# 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.

• You mean you are surprised Euclid didn't discover the sieve before Eratosthenes? Jun 27, 2016 at 5:51
• In general, Eratosthenes' life seems to be much better documented than Euclid's. It would be quite strange if it was just never mentioned that he also wrote the Elements. Assuming that he wrote them under the pseudonym of Euclid (unnoticed by either Strabo or the byzantine Suba) seems to be a little far-fetched. In particular because, as Mikhail Katz mentioned, it seems likely that the Elements were written before (or very early in) Erathostenes' lifetime. Jun 27, 2016 at 7:42
• I have voted to close this. History questions could well be appropriate here, but it's unclear to me what the reason is for believing that Euclid and Eratosthenes are the same. Maybe in 2000 years someone might think that Weyl, Weil and Wiles were all the same mathematician -- after all, in the distant future it might seem incomprehensible why one of them couldn't as well have proved what the others did. Jun 27, 2016 at 7:50
• There was some arguing about whether a recent edit is editorializing, that I removed as it was getting a bit heated, and I think the edit can be read more as clarifying what the question is about (in particular, that Eratosthenes might be just one of the authors, particularly regarding books dealing with arithmetic, not geometry). I do think the question is interesting or at least fair, but also that it is undoubtedly never going to be answered definitively (unless some buried papyri come to light, Nag Hammadi-like). In keeping with long-standing precedent in such matters, I've made it CW. Jun 28, 2016 at 12:27
• It seems worthwhile to record at least a reference to the standard view: www-history.mcs.st-and.ac.uk/Biographies/Euclid.html . Jun 28, 2016 at 13:35

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).

• This article seems to be unpublished. Is C. K. Raju a reputable scholar? Jun 27, 2016 at 8:29
• Are you aware of any scholarly reviews of that? Jun 27, 2016 at 8:52
• I don't at all see how this definitively answers the question, "is there any clear proof that Euclid and Eratosthenes are two different people?" If the question were "did Euclid actually exist?", then this answer might be an interesting data point, but the question seems to be about a stronger claim, that the author of The Elements might have been Eratosthenes, which this answer doesn't touch. (Please note that I am not criticizing Carlo Beenakker for this answer; I am criticizing acceptance of his response as a definitive answer.) Jun 28, 2016 at 3:11
• Mauro pointed out at MSE that there is a review of the book by Ferreiros here. Jun 28, 2016 at 7:33
• @ToddTrimble, I don't know much about the formal aspects of OM. Thus, because of this and due to the hostile attitude of some of the participants of this thread, I felt that I better accept the otherwise excellent answer by Carlo before the thread gets closed/removed, even if Carlo's answer was not definite. Jun 28, 2016 at 10:21

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.

• I like your notation for dates! Jun 27, 2016 at 7:09
• That's possible. Which of the dates do you wish to challenge? If you feel that the traditionally cited dates are incorrect you should discuss this explicitly in your question. Jun 27, 2016 at 7:11
• So your theory is that Eratosthenes wrote The Elements? This might be an interesting theory but perhaps more appropriate to the history of science and mathematics site HSM. Jun 27, 2016 at 7:40
• Your date notation is somewhat confusing, since there is no year 0. Does "$n$ BCE" correspond to $-n$ or $-(n-1)$? Jun 27, 2016 at 12:10
• @Argon --- I would imagine Mikhail Katz is referring to the astronomical year numbering -- where year 0 is the Julian year 1 BC; so $n$ BC/BCE is $-(n-1)$ Jun 27, 2016 at 13:11