Th e Elements and uncertainties in Heiberg’s edition 111
cases, by the fact that the aforementioned manuscripts have compiled the
proofs from diff erent versions which contained these proofs in isolation. If
what we have seen about Propositions x .107 and x .68–70 is recalled, the
process of transliteration and the desire to safeguard a fl ourishing tradi-
tion seems to us to constitute a propitious occasion for compiling proofs,
however incomplete.
Returning to discussions concerning the history of the text, we ought to
fi rst note that the double proofs do not fall within what is called authorial
variants. Euclid did not propose several proofs with the same results. Th us
the Greek manuscripts closest to the operation of transliteration ( P and B )
are most likely the results of a compilation of the tradition, rather than of
simple reproduction – changing only the writing – of a venerably aged
model.^107
Th e limits of Knorr’s criteria
It is oft en possible to perceive one or more reasons for the other types of
structural changes that I described earlier (additions, modifi cations of the
order). Th us Knorr thought it possible to order the diff erent states of the
text, if not according to authenticity, then at least relative to the degree of
alteration. We have already noted that this criterion of improvement applies
locally, and the example of changes in the order suggested to us that it does
not seem always to have been exercised for the benefi t of the one and the
same version. Th e phenomenon of the substitution of proofs evidences
another diffi culty.
Th e criterion of improvement works well enough as long as there is only
a single parameter (or even more, 108 but all acting in the same direction)
which governs the replacement of a proof or the modifi cation of a presen-
tation. But, when there are at least two acting in opposite directions, the
change which is more sophisticated from a certain point of view may be
less desirable from another point of view. Let us reconsider our example of
Proposition x .105. Admittedly, from a mathematical point of view, there is
an improvement (generalization), but from the logical, or metamathemati-
cal, point of view – and it is no doubt one of the points of view adopted by
107 See n. 33.
108 Th e most frequent parameters governing the replacement of proofs are the reinforcement of
the deductive structure, the substitution of a direct proof with an indirect proof (a criterion
notably explained by Heron – see Vitrac 2004 : 17–18 (regarding iii .9 aliter ) – and Menelaus),
the addition of the case of a fi gure and the level of discourse used (geometric objects versus
proportions; a criterion clearly noted by Pappus).