is true—are dubious; incidentally, an
  interesting question would be whether philologists find reason to
  think that the text of Elements was written by different authors.)
  Raju insists on the idea that Proclus’s views represent the original
  philosophy of mathematics in the Elements (p. 25), and he
  overemphasizes the connections between geometrical proof, Platonism,
  and Christian religion (see below).

