74 bernard vitrac
recent to the third century. In contrast to the manuscripts, the papyri have
the privileged position of being documents from Antiquity. An author
represented among the papyri is likely to have been used in teaching. In the
mathematical realm, the bulk of papyri preserved for us represent two cat-
egories: (1) very elementary school documents, and (2) astronomical texts.
It is therefore signifi cant that Euclid is the only geometer of the ‘scholarly’
tradition who appears in this type of text.Direct and indirect traditions
Nicolas Bourbaki probably did not consult the manuscripts of the Elements
to determine his opinion about the subject of the Euclidean ideal of proof,
and it is the same for the majority of Euclid’s modern readers. Generally,
they rely on a translation, or if they know ancient Greek, on a critical
edition produced by a modern philologist. In the case of the Greek texts of
the Elements , the critical edition was produced by J. L. Heiberg. If he reads
the work in Greek, the reader labours under the illusion that he has read
what Euclid has written. In this respect, the philological terminology and
its label ‘direct tradition’ can be misleading. Th e ‘direct’ tradition designates
the set of Greek manuscripts and papyri which contain the text either in its
totality or in part. Despite this label, we must not forget the considerable
number of intermediaries that came between us and the author, even in the
direct tradition. Th ese intermediaries include not only the copyists, who we
would like to believe did nothing more than passively reproduce the text,
but also, more importantly, those who took an active part in the transmis-
sion of the text – in particular ancient and medieval re-editors and, last
of all, the philologists who, beginning with the collection of the available
information, have constructed the critical edition that we read today. I have
thus reported, too briefl y, the several elements of the history of the preced-
ing ancient Greek texts to make the point that our knowledge about the
text of the Elements , like that of the majority of other ancient Greek texts,
is essentially indirect.
Classical philology is not without resources. It has developed methods to
‘reverse’ the course of time. Th ese methods make it possible to trace the rela-
tionships between manuscripts, to detect the mistakes of the copyists, and
in the ‘good’ cases to reconstitute an ancestor of the tradition, oft en imme-
diately before the transliteration, sometimes an ancient prototype from late
antiquity or from the Roman era. In the case of a Hellenistic author, this
result is still rather removed from the ‘original’ and thus necessitates appeals
to other sources. Th ese sources constitute the so-called indirect tradition