98 bernard vitrac
Sidon. Th e Euclidean proof of i .1 presupposes that there is not a common
segment for two distinct straight lines, 73 precisely what is here declared to be
impossible. Th e commentator denies the objection, using three arguments,
the fi rst and last of which are close to the contents of the two post-factum
explanations (in Th and P respectively), as well as to the scholia. 74
In this example, there is every reason to believe that the fi rst scenario was
the better one, that the ‘Euclidean’ proof of xi .1 was similar to that of the
indirect tradition. Heiberg could not have known the Gerard of Cremona
translation (discovered by A. A. Björnbo at the beginning of the twenti-
eth century), but he could have consulted Campanus’s edition, which has
neither of the post-factum explanations.
It goes without saying that the diff erence, from a mathematical point of
view, is minuscule. However, from the point of view of the history and use
of the text, it is the number of alterations of this type – in the hundreds^75 –
which is signifi cant. Additions like those which we have just seen regarding
xi .1 have been introduced on diff erent occasions, undoubtedly indepen-
dently of each other, since each version – including the Arabo-Latin trans-
lations which escape nearly uncorrupted by this phenomenon – has some
which are proper to it. 76 Th is work of improvement undoubtedly owes much
to the marginal annotations eventually integrated into the text itself. Yet it
partially blurs the distinction between ‘text’ and ‘commentary’.
For the majority of them, these additions ensure the ‘saturation’ of the
text. Th e interpretation of the Elements which the annotators presuppose is
more logical than mathematical. Indeed, for them, Euclid’s text represents
the very apprenticeship of deduction more than a means for the acquisition
of the fundamental results of geometry. Even if the role of the marginal
annotations has probably been less eff ective in the case of structural diver-
gences, we will see that the purpose which they pursue – when it can be
determined – is frequently the same.
From the point of view of the history of the text, the abundance of these
sometimes independent improvements implies that for the Elements and for
certain other mathematical texts the methods of transmission were much
more fl exible than those postulated by philologists whose model rests on the
tradition of poetic texts. It is not possible either to put the diff erent examples
of a text in a linearly ordered schema ( stemma ) or even to admit the simple
primacy accorded to a manuscript, such as Heiberg accorded to P. Clearly,75 For example, about 600 sentences are intended to point out a hypothesis or what was the object
of a previous proof. About twenty terminological explanations, mostly in Book x , may be added.
76 See Euclid/Vitrac 2001: iv 63.74 See Friedlein 1873 : 215. 17–216. 9.73 See Friedlein 1873 : 215.11–13, 215.15–16.