Euclid and the scientific thought in the third century B.C.

Renato Migliorato, Giuseppe Gentile
Department of Mathematics, University of Messina

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Abstract
The criticism on the texts of Euclid, even assuming different positions, starts generally from the previous assumption
that the author of the Elements is totally inside the Platonic-Aristotelian tradition. The thesis affirmed
in this paper is that many of the gaps and contradictions found by the criticism have their root in this
assumption. The authors assert that Euclid was a scientist that belonged in a full way to the new cultural
climate of the Hellenistic Kingdoms, and particularly of the Alexandria’s Museum. In this climate,
characterized by lively philosophical disputes, the scientists, and in particular Euclid, tend to obtain
coherent and stable results, voluntarily omitting to give their opinion on the real being of the scientific
object and on the truth of the principles.
1. Introduction
Even if important innovations in the critical studies on Euclidean geometry don’t less in a more
recent times, the period starting from the end of XIX century until the beginning of the XX century
is surely the more prolific one, that in which a critical order was constituted such to be considered
until now almost definitive. From 1850 to 1928 the Heiberg and Menge’s edition3 of the
Euclid’s works was published; this is considered the more reliable text and the nearest to the
original one. In the first half of the twentieth century there are many translations, with comments
and remarks, founded on the Heiberg’s text, as that one of Federigo Enriques4 or that one of
Heath5, and many critical elaborations by the same and other authors6. Among the more recent
published works that propose some new interpretative hypotheses, it seems suitable to us to mention two articles and a monograph of Lucio Russo7, a monograph of Francesca Incardona8 and a
book of Imre Toth9. The articles of Russo consider the first seven definitions in the first book of
the Elements, that he affirms to be posterior interpolations. With that, it would come to fall the
greatest residual reasons in favour of a supposed Platonism of Euclid. We believe that such hypothesis,
well argued and documented by the author, must be accepted, not only because it is historically
reliable, but also because it seems to us most suitable to answer to difficult problems,
remained unsolved, on the interpretation of the Euclidean work.
The book “La rivoluzione dimenticata” (The forgotten revolution), also by Russo, would demand
instead a more complex and articulated valuation that, in its entirety, is extraneous to the object
and the purpose of the present paper. However the fundamental hypotheses that are the nucleus
of the book, cannot remain excluded from our analysis.
For our purposes, it seems meaningful to us the introductory text by which Incardona accompanies
her translation of Euclid’s Optic. It seems interesting to us, in particular, the reasonings
that leads to interpret the work as a mathematical model of the phenomenon of the vision and
tend to insert Euclid in the context of a more wide evolution of the scientific and philosophical
ideas at the beginning of third century. Not well founded it appears instead the hypothesis, however
only fleetingly pointed out by the author, that Euclid could be absolutely assimilated to the
area of the Stoic philosophy. Finally the book of Imre Toth10 is interesting for us because it calls
our attention on some passages of Aristotle that evidence the existence, already before Euclid, of
an open problem on the parallel straight lines.
The purpose of the present paper is a re-examination of the Euclidean text to the light of the last
criticism’s history, with particular reference to the mentioned works, and of the most recent acquisitions
of knowledge on the Hellenistic society and culture of third century B.C. The conclusions
to which we will reach, seem to answer to some questions that still now remained opened.

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